# questions

tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch0.db
title:
A closed pipe is suddenly opened and changed to an open pipe of same length. The fundamental frequency of the resulting open pipe is less than that of 3rd harmonic of the earlier closed pipe by 55 Hz . Then, the value of fundamental frequency of the closed pipe is

slug:
a-closed-pipe-is-suddenly-opened-and-changed-to-an-open-pipe-of-same-length-the-fundamental-frequen-49856

question:
<div class="question">A closed pipe is suddenly opened and changed to an open pipe of same length. The fundamental frequency of the resulting open pipe is less than that of 3rd harmonic of the earlier closed pipe by $55 \mathrm{~Hz}$. Then, the value of fundamental frequency of the closed pipe is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$165 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$110 \mathrm{~Hz}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$55 \mathrm{~Hz}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$220 \mathrm{~Hz}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$55 \mathrm{~Hz}$</span> </div>

answer:
<div class="solution">Frequency of first mode of vibration<br/>$$<br/>n_1=\frac{3 v}{4 /}<br/>$$<br/>(closed pipe) and frequency of first mode of vibration<br/>$$<br/>n_2=\frac{v}{2 l}<br/>$$<br/>(open pipe)<br/>According to the question,<br/>$$<br/>n_1-n_2=\frac{3 v}{4 l}-\frac{v}{2 l}=55<br/>$$<br/>So $\quad \frac{3 v-2 v}{4 !}=55$<br/>$$<br/>\Rightarrow \quad \frac{v}{4 l}=55 \mathrm{~Hz}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'TS EAMCET', 'TS EAMCET 2014']
source_db: MarksBatch0.db
title:
A closed pipe is suddenly opened and changed to an open pipe of same length. The fundamental frequency of the resulting open pipe is less than that of 3rd harmonic of the earlier closed pipe by 55 Hz . Then, the value of fundamental frequency of the closed pipe is

slug:
a-closed-pipe-is-suddenly-opened-and-changed-to-an-open-pipe-of-same-length-the-fundamental-frequen-95723

question:
<div class="question">A closed pipe is suddenly opened and changed to an open pipe of same length. The fundamental frequency of the resulting open pipe is less than that of 3rd harmonic of the earlier closed pipe by $55 \mathrm{~Hz}$. Then, the value of fundamental frequency of the closed pipe is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$165 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$110 \mathrm{~Hz}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$55 \mathrm{~Hz}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$220 \mathrm{~Hz}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$55 \mathrm{~Hz}$</span> </div>

answer:
<div class="solution">Frequency of first mode of vibration<br/>$$<br/>n_1=\frac{3 v}{4 /}<br/>$$<br/>(closed pipe) and frequency of first mode of vibration<br/>$$<br/>n_2=\frac{v}{2 l}<br/>$$<br/>(open pipe)<br/>According to the question,<br/>$$<br/>n_1-n_2=\frac{3 v}{4 l}-\frac{v}{2 l}=55<br/>$$<br/>So $\quad \frac{3 v-2 v}{4 !}=55$<br/>$$<br/>\Rightarrow \quad \frac{v}{4 l}=55 \mathrm{~Hz}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch0.db
title:
A closed pipe of length 22 c m , when excited by a 1875 H z source forms standing waves. The number of pressure nodes formed in the pipe are [velocity of sound in air = 330 m s - 1 ]

slug:
a-closed-pipe-of-length-22-c-m-when-excited-by-a-1875-h-z-source-forms-standing-waves-the-number-25636

question:
<div class="question">A closed pipe of length <math><mn>22</mn><mi> </mi><mn>c</mn><mn>m</mn></math>, when excited by a <math><mn>1875</mn><mi> </mi><mn>H</mn><mn>z</mn></math> source forms standing waves. The number of pressure nodes formed in the pipe are [velocity of sound in air <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>330</mn><mi> </mi><mn>m</mn><mo> </mo><msup><mn>s</mn><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><mn>1</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mn>2</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>4</mn></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math><mn>3</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>3</mn></math></span> </div>

answer:
<div class="solution">Velocity of sound <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>330</mn><mi> </mi><mn>m</mn><mo> </mo><msup><mn>s</mn><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/><br/>Length of closed pipe <math><mo>=</mo><mn>22</mn><mi> </mi><mi>cm</mi><mo>=</mo><mn>22</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>2 </mn></mrow></msup><mn>m</mn></math><br/><br/>The fundamental frequency of the stationary wave,<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ν</mi><mn>0</mn></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>4</mn><mi>l</mi></mrow></mfrac><mo>=</mo><mfrac><mn>330</mn><mrow><mn>4</mn><mo>×</mo><mn>22</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfrac></math><br/><br/><math><mo>=</mo><mfrac><mrow><mn>330</mn><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn><mo>×</mo><mn>22</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3000</mn></mrow><mrow><mn>8</mn></mrow></mfrac></math><br/><br/><math><msub><mrow><mi>ν</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>375</mn><mi> </mi><mi>Hz</mi></math><br/><br/><math><mn>3</mn><msub><mrow><mi>ν</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>375</mn><mo>×</mo><mn>3</mn><mo>=</mo><mn>1125</mn><mi> </mi><mi>Hz</mi></math><br/><br/><math><mn>5</mn><msub><mrow><mi>ν</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>5</mn><mo>×</mo><mn>375</mn><mo>=</mo><mn>1875</mn><mi> </mi><mi>Hz</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>5</mn><mi>th</mi></msup></math> harmonic<br/><br/>Number of nodes <math><mo>=</mo><mn>3</mn></math><br/><br/>Note: In a closed pipe, odd harmonics are produced.</div>


title: A coin is tossed 5 times. The probability that tail appears an odd number of times, is
slug: a-coin-is-tossed-5-times-the-probability-that-tail-appears-an-odd-number-of-times-is-78722
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch0.db
question:
<div class="question">A coin is tossed 5 times. The probability that tail appears an odd number of times, is</div>

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

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

answer:
<div class="solution">Let $\mathrm{p}$ denote the probability of getting tail in a single of a coin.<br/>$\therefore \mathrm{p}=\frac{1}{2} \Rightarrow \mathrm{q}=\frac{1}{2} \&amp; \mathrm{n}=5$<br/>Let $X$ denot no. of tails in 5 tosses of coin. $\begin{aligned} \therefore \text { Required probability } &amp;=\mathrm{P}(\mathrm{x}=1)+\mathrm{P}(\mathrm{x}=3)+\mathrm{P}(\mathrm{x}=5) \\ &amp;=\frac{1}{2^{5}}\left[{ }^{5} \mathrm{C}_{1}+{ }^{5} \mathrm{C}_{3}+{ }^{5} \mathrm{C}_{5}\right] \\ &amp;=\frac{1}{2^{5}}[5+10+1]=\frac{16}{32}=\frac{1}{2} \end{aligned}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">11</span> </div>
source_db: MarksBatch0.db
title:
A coin is tossed 10 times. The number of heads minus the number of tails in 10 tosses is considered as the outcome of the experiment. What is the number of points in the sample space

slug:
a-coin-is-tossed-10-times-the-number-of-heads-minus-the-number-of-tails-in-10-tosses-is-considered-61934

question:
<div class="question">A coin is tossed 10 times. The number of heads minus the number of tails in 10 tosses is considered as the outcome of the experiment. What is the number of points in the sample space</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">10</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">C</span> <span class="option-data">21</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">99</span> </li> </ul>

answer:
<div class="solution">\begin{array}{c|c} <br/>Head &amp; Tail \\<br/>10 &amp; 0 \\<br/>9 &amp; 1 \\<br/>8 &amp; 2 \\<br/>7 &amp; 3 \\<br/>6 &amp; 4 \\<br/>5 &amp; 5 \\<br/>4 &amp; 6 \\<br/>3 &amp; 7 \\<br/>2 &amp; 8 \\<br/>1 &amp; 9 \\<br/>0 &amp; 10<br/>\end{array}<br/>Hence, total number of points in the sample space is<br/>$11 .$</div>


title: A coin is tossed five times. What is the probability that heads are observed more than three times?
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch0.db
slug:
a-coin-is-tossed-five-times-what-is-the-probability-that-heads-are-observed-more-than-three-times-62874

question:
<div class="question">A coin is tossed five times. What is the probability that heads are observed more than three times?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{3}{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{5}{16}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{32}$</span> </li> </ul>

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

answer:
<div class="solution">Let P denote the probability of getting head in a single toss of a coin.<br/>$\mathrm{p}=\frac{1}{2} \Rightarrow \mathrm{q}=\frac{1}{2}$<br/>Let $X$ denote the no. of heads in 5 tosses of a coin. then, $\mathrm{X}$ is a binomial variate with parameters; $\mathrm{n}=5 \&amp;$<br/>$\mathrm{p}=\frac{1}{2}$.<br/>$\therefore$ Req. probability $=\mathrm{P}(\mathrm{x}&gt;3)$<br/>$=1-\mathrm{P}(\mathrm{x} \leq 3)$<br/>$=1-[\mathrm{P}(\mathrm{x}=0)+\mathrm{P}(\mathrm{x}=1)+\mathrm{P}(\mathrm{x}=2)+\mathrm{P}(\mathrm{x}=3)]$<br/>$=1-\left[{ }^{5} \mathrm{C}_{0}+{ }^{5} \mathrm{C}_{1}+{ }^{5} \mathrm{C}_{2}+{ }^{5} \mathrm{C}_{3}\right] \frac{1}{2^{5}}$<br/>$=1-[1+5+10+10] \frac{1}{32}$<br/>$=\frac{32}{32}-\frac{26}{32}=\frac{6}{32}=\frac{3}{16}$</div>


tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch0.db
title:
A coin is tossed three times. Consider the following events: A: No head appears B: Exactly one head appears C: At least two heads appear Which one of the following is correct?

slug:
a-coin-is-tossed-three-times-consider-the-following-events-a-no-head-appears-b-exactly-one-head-17923

question:
<div class="question">A coin is tossed three times. Consider the following events:<br/>A: No head appears<br/>B: Exactly one head appears<br/>C: At least two heads appear Which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(\mathrm{A} \cup \mathrm{B}) \cap(\mathrm{A} \cup \mathrm{C})=\mathrm{B} \cup \mathrm{C}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right) \cup\left(\mathrm{A} \cap \mathrm{C}^{\prime}\right)=\mathrm{B}^{\prime} \cup \mathrm{C}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{A} \cap\left(\mathrm{B}^{\prime} \cup \mathrm{C}^{\prime}\right)=\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$A \cap\left(B^{\prime} \cup C^{\prime}\right)=B^{\prime} \cap C$</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">$A \cap\left(B^{\prime} \cup C^{\prime}\right)=B^{\prime} \cap C$</span> </div>

answer:
<div class="solution">$\quad U=\{(H H H)(H H T)(H T H)(H T T)(T H H)(T H T)(T T H)(T T T)\}$<br/>$A=\{(T T T)\}$<br/>$B=\{(H T T)(T H T)(T T H)\}$<br/>$C=\{(H H H)(H H T)(H T H)(T H H)\}$<br/>By checking the options<br/>(d) $A \cap\left(B^{\prime} \cup C^{\prime}\right)=B^{\prime} \cap C^{\prime}$ is correct.</div>


title: A coin is tossed three times. What is the probability of getting head and tail alternately?
slug: a-coin-is-tossed-three-times-what-is-the-probability-of-getting-head-and-tail-alternately-81351
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch0.db
question:
<div class="question">A coin is tossed three times. What is the probability of getting head and tail alternately?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{8}$</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">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{4}$</span> </li> </ul>

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

answer:
<div class="solution">Coin is tossed three times i.e. total outcomes $=2^{3}=8$ $[(\mathrm{H}, \mathrm{H}, \mathrm{H}),(\mathrm{H}, \mathrm{H}, \mathrm{T}),(\mathrm{H}, \mathrm{T}, \mathrm{H}),(\mathrm{H}, \mathrm{T}, \mathrm{T}),(\mathrm{T}, \mathrm{H}, \mathrm{H}),$,<br/>$(\mathrm{T}, \mathrm{H}, \mathrm{T}),(\mathrm{T}, \mathrm{T}, \mathrm{H}),(\mathrm{T}, \mathrm{T}, \mathrm{T})]$<br/>Alternate head and tail are coming two times only.<br/>Thus prob. of getting head and tail alternately $=\frac{2}{8}=\frac{1}{4}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch0.db
title:
A coin is tossed three times. What is the probability of getting head and tail (HTH) or tail and head (THT) alternatively?

slug:
a-coin-is-tossed-three-times-what-is-the-probability-of-getting-head-and-tail-hth-or-tail-and-hea-14947

question:
<div class="question">A coin is tossed three times. What is the probability of getting head and tail (HTH) or tail and head (THT) alternatively?</div>

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

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

answer:
<div class="solution">Total possible outcomes, $\mathrm{S}=\{\mathrm{HHH}, \mathrm{HHT}, \mathrm{HTH}, \mathrm{THT}$.<br/>TTH. THH, TTT, HTT $\}$ and desired outcomes E $=\{\mathrm{HTH}, \mathrm{THT}\}$<br/>$\Rightarrow \mathrm{n}(\mathrm{E})=2$ and $\mathrm{n}(\mathrm{S})=8$<br/>Hence, required probability $=\mathrm{P}(\mathrm{E})=\frac{\mathrm{n}(\mathrm{E})}{\mathrm{n}(\mathrm{S})}=\frac{2}{8}=\frac{1}{4}$</div>


tags: ['Chemistry', 'Surface Chemistry', 'AIIMS', 'AIIMS 2015']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch0.db
title:
A colloidal solution is kept in dark and is illuminated by a beam of light then, brightness appears at the right angle of direction of light. This effect is called

slug:
a-colloidal-solution-is-kept-in-dark-and-is-illuminated-by-a-beam-of-light-then-brightness-appears-35107

question:
<div class="question">A colloidal solution is kept in dark and is illuminated by a beam of light then, brightness appears at the right angle of direction of light. This effect is called</div>

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

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


tags: ['Chemistry', 'Surface Chemistry', 'AIIMS', 'AIIMS 2012']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch0.db
title:
A colloidal solution is kept in dark and is illuminated by a beam of light then brightness appears at the right angle of direction of light. This effect is called

slug:
a-colloidal-solution-is-kept-in-dark-and-is-illuminated-by-a-beam-of-light-then-brightness-appears-a-10333

question:
<div class="question">A colloidal solution is kept in dark and is illuminated by a beam of light then brightness appears at the right angle of direction of light. This effect is called</div>

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

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


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch0.db
title:
A committee of three has to be chosen from a group of 4 men and 5 women. If the selection is made at random, what is the probability that exactly two members are men?

slug:
a-committee-of-three-has-to-be-chosen-from-a-group-of-4-men-and-5-women-if-the-selection-is-made-at-51090

question:
<div class="question">A committee of three has to be chosen from a group of 4 men and 5 women. If the selection is made at random, what is the probability that exactly two members are men?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{5}{14}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{21}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{3}{14}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{8}{21}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{14}$</span> </div>

answer:
<div class="solution">$\mathrm{n}(\mathrm{E})={ }^{4} \mathrm{C}_{2} \times{ }^{5} \mathrm{C}_{1}=6 \times 5=30$.<br/>$\mathrm{n}(\mathrm{S})={ }^{9} \mathrm{C}_{3}=84$<br/>$\therefore$ Probability, $\mathrm{P}(\mathrm{E})=\frac{\mathrm{n}(\mathrm{E})}{\mathrm{n}(\mathrm{S})}=\frac{30}{84}=\frac{5}{14}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch0.db
title:
A committee of two persons is contituted from two men and two women. What is the probability that the committee will have only women?

slug:
a-committee-of-two-persons-is-contituted-from-two-men-and-two-women-what-is-the-probability-that-th-29080

question:
<div class="question">A committee of two persons is contituted from two men and two women. What is the probability that the committee will have only women?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{6}$</span> </div>

answer:
<div class="solution">$2 \mathrm{Men}+2$ Women Required probability $=\frac{{ }^{2} \mathrm{C}_{0} \times{ }^{2} \mathrm{C}_{2}}{{ }^{4} \mathrm{C}_{2}}=\frac{1 \times 1}{6}=\frac{1}{6}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch0.db
title:
A committee of two persons is selected from two men and two women. The probability that the committee will have exactly one woman is

slug:
a-committee-of-two-persons-is-selected-from-two-men-and-two-women-the-probability-that-the-committe-27009

question:
<div class="question">A committee of two persons is selected from two men and two women. The probability that the committee will have exactly one woman is</div>

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

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

answer:
<div class="solution">We know, $\mathrm{P}(\mathrm{E})=\frac{\text { No. of favourable outcomes }}{\text { Total no. of outcomes }}$<br/>Number of possible outcomes $={ }^{4} \mathrm{C}_{2}=\frac{4 \times 3}{2 \times 1}=6$<br/>(selecting 2 people from 4 people $)$ Number of favourable outcomes<br/>$={ }^{2} \mathrm{C}_{1} \times{ }^{2} \mathrm{C}_{1}=2 \times 2=4$<br/>(selecting 1 from 2 men, 1 from 2 women)<br/>$\mathrm{P}(\mathrm{E})=\frac{4}{6}=\frac{2}{3}$</div>


title: A common structural feature of vessel elements and sieve tube elements is:
slug: a-common-structural-feature-of-vessel-elements-and-sieve-tube-elements-is-94374
tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET', 'NEET 2006']
source_db: MarksBatch0.db
question:
<div class="question">A common structural feature of vessel elements and sieve tube elements is:</div>

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

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

answer:
<div class="solution">Enucleate condition means cells without nuclei. This is found in vessel elements and sieve tube elements. Thick secondary wall is the inner most layer of cell wall, it is found in the cell wall of vessel elements. The thickened areas present on the lateral walls of the vessel elements are called pores. P- proteins are the small tubular fibrillar, granular or crystalline sub microscopic structures appear in the cytoplasm of sieve tubes.<br/>Caution<br/>Xylem vessels become dead and lose their protoplasm due to deposition of lignified secondary wall mature sieve tube elements are living cells without nucleus.</div>


title: A common structural feature of vessel elements and sieve tube elements is ________.
slug: a-common-structural-feature-of-vessel-elements-and-sieve-tube-elements-is-88145
tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch0.db
question:
<div class="question">A common structural feature of vessel elements and sieve tube elements is ________.</div>

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

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

answer:
<div class="solution">The food conducting tissue of vascular plants is phloem. The components of phloem are sieve elements, companion cells, phloem parenchyma, phloem fibers, and sclereids. Whereas xylem is the water-conducting tissue of vascular plants. The components of xylem are Tracheids, vessel, xylem parenchyma, and xylem fiber. There is one characteristic that is common in vessel element and sieve tube element is that they both are non-living at maturity or nucleus is absent in both of them.</div>


slug: a-common-transistor-radio-set-requires-12-v-d-c-for-its-operation-the-d-c-source-is-constru-57529
tags: ['Physics', 'Alternating Current', 'JEE Main', 'JEE Main 2021 (24 Feb Shift 1)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">440</span> </div>
source_db: MarksBatch0.db
title:
A common transistor radio set requires 12 V D . C . for its operation. The D . C . source is constructed by using a transformer and a rectifier circuit, which are operated at 220 V A . C . on standard domestic A . C . supply. The number of turns of secondary coil are 24 , then the number of turns of primary are

question:
<div class="question">A common transistor radio set requires <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo> </mo><mi mathvariant="normal">V</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">D</mi><mo>.</mo><mi mathvariant="normal">C</mi><mo>.</mo></mrow></mfenced></math> for its operation. The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">D</mi><mo>.</mo><mi mathvariant="normal">C</mi><mo>.</mo></math> source is constructed by using a transformer and a rectifier circuit, which are operated at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>220</mn><mo> </mo><mi mathvariant="normal">V</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">A</mi><mo>.</mo><mi mathvariant="normal">C</mi><mo>.</mo></mrow></mfenced></math> on standard domestic <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi><mo>.</mo><mi mathvariant="normal">C</mi><mo>.</mo></math> supply. The number of turns of secondary coil are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>24</mn><mo>,</mo></math> then the number of turns of primary are</div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>N</mi><mi>P</mi></msub><msub><mi>N</mi><mi>S</mi></msub></mfrac><mo>=</mo><mfrac><msub><mi>V</mi><mi>P</mi></msub><msub><mi>V</mi><mi>S</mi></msub></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>N</mi><mi>P</mi></msub><mn>24</mn></mfrac><mo>=</mo><mfrac><mn>220</mn><mn>12</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msub><mi>N</mi><mi>P</mi></msub><mo>=</mo><mfrac><mrow><mn>220</mn><mo>×</mo><mn>24</mn></mrow><mn>12</mn></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msub><mi>N</mi><mi>P</mi></msub><mo>=</mo><mn>440</mn></math></p></div>


slug: a-complex-compound-of-co-3-with-molecular-formula-cocl-x-ynh-3-gives-a-total-of-3-ions-42488
tags: ['Chemistry', 'Coordination Compounds', 'AP EAMCET', 'AP EAMCET 2005']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch0.db
title:
A complex compound of Co 3 + with molecular formula CoCl x â â yNH 3 â gives a total of 3 ions when dissolved in water. How many Cl â ions satisfy both primary and secondary valencies in this complex?

question:
<div class="question">A complex compound of $\mathrm{Co}^{3+}$ with molecular formula $\mathrm{CoCl}_{\mathrm{x}} \cdot \mathrm{yNH}_3$ gives a total of 3 ions when dissolved in water. How many $\mathrm{Cl}^{-}$ions satisfy both primary and secondary valencies in this complex?</div>

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

answer:
<div class="solution">As cobalt is present as $\mathrm{Co}^{3+}$ and co-ordination number of cobalt is 6 , the molecular formula of compound should be $\mathrm{CoCl}_3 \cdot \mathrm{yNH}_3$. Now, as it gives a total of 3 ions when dissolved in water, its structural formula must be $\left[\mathrm{CoCl}\left(\mathrm{NH}_3\right)_5\right] \mathrm{Cl}_2$<br/>$\left[\mathrm{CoCl}\left(\mathrm{NH}_3\right)_5\right] \mathrm{Cl}_2 \rightleftharpoons\left[\mathrm{CoCl}\left(\mathrm{NH}_3\right)_5\right]^{2+}+2 \mathrm{Cl}^{-}$<br/>Thus only one $\mathrm{Cl}^{-}$ion is satisfying both primary and secondary valency of $\mathrm{Co}^{3+}$ in this compound.</div>


title: A complex number is given by z = 1 â ( 1 â i ) 2 1 + 2 i â What is the modulus of z ?
slug: a-complex-number-is-given-by-z-1-1-i-2-1-2-i-what-is-the-modulus-of-z-99051
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch0.db
question:
<div class="question">A complex number is given by $z=\frac{1+2 i}{1-(1-i)^{2}}$<br/>What is the modulus of $z$ ?</div>

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

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


slug: a-complex-number-is-given-by-z-1-1-i-2-1-2-i-what-is-the-principal-argument-of-z-77752
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch0.db
title:
A complex number is given by z = 1 â ( 1 â i ) 2 1 + 2 i â What is the principal argument of z ?

question:
<div class="question">A complex number is given by $z=\frac{1+2 i}{1-(1-i)^{2}}$<br/>What is the principal argument of $z$ ?</div>

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

answer:
<div class="solution">$\begin{aligned} &amp; z=\frac{1+2 i}{1-(1-i)^{2}}=1=1+0 . i \\ &amp; \therefore \text { Principal argument of } z=\tan \theta \\ &amp;=\frac{0}{1}=0 \\ &amp; \therefore \theta=0^{\circ} \end{aligned}$</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'JEE Main']
source_db: MarksBatch0.db
title:
A complex of cobalt with ammonia is analyzed for determining its formula, by titrating it against a strandarized acid as follows : Co NH 3 x Cl y aq + HCl â NH 4 + aq + Co y + + aq + Cl â aq A 1.8 g complex required 20.00 mL 1.54 M HCl to reach the equivalence point. Also, if the reaction mixture at equivalence point is treated with excess of AgNO 3 solution, 7.735 g of AgCl precipitate was produced. What is the formula of this complex? [Given: atomic weight of CO = 59 gmol -1 ]

slug:
a-complex-of-cobalt-with-ammonia-is-analyzed-for-determining-its-formula-by-titrating-it-against-a-30942

question:
<div class="question">A complex of cobalt with ammonia is analyzed for determining its formula, by titrating it against a strandarized acid as follows :<br/>
 <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Co</mi><msub><mfenced><msub><mi>NH</mi><mn>3</mn></msub></mfenced><mi mathvariant="normal">x</mi></msub><msub><mi>Cl</mi><mi mathvariant="normal">y</mi></msub><mfenced><mi>aq</mi></mfenced><mo>+</mo><mi>HCl</mi><mo>â</mo><msubsup><mi>NH</mi><mn>4</mn><mo>+</mo></msubsup><mfenced><mi>aq</mi></mfenced><mo>+</mo><msup><mi>Co</mi><mrow><mi mathvariant="normal">y</mi><mo>+</mo></mrow></msup><mo>+</mo><mfenced><mi>aq</mi></mfenced><mo>+</mo><msup><mi>Cl</mi><mo>â</mo></msup><mfenced><mi>aq</mi></mfenced></math>  A 1.8 g complex required 20.00 mL 1.54 M HCl to reach the equivalence point. Also, if the reaction mixture at equivalence point is treated with excess of AgNO<sub>3</sub>  solution, 7.735 g of AgCl precipitate was produced. What is the formula of this complex?<br/>
[Given: atomic weight of CO = 59 gmol<sup>-1</sup>]</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math> <mrow> <mtext>Co</mtext> <msub> <mrow> <mfenced close=")" open="(" separators=","> <mrow> <msub> <mtext>NH</mtext> <mn>3</mn> </msub> </mrow> </mfenced> </mrow> <mn>4</mn> </msub> <msub> <mtext>Cl</mtext> <mn>3</mn> </msub> </mrow> </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> <mrow> <mtext>Co</mtext> <msub> <mrow> <mfenced close=")" open="(" separators=","> <mrow> <msub> <mtext>NH</mtext> <mn>4</mn> </msub> </mrow> </mfenced> </mrow> <mn>4</mn> </msub> <msub> <mtext>Cl</mtext> <mn>3</mn> </msub> </mrow> </math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math> <mrow> <mtext>Co</mtext> <msub> <mrow> <mfenced close=")" open="(" separators=","> <mrow> <msub> <mtext>NH</mtext> <mn>3</mn> </msub> </mrow> </mfenced> </mrow> <mn>3</mn> </msub> <msub> <mtext>Cl</mtext> <mn>4</mn> </msub> </mrow> </math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math> <mrow> <mtext>Co</mtext> <msub> <mrow> <mfenced close=")" open="(" separators=","> <mrow> <msub> <mtext>NH</mtext> <mn>4</mn> </msub> </mrow> </mfenced> </mrow> <mn>3</mn> </msub> <msub> <mtext>Cl</mtext> <mn>4</mn> </msub> </mrow> </math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math> <mrow> <mtext>Co</mtext> <msub> <mrow> <mfenced close=")" open="(" separators=","> <mrow> <msub> <mtext>NH</mtext> <mn>3</mn> </msub> </mrow> </mfenced> </mrow> <mn>4</mn> </msub> <msub> <mtext>Cl</mtext> <mn>3</mn> </msub> </mrow> </math></span> </div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>Co</mi><msub><mrow><mo>(</mo><msub><mi>NH</mi><mn>3</mn></msub><mo>)</mo></mrow><mi mathvariant="normal">x</mi></msub><msub><mi>Cl</mi><mi mathvariant="normal">y</mi></msub></mrow><mfrac><mrow><mn>1</mn><mo>.8</mo></mrow><mi mathvariant="normal">M</mi></mfrac></munder><mo>+</mo><munder><mi>xHCL</mi><mfrac><mrow><mn>1</mn><mo>.8</mo></mrow><mi mathvariant="normal">M</mi></mfrac></munder><mo>â</mo><mi mathvariant="normal">x</mi><munder><msubsup><mi>NH</mi><mn>4</mn><mo>+</mo></msubsup><mi>aq</mi></munder><mo>+</mo><msup><mi>Co</mi><mrow><mi mathvariant="normal">y</mi><mo>+</mo></mrow></msup><mo>+</mo><munder><mrow><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>)</mo><msup><mi>Cl</mi><mo>-</mo></msup></mrow><mfrac><mrow><mn>1</mn><mo>.8</mo><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">r</mi><mo>)</mo></mrow><mi mathvariant="normal">M</mi></mfrac></munder></math><br/><br/><math> <mrow> <mfrac> <mrow> <mn>1</mn> <mtext>.</mtext> <mn>8</mn> <mtext>x</mtext> </mrow> <mrow> <mtext>M</mtext> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mn>0</mn> <mo>Ã</mo> <mn>1</mn> <mtext>.</mtext> <mn>5</mn> <mn>4</mn> <mo>Ã</mo> <mn>1</mn> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> <mo>=</mo> <mn>3</mn> <mn>0</mn> <mtext>.</mtext> <mn>8</mn> <mo>Ã</mo> <mn>1</mn> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </mrow> </math>       ........(I)<br/><br/><math> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>Â·</mo> <mn>8</mn> <mfenced close=")" open="(" separators=","> <mrow> <mtext>x</mtext> <mo>+</mo> <mtext>y</mtext> </mrow> </mfenced> </mrow> <mrow> <mtext>M</mtext> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>7</mn> <mtext>.</mtext> <mn>7</mn> <mn>3</mn> <mn>5</mn> </mrow> <mrow> <mn>1</mn> <mn>4</mn> <mn>3</mn> <mtext>.</mtext> <mn>5</mn> </mrow> </mfrac> <mo>=</mo> <mn>5</mn> <mn>3</mn> <mtext>.</mtext> <mn>9</mn> <mo>Ã</mo> <mn>1</mn> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </mrow> </math>           ........(II)<br/><br/><math> <mrow> <mfrac> <mrow> <mtext>II</mtext> </mrow> <mrow> <mtext>I</mtext>  </mrow> </mfrac> <mo>â </mo> <mfrac> <mrow> <mn>5</mn> <mn>3</mn> <mtext>.</mtext> <mn>9</mn> <mo>Ã</mo> <mn>1</mn> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </mrow> <mrow> <mn>3</mn> <mn>0</mn> <mtext>.</mtext> <mn>8</mn> <mo>Ã</mo> <mn>1</mn> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> <mtext>.</mtext> <mn>7</mn> <mn>5</mn> <mo>=</mo> <mfrac> <mrow> <mtext>x</mtext> <mo>+</mo> <mtext>y</mtext> </mrow> <mrow> <mtext>x</mtext> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mtext>y</mtext> </mrow> <mrow> <mtext>x</mtext> </mrow> </mfrac> </mrow> </math><br/><br/><math> <mrow> <mfrac> <mrow> <mtext>y</mtext> </mrow> <mrow> <mtext>x</mtext> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mtext>.</mtext> <mn>7</mn> <mn>5</mn> </mrow> </math><br/><br/>Substituting in (I) <math> <mrow> <mo>â</mo> </mrow> </math><br/><br/><math> <mrow> <mfrac> <mrow> <mn>1</mn> <mtext>.</mtext> <mn>8</mn> <mtext>x</mtext> </mrow> <mrow> <mn>5</mn> <mn>9</mn> <mo>+</mo> <mn>1</mn> <mn>7</mn> <mtext>x</mtext> <mo>+</mo> <mn>3</mn> <mn>5</mn> <mtext>.</mtext> <mn>5</mn> <mo>Ã</mo> <mtext>0</mtext> <mtext>.</mtext> <mn>7</mn> <mn>5</mn> <mtext>x</mtext> </mrow> </mfrac> <mo>=</mo> <mn>3</mn> <mn>0</mn> <mtext>.</mtext> <mn>8</mn> <mo>Ã</mo> <mn>1</mn> <msup> <mn>0</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </mrow> </math><br/><br/><math> <mrow> <mo>â´</mo> </mrow> </math>     x = 3.98 = 4        <math> <mrow> <mo>â´</mo> </mrow> </math>    y = 3</div>


tags: ['Chemistry', 'Amines', 'NEET']
source_db: MarksBatch0.db
title:
A compound (X) with molecular formula C 3 H 9 N reacts with C 6 H 5 S O 2 C l to give a solid which is insoluble in alkali (X) is

slug:
a-compound-x-with-molecular-formula-c-3-h-9-n-reacts-with-c-6-h-5-s-o-2-c-l-to-give-a-solid-which-76015

question:
<div class="question">A compound (X) with molecular formula <math><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>9</mn></mrow></msub><mi mathvariant="normal">N</mi></math> reacts with <math><msub><mrow><mi mathvariant="normal">C</mi></mrow><mrow><mn>6</mn></mrow></msub><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>5</mn></mrow></msub><mi mathvariant="normal">S</mi><msub><mrow><mi mathvariant="normal">O</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal">C</mi><mi mathvariant="normal">l</mi></math> to give a solid which is insoluble in alkali (X) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>3</mn></mrow></msub><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal">N</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/nta_abhyas/neet/images/eda1790e-bd40-4b85-b0ef-1b53624777c1.png" style="width:0.79125in;height:0.91913in"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">N</mi><mi mathvariant="normal">H</mi><mo>-</mo><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/nta_abhyas/neet/images/e1d6cb43-54f3-4630-996c-8bea4990cd94.png" style="width:1.08681in;height:0.64653in"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>-</mo><mi mathvariant="normal">N</mi><mi mathvariant="normal">H</mi><mo>-</mo><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal">C</mi><msub><mrow><mi mathvariant="normal">H</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> </div>

answer:
<div class="solution">Since the compound reacts with benzenesulphonyl chloride to give a product which is insoluble in alkali, it shows there is no H attached to N in the product. <br/>Hence, the compound X is a secondary amine. <br/><img src="https://cdn-question-pool.getmarks.app/nta_abhyas/neet/images/41c8e5c6-46d2-4c8f-901e-7a4bd79b61ed.png" style="width:3.82708in;height:0.95347in"/></div>


slug: a-compound-z-with-molecular-formula-c-3-h-9-n-reacts-with-c-6-h-5-so-2-cl-to-giv-57262
tags: ['Chemistry', 'Amines']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
A compound Z with molecular formula C 3 â H 9 â N reacts with C 6 â H 5 â SO 2 â Cl to give a solid, insoluble in alkali. Idenfity Z .

question:
<div class="question">A compound $\mathrm{Z}$ with molecular formula $\mathrm{C}_3 \mathrm{H}_9 \mathrm{~N}$ reacts with $\mathrm{C}_6 \mathrm{H}_5 \mathrm{SO}_2 \mathrm{Cl}$ to give a solid, insoluble in alkali. Idenfity $\mathrm{Z}$.</div>

answer:
<div class="solution">Since, $\mathrm{Z}$ on reaction with $\mathrm{C}_6 \mathrm{H}_5 \mathrm{SO}_2 \mathrm{Cl}$ gives a product insoluble in alkali. Therefore, the product does not have a replaceable shows that $\mathrm{Z}$ is a secondary amine.<br/><br/>![rendered image](https://cdn.mathpix.com/snip/images/kY6f1uLbfmvn5fXFwz0vZHkPniN4tEBwFKmyZNHPX5g.rendered.fullsize.png)</div>


title: A condition for a function y = f ( x ) to have an inverse is that it should be
slug: a-condition-for-a-function-y-f-x-to-have-an-inverse-is-that-it-should-be-58861
tags: ['Mathematics', 'Functions', 'JEE Main']
answer: <div class="solution">It is a fundamental concept.</div>
source_db: MarksBatch0.db
question:
<div class="question">A condition for a function $y=f(x)$ to have an inverse is that it should be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Strictly monotonic and continuous in the domain</span> </div>


title: A condition where the protoxylem lies towards periphery and metaxylem lies towards the center is
tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Exarch</span> </div>
source_db: MarksBatch0.db
slug:
a-condition-where-the-protoxylem-lies-towards-periphery-and-metaxylem-lies-towards-the-center-is-22927

question:
<div class="question">A condition where the protoxylem lies towards periphery and metaxylem lies towards the center is</div>

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

answer:
<div class="solution">In exarch, the development of protoxylem occurs from the outermost part of the procambium, i.e., the development is from outside to inside. The protoxylem is found outside of the metaxylem (centripetal pattern). It is found in roots.</div>


title: A conjoint and open vascular bundle will be observed in the transverse section of
slug: a-conjoint-and-open-vascular-bundle-will-be-observed-in-the-transverse-section-of-35210
tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch0.db
question:
<div class="question">A conjoint and open vascular bundle will be observed in the transverse section of</div>

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

answer:
<div class="solution">Each vascular bundle is conjoint, collateral, open, and with endarch in protoxylem dicot stem</div>


tags: ['Physics', 'Ray Optics', 'BITSAT', 'BITSAT 2018']
source_db: MarksBatch0.db
title:
A convex lens shown in the figure is made up of two types of transparent materials. A point sources of light is placed on its principal axis. If reflections from the boundaries between layers are ignored, the lens will form

slug:
a-convex-lens-shown-in-the-figure-is-made-up-of-two-types-of-transparent-materials-a-point-sources-66864

question:
<div class="question">A convex lens shown in the figure is made up of two types of transparent materials. A point sources of light is placed on its principal axis. If reflections from the boundaries between layers are ignored, the lens will form<br/><img src="https://cdn-question-pool.getmarks.app/pyq/bitsat/9d490464-232a-4a92-a368-eb213a0feddd-image.png" style="width: 90px; height: 261px;"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">only one image</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">two images</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">infinite images</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">no image at all</span> </li> </ul>

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

answer:
<div class="solution">Since, the lens is made up of two kinds of transparent material, it has two refractive indices for the incident beam of light. Hence, there will be two focal lengths of the lens and therefore two images will be observed.</div>


slug: a-coordination-compound-is-made-of-co-3-nh-3-and-cl-0-1-m-solution-of-this-complex-w-85341
tags: ['Chemistry', 'Coordination Compounds', 'AP EAMCET', 'AP EAMCET 2018 (23 Apr Shift 2)']
source_db: MarksBatch0.db
title:
A coordination compound is made of Co 3 + , NH 3 â , and Cl â , 0.1 M solution of this complex when treated with excess silver nitrate gave no precipitate. The formula of the complex and secondary valency of metal are respectively.

question:
<div class="question">A coordination compound is made of $\mathrm{Co}^{3+}$, $\mathrm{NH}_3$, and $\mathrm{Cl}^{-}, 0.1 \mathrm{M}$ solution of this complex when treated with excess silver nitrate gave no precipitate. The formula of the complex and secondary valency of metal are respectively.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_3 \mathrm{Cl}_3\right], 6$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_5 \mathrm{Cl}_3 \mathrm{Cl}_2, 6\right.$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_3 \mathrm{Cl}_3\right], 3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_4 \mathrm{Cl}_2\right] \mathrm{Cl}, 6$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_3 \mathrm{Cl}_3\right], 6$</span> </div>

answer:
<div class="solution">Since, no precipitate of $\mathrm{AgCl}$ is obtained on treating the compound with silver nitrate that means no $\mathrm{Cl}^{-}$ions are available outside the coordination sphere. So, correct formula of coordination compound is $\left[\mathrm{Co}\left(\mathrm{NH}_3\right)_3 \mathrm{Cl}_3\right.$. Since, central metal $\mathrm{Co}$ forms six coordination bonds thus the secondary valency is 6 .</div>


title: A corner point of a feasible region is a point in the region which is the of two boundary lines.
tags: ['Mathematics', 'Linear Programming']
options: None
correct_option: None
answer: <div class="solution">Intersection</div>
source_db: MarksBatch0.db
slug:
a-corner-point-of-a-feasible-region-is-a-point-in-the-region-which-is-the-of-two-boundary-lines-72284

question:
<div class="question">A corner point of a feasible region is a point in the region which is the of two boundary lines.</div>


slug: a-cos-sin-sin-cos-a-cos-sin-62570
tags: ['Mathematics', 'Matrices', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch0.db
title:
A ( Î± ) â = [ cos Î± sin Î± â â sin Î± cos Î± â ] , A ( Î² ) â = [ cos Î² sin Î² â â sin Î² cos Î² â ] Which one of the following is correct?

question:
<div class="question">$A_{(\alpha)}=\left[\begin{array}{cc}\cos \alpha &amp; -\sin \alpha \\ \sin \alpha &amp; \cos \alpha\end{array}\right], A_{(\beta)}=\left[\begin{array}{cc}\cos \beta &amp; -\sin \beta \\ \sin \beta &amp; \cos \beta\end{array}\right]$<br/>Which one of the following is correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{A}_{(-\alpha)} \mathrm{A}_{(-\beta)}=\mathrm{A}_{(\alpha+\beta)}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{A}_{(-\alpha)} \mathrm{A}_{(\beta)}=\mathrm{A}_{(\beta-\alpha)}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$A_{(\alpha)}+A_{(-\beta)}=A_{\{-(\beta-\alpha)\}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$A_{(\alpha)}+A_{(\beta)}=A_{(\alpha+\beta)}$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$A_{(\alpha)}+A_{(\beta)}=A_{(\alpha+\beta)}$</span> </div>

answer:
<div class="solution">As given $\mathrm{A}(\alpha)=\left[\begin{array}{cc}\cos \alpha &amp; -\sin \alpha \\ \sin \alpha &amp; \cos \alpha\end{array}\right]$<br/>and $\quad A(\beta)=\left[\begin{array}{cc}\cos \beta &amp; -\sin \beta \\ \sin \beta &amp; \cos \beta\end{array}\right]$<br/>Hence,<br/>$\mathrm{A}(\alpha) \cdot \mathrm{A}(\beta)=\left[\begin{array}{cc}\cos \alpha &amp; -\sin \alpha \\ \sin \alpha &amp; \cos \alpha\end{array}\right]\left[\begin{array}{cc}\cos \beta &amp; -\sin \beta \\ \sin \beta &amp; \cos \beta\end{array}\right]$<br/>$=\left[\begin{array}{cc}\cos \alpha \cos \beta-\sin \alpha \sin \beta &amp; -\cos \alpha \sin \beta-\sin \alpha \cos \beta \\ \sin \alpha \cos \beta+\cos \alpha \sin \beta &amp; \sin \alpha \sin \beta+\cos \alpha \cos \beta\end{array}\right]$<br/>$=\left|\begin{array}{cc}\cos (\alpha+\beta) &amp; -\sin (\alpha+\beta) \\ \sin (\alpha+\beta) &amp; \cos (\alpha+\beta)\end{array}\right|=\mathbf{A}_{(\alpha+\beta)}$</div>


tags: ['Physics', 'Alternating Current', 'NEET']
source_db: MarksBatch0.db
title:
A current of 5 A is flowing at 220 V in the primary coil of a transformer. If the voltage produced in the secondary coil is 2200 V and 50 % of power is lost, then the current in the secondary coil is

slug:
a-current-of-5-a-is-flowing-at-220-v-in-the-primary-coil-of-a-transformer-if-the-voltage-produced-i-21046

question:
<div class="question">A current of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</mi></math> is flowing at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>220</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math> in the primary coil of a transformer. If the voltage produced in the secondary coil is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2200</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math> and <math><mn>50</mn><mi>%</mi></math> of power is lost, then the current in the secondary coil is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.25</mo><mi mathvariant="normal">A</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>2</mn><mo>.5</mo><mi mathvariant="normal">A</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>.5</mo><mi mathvariant="normal">A</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>5</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</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>.25</mo><mi mathvariant="normal">A</mi></math></span> </div>

answer:
<div class="solution">Given, power output<br/><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>Ã</mo></math> power input (or <math><mn>50</mn><mi>%</mi></math> of input power)<br/><br/><math><mo>â´</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>V</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>I</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>â</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub></math> <math><mfenced close="]" open="[" separators="|"><mrow><mo>âµ</mo><mi>P</mi><mo>=</mo><mi>V</mi><mi>I</mi></mrow></mfenced></math><br/><br/>Here, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>p</mi></msub><mo>=</mo><mn>5</mn><mi> </mi><mi mathvariant="normal">A</mi><mo>,</mo><mi> </mi><msub><mi>V</mi><mi>p</mi></msub><mo>=</mo><mn>220</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math><br/><br/>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mn>2200</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â´</mo><msub><mi>I</mi><mi>s</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>I</mi><mi>p</mi></msub><mo>â</mo><msub><mi>V</mi><mi>p</mi></msub></mrow><mrow><mn>2</mn><msub><mi>V</mi><mi>s</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn><mo>Ã</mo><mn>220</mn></mrow><mrow><mn>2</mn><mo>Ã</mo><mn>2200</mn></mrow></mfrac><mo>=</mo><mn>0.</mn><mn>25</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</mi></math></div>


tags: ['Physics', 'Alternating Current', 'COMEDK', 'COMEDK 2018']
source_db: MarksBatch0.db
title:
A current of 5 A is flowing at 220 V in the primary coil of a transformer. If the voltage produced in the secondary coil is 2200 V and 50% of power is lost, then the current in the secondary will be

slug:
a-current-of-5-a-is-flowing-at-220-v-in-the-primary-coil-of-a-transformer-if-the-voltage-produced-i-72170

question:
<div class="question">A current of $5 \mathrm{~A}$ is flowing at $220 \mathrm{~V}$ in the primary coil of a transformer. If the voltage produced in the secondary coil is $2200 \mathrm{~V}$ and $50 \%$ of power is lost, then the current in the secondary will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.5 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.25 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.5 \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.25 \mathrm{~A}$</span> </div>

answer:
<div class="solution">For a transformer,<br/>$\begin{aligned} I_{P}=5 \mathrm{~A}, V_{P} &amp;=220 \mathrm{~V} \\ V_{S}=2200 \mathrm{~V} \end{aligned}$<br/>Since, $50 \%$ power is lost, hence power at secondary side,<br/>$$<br/>\begin{array}{cl} <br/>&amp; P_{0}=50 \% \text { of } P_{i}=\frac{50}{100} \times 1100 \\<br/>\Rightarrow &amp; P_{0}=550 \mathrm{~W} \\<br/>\Rightarrow &amp; V_{S} I_{S}=550 \\<br/>\Rightarrow &amp; 2200 \times I_{S}=550 \\<br/>\Rightarrow &amp; I_{S}=\frac{550}{2200}=\frac{1}{4}=0.25 \mathrm{~A}<br/>\end{array}<br/>$$</div>


tags: ['Physics', 'Alternating Current', 'KCET', 'KCET 2008']
source_db: MarksBatch0.db
title:
A current of 5 A is flowing at 220 V in the primary coil of a transformer. If the voltage produced in the secondary coil is 2200 V and 50% of power is lost, then the current in secondary will be

slug:
a-current-of-5-a-is-flowing-at-220-v-in-the-primary-coil-of-a-transformer-if-the-voltage-produced-i-61707

question:
<div class="question">A current of $5 \mathrm{~A}$ is flowing at $220 \mathrm{~V}$ in the primary coil of a transformer. If the voltage produced in the secondary coil is $2200 \mathrm{~V}$ and $50 \%$ of power is lost, then the current in secondary will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2.5 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$5 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.25 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.5 \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.25 \mathrm{~A}$</span> </div>

answer:
<div class="solution">$\mathrm{V}_{\mathrm{p}}=220 \mathrm{~V}, \mathrm{~V}_{\mathrm{s}}=2200 \mathrm{~V}, \mathrm{I}_{\mathrm{p}}=5 \mathrm{~A}, \mathrm{I}_{\mathrm{s}}=$ ?<br/>Power loss $=50 \%$<br/>Efficiency of transformer ( $\eta$ ) is defined as the ratio of output power and input power<br/>$$<br/>\eta \%=\frac{\mathrm{P}_{\mathrm{out}}}{\mathrm{P}_{\mathrm{in}}} \times 100=\frac{\mathrm{V}_{\mathrm{s}} \mathrm{I}_{\mathrm{s}}}{\mathrm{V}_{\mathrm{p}} \mathrm{I}_{\mathrm{p}}} \times 100<br/>$$<br/>$$<br/>\begin{aligned}<br/>50 &amp;=\frac{2200 \times \mathrm{I}_{\mathrm{s}}}{220 \times 5} \times 100 \\<br/>\mathrm{I}_{\mathrm{s}} &amp;=0.25 \mathrm{~A}<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Alternating Current', 'JEE Main']
source_db: MarksBatch0.db
title:
A current of 5 A is flowing at 220 V in the primary coil of a transformer. If the voltage produced in the secondary coil is 2200 V and 50 % of power is lost, then the current in the secondary coil is

slug:
a-current-of-5-a-is-flowing-at-220-v-in-the-primary-coil-of-a-transformer-if-the-voltage-produced-i-78948

question:
<div class="question">A current of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</mi></math> is flowing at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>220</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math> in the primary coil of a transformer. If the voltage produced in the secondary coil is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2200</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math> and <math><mn>50</mn><mi>%</mi></math> of power is lost, then the current in the secondary coil is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.25</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</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>2</mn><mo>.5</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</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>.5</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</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>5</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</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>.25</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</mi></math></span> </div>

answer:
<div class="solution">Given, power output<br/><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>×</mo></math> power input (or <math><mn>50</mn><mi>%</mi></math> of input power)<br/><br/><math><mo>∴</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>V</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>I</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⋅</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub></math> <math><mfenced close="]" open="[" separators="|"><mrow><mo>∵</mo><mi>P</mi><mo>=</mo><mi>V</mi><mi>I</mi></mrow></mfenced></math><br/><br/>Here, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>p</mi></msub><mo>=</mo><mn>5</mn><mi> </mi><mi mathvariant="normal">A</mi><mo>,</mo><mi> </mi><msub><mi>V</mi><mi>p</mi></msub><mo>=</mo><mn>220</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math><br/><br/>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mn>2200</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><msub><mi>I</mi><mi>s</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>I</mi><mi>p</mi></msub><mo>⋅</mo><msub><mi>V</mi><mi>p</mi></msub></mrow><mrow><mn>2</mn><msub><mi>V</mi><mi>s</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn><mo>×</mo><mn>220</mn></mrow><mrow><mn>2</mn><mo>×</mo><mn>2200</mn></mrow></mfrac><mo>=</mo><mn>0.</mn><mn>25</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</mi></math></div>


tags: ['Physics', 'Alternating Current', 'NEET']
source_db: MarksBatch0.db
title:
A current of 5 A is flowing at 220 V in the primary coil of a transformer. If the voltage produced in the secondary coil is 2200 V and 50 % of power is lost, then the current in the secondary coil is

slug:
a-current-of-5-a-is-flowing-at-220-v-in-the-primary-coil-of-a-transformer-if-the-voltage-produced-i-10300

question:
<div class="question">A current of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</mi></math> is flowing at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>220</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math> in the primary coil of a transformer. If the voltage produced in the secondary coil is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2200</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math> and <math><mn>50</mn><mi>%</mi></math> of power is lost, then the current in the secondary coil is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.25</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</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>2</mn><mo>.5</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</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>.5</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</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>5</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</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>.25</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</mi></math></span> </div>

answer:
<div class="solution">Given, power output<br/><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>×</mo></math> power input (or <math><mn>50</mn><mi>%</mi></math> of input power)<br/><br/><math><mo>∴</mo><msub><mrow><mi>I</mi></mrow><mrow><mi>s</mi></mrow></msub><msub><mrow><mi>V</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>I</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>⋅</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>p</mi></mrow></msub></math> <math><mfenced close="]" open="[" separators="|"><mrow><mo>∵</mo><mi>P</mi><mo>=</mo><mi>V</mi><mi>I</mi></mrow></mfenced></math><br/><br/>Here, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>p</mi></msub><mo>=</mo><mn>5</mn><mi> </mi><mi mathvariant="normal">A</mi><mo>,</mo><mi> </mi><msub><mi>V</mi><mi>p</mi></msub><mo>=</mo><mn>220</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math><br/><br/>and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>s</mi></msub><mo>=</mo><mn>2200</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">V</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><msub><mi>I</mi><mi>s</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>I</mi><mi>p</mi></msub><mo>⋅</mo><msub><mi>V</mi><mi>p</mi></msub></mrow><mrow><mn>2</mn><msub><mi>V</mi><mi>s</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>5</mn><mo>×</mo><mn>220</mn></mrow><mrow><mn>2</mn><mo>×</mo><mn>2200</mn></mrow></mfrac><mo>=</mo><mn>0.</mn><mn>25</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">A</mi></math></div>


tags: ['Physics', 'Alternating Current', 'MHT CET', 'MHT CET 2022 (08 Aug Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.25 A</span> </div>
source_db: MarksBatch0.db
title:
A current of 5 A is flowing at 220 V in the primary coil of transformer. If the voltage produced in the secondary coil is 2200 V and 50% of power is lost, then the current in secondary will be

slug:
a-current-of-5-a-is-flowing-at-220-v-in-the-primary-coil-of-transformer-if-the-voltage-produced-in-89828

question:
<div class="question">A current of $5 \mathrm{~A}$ is flowing at $220 \mathrm{~V}$ in the primary coil of transformer. If the voltage produced in the secondary coil is $2200 \mathrm{~V}$ and $50 \%$ of power is lost, then the current in secondary will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2.5 A</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.5 A</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0.25 A</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5 A</span> </li> </ul>

answer:
<div class="solution">Let $I$ be the current flowing through secondary coil.<br/>Current on primary (input) coil $i=5 \mathrm{~A}$<br/>Voltage in primary coil $V=220 \mathrm{~V}$<br/>Voltage in secondary coil $V^{\prime}=2200 \mathrm{~V}$<br/>Considering only $50 \%$ power is transmitted:<br/>$\begin{aligned}<br/>&amp; 0.5(\mathrm{Vi})=V^{\prime} I \\<br/>&amp; \therefore 0.5 \times 220 \times 5=2200 \times I \\<br/>&amp; \Rightarrow I=0.25 \mathrm{~A}<br/>\end{aligned}$</div>


tags: ['Physics', 'Dual Nature of Matter', 'TS EAMCET', 'TS EAMCET 2020 (09 Sep Shift 1)']
source_db: MarksBatch0.db
title:
A current of 6 A is flowing at 220 V in the primary coil of a transformer. If the voltage produced in the secondary coil is 1100 V and 40 % of power is lost, then the current in the secondary coil will be:

slug:
a-current-of-6-a-is-flowing-at-220-v-in-the-primary-coil-of-a-transformer-if-the-voltage-produced-i-23967

question:
<div class="question">A current of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo> </mo><mi mathvariant="normal">A</mi></math> is flowing at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>220</mn><mo> </mo><mi mathvariant="normal">V</mi></math> in the primary coil of a transformer. If the voltage produced in the secondary coil is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1100</mn><mo> </mo><mi mathvariant="normal">V</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo>%</mo></math> of power is lost, then the current in the secondary coil will be:</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>0</mn><mo>.</mo><mn>28</mn><mo> </mo><mi mathvariant="normal">A</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>0</mn><mo>.</mo><mn>36</mn><mo> </mo><mi mathvariant="normal">A</mi></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>0</mn><mo>.</mo><mn>48</mn><mo> </mo><mi mathvariant="normal">A</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">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>42</mn><mo> </mo><mi mathvariant="normal">A</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>48</mn><mo> </mo><mi mathvariant="normal">A</mi></math></span> </div>

answer:
<div class="solution"><p>Given: Voltage in the primary coil <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>i</mi></msub><mo>=</mo><mn>220</mn><mo> </mo><mi mathvariant="normal">V</mi></math>, voltage in the secondary coil <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>O</mi></msub><mo>=</mo><mn>1100</mn><mo> </mo><mi mathvariant="normal">V</mi></math>, current in the primary coil, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>i</mi></msub><mo>=</mo><mn>6</mn><mo> </mo><mi mathvariant="normal">A</mi></math>, Power lost<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>40</mn><mo>%</mo></math></p><p>Let the current in the secondary coil be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>O</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi></math> be the efficiency of the transformer.</p><p>Efficiency <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>η</mi><mo>=</mo><mfrac><msub><mi>P</mi><mi>OUT</mi></msub><msub><mi>P</mi><mi>IN</mi></msub></mfrac></math></p><p>Then, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>OUT</mi></msub><mo>=</mo><msub><mi>P</mi><mi>IN</mi></msub><mo>×</mo><mi>η</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>O</mi></msub><mo>×</mo><msub><mi>I</mi><mi>O</mi></msub><mo>=</mo><msub><mi>V</mi><mi>i</mi></msub><mo>×</mo><msub><mi>I</mi><mi>i</mi></msub><mo>×</mo><mi>η</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>O</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>V</mi><mi>i</mi></msub><mo>×</mo><msub><mi>I</mi><mi>i</mi></msub><mo>×</mo><mi>η</mi></mrow><msub><mi>V</mi><mi>O</mi></msub></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mi>O</mi></msub><mo>=</mo><mfrac><mrow><mn>220</mn><mo>×</mo><mn>6</mn><mo>×</mo><mn>40</mn></mrow><mrow><mn>110</mn><mo>×</mo><mn>100</mn></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>48</mn><mo> </mo><mi mathvariant="normal">A</mi></math></p></div>


tags: ['Physics', 'Electromagnetic Induction', 'NEET']
source_db: MarksBatch0.db
title:
A current of 10 A in the primary coil of a circuit is reduced to zero. If the coefficient of mutual inductance is 3 H and emf induced in the secondary coil is 30 kV, time taken for the change of current is

slug:
a-current-of-10-a-in-the-primary-coil-of-a-circuit-is-reduced-to-zero-if-the-coefficient-of-mutual-71066

question:
<div class="question">A current of <math> <mrow> <mn>10</mn><mtext>â</mtext><mtext>A</mtext></mrow> </math> in the primary coil of a circuit is reduced to zero. If the coefficient of mutual inductance is <math> <mrow> <mn>3</mn><mtext>â</mtext><mtext>H</mtext></mrow> </math> and emf induced in the secondary coil is <math> <mrow> <mn>30</mn><mtext>â</mtext><mtext>kV,</mtext></mrow> </math> time taken for the change of current is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><msup><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></msup><mi> s</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><msup><mrow><mn>10</mn></mrow><mrow><mn>2</mn></mrow></msup><mn> s</mn></math> </span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>3</mn></mrow></msup><mn> s</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><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>2</mn></mrow></msup><mn> s</mn></math> </span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>3</mn></mrow></msup><mn> s</mn></math></span> </div>

answer:
<div class="solution">As <math><mi>e</mi><mo>=</mo><mi>M</mi><mfrac><mrow><mn>d</mn><mi>i</mi></mrow><mrow><mn>d</mn><mi>t</mi></mrow></mfrac></math><br/><br/><math><mo>â´</mo><mi> </mi><mn>30</mn><mo>Ã</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mn>3</mn><mo>Ã</mo><mfrac><mrow><mn>10</mn></mrow><mrow><mn>d</mn><mi>t</mi></mrow></mfrac><mo>,</mo></math><br/><br/><math><mn>d</mn><mi>t</mi><mo>=</mo><mi> </mi><mfrac><mrow><mn>30</mn></mrow><mrow><mn>30</mn><mo>Ã</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></msup></mrow></mfrac><mo>=</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>-</mo><mn>3</mn></mrow></msup><mi> s</mi></math></div>


tags: ['Physics', 'Thermodynamics']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase, if the gas is compressed to half its original volume?

slug:
a-cylinder-with-a-movable-piston-contains-3-moles-of-hydrogen-at-standard-temperature-and-pressure-59631

question:
<div class="question">A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase, if the gas is compressed to half its original volume?</div>

answer:
<div class="solution">The process is adiabatic, as no heat in exchanged<br/>$$<br/>\therefore \quad P_2 V_2^\gamma=P_1 V_1^\gamma \quad \therefore \frac{P_2}{P_1}=\left(\frac{V_1}{V_2}\right)^\gamma<br/>$$<br/>Given $V_2=1 / 2 V_1$<br/>$$<br/>\therefore \quad \frac{P_2}{P_1}=\left(\frac{V_1}{V_1 / 2}\right)^{1.4}=2.64<br/>$$</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch0.db
title:
A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be k times the height of the jar. The value of k is

slug:
a-cylindrical-jar-without-a-lid-has-to-be-constructed-using-a-given-surface-area-of-a-metal-sheet-i-83448

question:
<div class="question">A cylindrical jar without a lid has to be constructed using a given surface area of a metal sheet. If the capacity of the jar is to be maximum, then the diameter of the jar must be $\mathrm{k}$ times the height of the jar. The value of k is</div>

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

answer:
<div class="solution">The capacity of the jar will be maximum if height and radius of the cylinder are equal. $\therefore$ height $=$ radius<br/>(1) Given diameter is $\mathrm{k}$ times the height of the jar. Diameter $=2 \times$ radius $=2 \times$ height $\quad \ldots .($ from $(1))$ $\therefore \mathrm{k}=2$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2019 (21 Apr Shift 2)']
source_db: MarksBatch0.db
title:
A cylindrical tube open at both ends has a fundamental frequency f in air. The tube is dipped vertically in water so that 60 % of the tube is in water. Then the fundamental frequency of air column is

slug:
a-cylindrical-tube-open-at-both-ends-has-a-fundamental-frequency-f-in-air-the-tube-is-dipped-vertic-74981

question:
<div class="question">A cylindrical tube open at both ends has a fundamental frequency <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> in air. The tube is dipped vertically in water so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><mo>%</mo></math> of the tube is in water. Then the fundamental frequency of air column 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"><mfrac><mi>f</mi><mn>2</mn></mfrac></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"><mfrac><mrow><mn>5</mn><mi>f</mi></mrow><mn>4</mn></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 xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><mi>f</mi></mrow><mn>4</mn></mfrac></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>2</mn><mi>f</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>5</mn><mi>f</mi></mrow><mn>4</mn></mfrac></math></span> </div>

answer:
<div class="solution"><blockquote><p>Consider the length of the tube be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math></p><p>The fundamental frequency for the open organ pipe is,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mfrac><mi>V</mi><mrow><mn>2</mn><mo> </mo><mi mathvariant="normal">L</mi></mrow></mfrac></math></p><p>The figure below represents the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><mo>%</mo></math> of tube is dipped<br/>vertically in water.</p><p><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/b52a9dee-2b03-41a1-ba10-f55730a990b6-image.png" style="width: 292px; height: 271px;"/></p><p>The length of the closed organ pipe is,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>l</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>40</mn><mn>100</mn></mfrac><mo> </mo><mi>L</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>l</mi><mo>'</mo></msup><mo>=</mo><mfrac><mrow><mn>2</mn><mo> </mo><mi mathvariant="normal">L</mi></mrow><mn>5</mn></mfrac></math></p><p>The fundamental frequency of air column is,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mfrac><mi>V</mi><mrow><mn>4</mn><msup><mi mathvariant="normal">l</mi><mo>'</mo></msup></mrow></mfrac></math></p><p>Substitute the values in the above equation.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mfrac><mi>V</mi><mrow><mn>4</mn><mo> </mo><mfenced><mfrac><mrow><mn>2</mn><mo> </mo><mi mathvariant="normal">L</mi></mrow><mn>5</mn></mfrac></mfenced></mrow></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>5</mn><mn>4</mn></mfrac><mfenced><mfrac><mi>V</mi><mrow><mn>2</mn><mo> </mo><mi mathvariant="normal">L</mi></mrow></mfrac></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mo>'</mo></msup><mo>=</mo><mfrac><mn>5</mn><mn>4</mn></mfrac><mi>f</mi></math></p></blockquote></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2012 (Offline)']
answer: <div class="solution">$f_o=\frac{v}{2 \ell}$<br/>$f_c=\frac{v}{2 \ell}$</div>
source_db: MarksBatch0.db
title:
A cylindrical tube, open at both ends, has a fundamental frequency, f , in air. The tube is dipped vertically in water so that half of it is in water. The fundamental frequency of the air-column is now

slug:
a-cylindrical-tube-open-at-both-ends-has-a-fundamental-frequency-f-in-air-the-tube-is-dipped-v-59020

question:
<div class="question">A cylindrical tube, open at both ends, has a fundamental frequency, $\mathrm{f}$, in air. The tube 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="correct"> <span class="option-label">A</span> <span class="option-data"><br/>$f$<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/>$\frac{f}{2}$<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>$\frac{3 f}{4}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$2 f$</span> </li> </ul>

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


tags: ['Physics', 'Waves and Sound', 'TS EAMCET', 'TS EAMCET 2022 (20 Jul Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">f</span> </div>
source_db: MarksBatch0.db
title:
A cylindrical tube open at both ends has a fundamental frequency 'f' in air. The tube is dipped vertically in water so that half of it is in water. The new fundamental frequency is

slug:
a-cylindrical-tube-open-at-both-ends-has-a-fundamental-frequency-f-in-air-the-tube-is-dipped-vert-91812

question:
<div class="question">A cylindrical tube open at both ends has a fundamental<br/>frequency 'f' in air. The tube is dipped vertically in water so<br/>that half of it is in water. The new fundamental frequency is</div>

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

answer:
<div class="solution">Initially<br/>$$<br/>\mathrm{f}=\frac{\mathrm{V}}{2 \ell}<br/>$$<br/>Finally<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/Ot8N3cebrtqcSIfR_8GYmZy20ecKUhNwPxquRW6lAjI.original.fullsize.png"/><br/><br/>From diagram, we can see<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{\lambda}{4}=\frac{\ell}{2} \\<br/>&amp; \Rightarrow \lambda=21<br/>\end{aligned}<br/>$$<br/>Now, fundamental frequency<br/>$$<br/>=\frac{\mathrm{V}}{\lambda}=\frac{\mathrm{V}}{2 \ell}=\mathrm{f}<br/>$$<br/>So, there will be no change in fundamental frequency.</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2022 (08 Aug Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">f</span> </div>
source_db: MarksBatch0.db
title:
A cylindrical tube open at both ends, has a vibrating air column of fundamental frequency ' f ' in air. The tube is dipped vertically in water so that half of its is in water. The fundamental frequency of the vibrating air column is now

slug:
a-cylindrical-tube-open-at-both-ends-has-a-vibrating-air-column-of-fundamental-frequency-f-in-a-59614

question:
<div class="question">A cylindrical tube open at both ends, has a vibrating air column of fundamental frequency ' $\mathrm{f}$ ' in air. The tube is dipped vertically in water so that half of its is in water. The fundamental frequency of the vibrating air column is now</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/mht_cet/UxqHgqwid1NmekK3JPUyzBSLiEFwfoTePN00SElba-A.original.fullsize.png"/><br/><br/>$v=\lambda f$<br/>Fundamental frequency's $\lambda=\frac{v}{f} \Rightarrow f_0=\frac{v}{2 L}=f$<br/>If the tube is dipped half into liquid then it acts as if closed at $\frac{\mathrm{L}}{2}$.<br/>See figure<img src="https://cdn-question-pool.getmarks.app/pyq/mht_cet/jaHyZdtALYyuizLlAhNeNHSUgva7lLNdXAmz-7jZZro.original.fullsize.png"/><br/><br/>The frequency can be written as $\mathrm{f}_{\mathrm{c}}=\frac{\mathrm{v}}{\lambda_1}=\frac{\mathrm{v}}{2 \mathrm{~L}}=\mathrm{f}$<br/>$\therefore$ Fundamental frequency would remain the same ' $\mathrm{f}$ '.</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2022 (10 Aug Shift 1)']
source_db: MarksBatch0.db
title:
A cylindrical tube open at both ends has fundamental frequency f in air. When the tube is dipped vertically in water so that one-third part of the tube is in water, the fundamental frequency of air column becomes (neglect end correction)

slug:
a-cylindrical-tube-open-at-both-ends-has-fundamental-frequency-f-in-air-when-the-tube-is-dipped-ver-26253

question:
<div class="question">A cylindrical tube open at both ends has fundamental frequency $f$ in air. When the tube is dipped vertically in water so that one-third part of the tube is in water, the fundamental frequency of air column becomes (neglect end correction)</div>

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

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

answer:
<div class="solution">We know. Frequency is ratio of velocity of sound and wavelength:<br/>$f=\frac{v}{\lambda}$<br/>For tube of length , in fundamental mode wavelength is given by $\frac{\lambda}{2}=L$<br/>$\frac{\lambda}{2}=L$.<br/>$\therefore f=\frac{v}{2 L}$<br/>Now, if tube is dipped one third, then for fundamental mode in air column has:<br/>$\begin{aligned} &amp; \frac{\lambda^{\prime}}{2}=\left(\frac{2}{3} L\right) \\ &amp; \therefore f^{\prime}=\frac{3 v}{4 L}=\frac{3}{2} f\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2021 (24 Sep Shift 2)']
source_db: MarksBatch0.db
title:
A cylindrical tube open at both ends has fundamental frequency ' n ' in air. The tube is dipped vertically in water so that one-fourth of it is in water. The fundamental frequency of the air column becomes

slug:
a-cylindrical-tube-open-at-both-ends-has-fundamental-frequency-n-in-air-the-tube-is-dipped-vert-97031

question:
<div class="question">A cylindrical tube open at both ends has fundamental frequency ' $n$ ' in air. The tube is dipped vertically in water so that one-fourth of it is in water. The fundamental frequency of the air column becomes</div>

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

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

answer:
<div class="solution">The fundamental frequency of open tube is<br/>$\mathrm{n}_1=\frac{\mathrm{v}}{2 \ell_1}$<br/>When tube is dipped in water, one-fourth of it is in water and three-fourth is in air.<br/><br/>Hence, it becomes a tube closed at one end with length $\ell_2=\frac{3}{4} \ell_1$ The fundamental frequency of closed tube is<br/>$\begin{aligned}<br/>&amp; \mathrm{n}_2=\frac{\mathrm{v}}{4 \ell_2} \\<br/>&amp; \therefore \frac{\mathrm{n}_2}{\mathrm{n}_1}=\frac{1}{4 \ell_2} \times 2 \ell_1=\frac{\ell_1}{2 \ell_2}=\frac{4}{2 \times 3} \quad\left[\because \frac{\ell_1}{\ell_2}=\frac{4}{3}\right] \\<br/>&amp; \therefore \mathrm{n}_2=\frac{2}{3} \mathrm{n}_1=\frac{2}{3} \mathrm{n}<br/>\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'KCET', 'KCET 2009']
source_db: MarksBatch0.db
title:
A cylindrical tube open at both the ends has a fundamental frequency of 390 Hz in air. If 4 1 â th of the tube is immersed vertically in water the fundamental frequency of air column is

slug:
a-cylindrical-tube-open-at-both-the-ends-has-a-fundamental-frequency-of-390-hz-in-air-if-4-1-th-51860

question:
<div class="question">A cylindrical tube open at both the ends has a fundamental frequency of $390 \mathrm{~Hz}$ in air. If $\frac{1}{4}$ th of the tube is immersed vertically in water the fundamental frequency of air column is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$260 \mathrm{~Hz}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$130 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$390 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$520 \mathrm{~Hz}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$260 \mathrm{~Hz}$</span> </div>

answer:
<div class="solution">Fundamental frequency of cylindrical open tube<br/>$$<br/>\mathrm{n}=\frac{\mathrm{v}}{2 \mathrm{~L}}=390 \mathrm{~Hz}<br/>$$<br/>When it is immersed in water it becomes a closed tube of length $\frac{3}{4}$ th of the initial length.<br/>Therefore, its fundamental frequency is<br/>$$<br/>\begin{aligned}<br/>\mathrm{n}^{\prime} &amp;=\frac{\mathrm{V}}{4\left(\frac{3}{4} \mathrm{~L}\right)}=\frac{\mathrm{v}}{3 \mathrm{~L}}=\frac{2}{3}\left(\frac{\mathrm{V}}{2 \mathrm{~L}}\right) \\<br/>&amp;=\frac{2}{3} \times 390 \mathrm{~Hz}=260 \mathrm{~Hz}<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch0.db
title:
A cylindrical tube open at both the ends has a fundamental frequency of 390 Hz in the air. If 1 4 of the tube is immersed vertically in water, the fundamental frequency of air column is

slug:
a-cylindrical-tube-open-at-both-the-ends-has-a-fundamental-frequency-of-390-hz-in-the-air-if-1-4-of-25903

question:
<div class="question">A cylindrical tube open at both the ends has a fundamental frequency of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>390</mn><mo> </mo><mi>Hz</mi></math> in the air. If <math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math> of the tube is immersed vertically in water, the fundamental frequency of air column is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>260</mn><mo> </mo><mi>Hz</mi></math><div id="gtx-trans" style="position: absolute; left: 129px; top: 37.6667px;"><div class="gtx-trans-icon"> </div></div></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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>130</mn><mo> </mo><mi>Hz</mi></math><div id="gtx-trans" style="position: absolute; left: 129px; top: 37.6667px;"><div class="gtx-trans-icon"> </div></div></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>390</mn><mo> </mo><mi>Hz</mi></math><div id="gtx-trans" style="position: absolute; left: 129px; top: 37.6667px;"><div class="gtx-trans-icon"> </div></div></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>520</mn><mo> </mo><mi>Hz</mi></math><div id="gtx-trans" style="position: absolute; left: 129px; top: 37.6667px;"><div class="gtx-trans-icon"> </div></div></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>260</mn><mo> </mo><mi>Hz</mi></math><div id="gtx-trans" style="position: absolute; left: 129px; top: 37.6667px;"><div class="gtx-trans-icon"> </div></div></span> </div>

answer:
<div class="solution">The fundamental frequency of cylindrical open tube<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mi>L</mi></mrow></mfrac><mo>=</mo><mn>390</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">H</mi><mi mathvariant="normal">z</mi></math><br/><br/>When it is immersed in water it becomes a closed tube of length, <math><msup><mrow><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mrow><mrow><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi></mrow></msup></math> of the initial length.<br/><br/>Therefore, its fundamental frequency is<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mi mathvariant="normal">′</mi></msup><mo>=</mo><mfrac><mi>v</mi><mrow><mn>4</mn><mfenced separators="|"><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mi>L</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mi>v</mi><mrow><mn>3</mn><mi>L</mi></mrow></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mfenced separators="|"><mfrac><mi>v</mi><mrow><mn>2</mn><mi>L</mi></mrow></mfrac></mfenced></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>×</mo><mn>390</mn><mo> </mo><mi mathvariant="normal">H</mi><mi mathvariant="normal">z</mi><mo>=</mo><mn>260</mn><mo> </mo><mi mathvariant="normal">H</mi><mi mathvariant="normal">z</mi></math></div>


tags: ['Physics', 'Oscillations', 'NEET']
source_db: MarksBatch0.db
title:
A cylindrical tube, open at both the ends has fundamental frequency n . If one of the ends is closed, the fundamental frequency will become

slug:
a-cylindrical-tube-open-at-both-the-ends-has-fundamental-frequency-n-if-one-of-the-ends-is-closed-17754

question:
<div class="question">A cylindrical tube, open at both the ends has fundamental frequency <math><mi>n</mi></math> . If one of the ends is closed, the fundamental frequency will become</div>

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

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

answer:
<div class="solution">For a cylindrical tube <br/><br/>If both the ends is closed then<br/><br/>Fundamental frequency,  <math><mi>n</mi><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></math>              .....(i)<br/><br/>If one of the ends is closed then fundamental frequency,  <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><mi>l</mi></mrow></mfrac></math>    ....(ii)<br/><br/>from equation (i) and (ii)<br/><br/> <math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></div>


slug: a-data-consists-of-n-observations-x-1-x-2-x-n-if-i-1-n-x-i-1-2-9-n-and-i-26192
tags: ['Mathematics', 'Statistics', 'JEE Main', 'JEE Main 2019 (09 Jan Shift 2)']
source_db: MarksBatch0.db
title:
A data consists of n observations: x 1 , x 2 , â¦ , x n . If â i = 1 n x i + 1 2 = 9 n and â i = 1 n x i - 1 2 = 5 n , then the standard deviation of this data is

question:
<div class="question">A data consists of <math><mi>n</mi></math> observations: <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mi> </mi><msub><mi>x</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><mo>…</mo><mo> </mo><mo>,</mo><mi> </mi><msub><mi>x</mi><mi>n</mi></msub><mo>.</mo></math> If <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msup><mfenced><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>9</mn><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo stretchy="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>5</mn><mi>n</mi></math>, then the standard deviation of this data is</div>

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

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

answer:
<div class="solution"><blockquote><p>Given:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>9</mn><mi>n</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced separators="|"><mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>9</mn><mi>n</mi><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>1</mn></mfenced></math></p><p>And, <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msup><mfenced separators="|"><mrow><msub><mi>x</mi><mi>i</mi></msub><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>5</mn><mi>n</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced separators="|"><mrow><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>-</mo><mn>2</mn><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>5</mn><mi>n</mi><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>2</mn></mfenced></math> </p><p>Form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>1</mn></mfenced><mo>+</mo><mfenced><mn>2</mn></mfenced></math>, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced><mrow><mn>2</mn><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>14</mn><mi>n</mi></math> </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msubsup><mi>x</mi><mi>i</mi><mn>2</mn></msubsup><mo>=</mo><mn>6</mn><mi>n</mi><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>3</mn></mfenced><mo> </mo></math></p><p>From <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>1</mn></mfenced><mo>-</mo><mfenced><mn>2</mn></mfenced></math>, we get </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><mo>=</mo><mi>n</mi><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>4</mn></mfenced></math></p><p><math><mo>∴</mo></math> Variance, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mi>E</mi><mfenced separators="|"><msup><mi>x</mi><mn>2</mn></msup></mfenced><mo>-</mo><msup><mfenced separators="|"><mrow><mi>E</mi><mfenced separators="|"><mi>x</mi></mfenced></mrow></mfenced><mn>2</mn></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo>∑</mo><msup><mi>x</mi><mn>2</mn></msup></mrow><mi>n</mi></mfrac><mo>-</mo><mfrac><mrow><mo>∑</mo><mi>x</mi></mrow><mi>n</mi></mfrac><mo> </mo></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>σ</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>6</mn><mi>n</mi></mrow><mi>n</mi></mfrac><mo>-</mo><msup><mfenced separators="|"><mfrac><mi>n</mi><mi>n</mi></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mn>6</mn><mo>-</mo><mn>1</mn><mo>=</mo><mn>5</mn></math></p><p><math><mo>∴</mo></math> Standard deviation, <math><mi>σ</mi><mo>=</mo><msqrt><mn>5</mn></msqrt></math>.</p></blockquote></div>


title: A die is thrown 100 times, then the standard deviation of getting an even number is
slug: a-die-is-thrown-100-times-then-the-standard-deviation-of-getting-an-even-number-is-75291
tags: ['Mathematics', 'Statistics', 'MHT CET', 'MHT CET 2020 (14 Oct Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch0.db
question:
<div class="question">A die is thrown 100 times, then the standard deviation of getting an even number is</div>

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

answer:
<div class="solution">Let $x$ denote the number of succeses in 100 Then $x$ follows binomial distribution with $n=1008$ $p=y_{2}, q=\frac{1}{2}$<br/>$\therefore$ variance of $x=n p q=100 \times \frac{1}{2} \times \frac{1}{2}=25$<br/>$\begin{array}{l}\text { Standard } \\ \text { Deviation }\end{array}=\sqrt{\text { vatiance }}=\sqrt{25}$<br/>$=5$</div>


title: A die is tossed twice. What is the probability of getting a sum of 10 ?
slug: a-die-is-tossed-twice-what-is-the-probability-of-getting-a-sum-of-10-78657
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch0.db
question:
<div class="question">A die is tossed twice. What is the probability of getting a sum of $10 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{18}$</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">$\frac{1}{12}$</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">$\frac{5}{12}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{12}$</span> </div>

answer:
<div class="solution">Number of possible outcomes $=36$ When sum is 10 , samples are $(5,5),(4,6)$ and $(6,4)$ Required probability $=\frac{3}{36}=\frac{1}{12}$</div>


title: A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch0.db
slug:
a-fair-coin-is-tossed-100-times-what-is-the-probability-of-getting-tails-an-odd-number-of-times-95461

question:
<div class="question">A fair coin is tossed 100 times. What is the probability of getting tails an odd number of times?</div>

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

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

answer:
<div class="solution">Let $\mathrm{x}$ denote number of tails. Then, $\mathrm{X}$ is a binomial variate with parameters:<br/>$\mathrm{x}=100 \&amp; \mathrm{p}=\frac{1}{2}$<br/>$\therefore \mathrm{p}(\mathrm{x}=\mathrm{r})={ }^{100} \mathrm{C}_{\mathrm{r}}\left(\frac{1}{2}\right)^{100} ;(\mathrm{r}=0,1,2, \ldots \ldots .100)$<br/>Req. probability $=\mathrm{P}(\mathrm{x}=1)+\mathrm{P}(\mathrm{x}=3)+\ldots .+\mathrm{P}(\mathrm{x}=99)$<br/>$=\left[\frac{1}{2}\right]^{100}\left[{ }^{100} \mathrm{C}_{1}+{ }^{100} \mathrm{C}_{3}+\ldots .+{ }^{100} \mathrm{C}_{99}\right]$<br/>$=\frac{1}{2^{100}} 2^{99}=\frac{1}{2}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch0.db
title:
A fair coin is tossed and an unbiased dice is rolled together. What is the probability of getting a 2 or 4 or 6 along with head?

slug:
a-fair-coin-is-tossed-and-an-unbiased-dice-is-rolled-together-what-is-the-probability-of-getting-a-19668

question:
<div class="question">A fair coin is tossed and an unbiased dice is rolled together. What is the probability of getting a 2 or 4 or 6 along with head?</div>

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

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

answer:
<div class="solution">Total number of events with dice and coin $=6 \times 2=12$ Number of possibilities $=(2, \mathrm{H}),(4, \mathrm{H})$ and $(6, \mathrm{H})$ i.e., 3<br/>$\therefore$ Probability $=\frac{3}{12}=\frac{1}{4}$</div>


title: A fair coin is tossed four times. What is the probability that at most three tails occur ?
slug: a-fair-coin-is-tossed-four-times-what-is-the-probability-that-at-most-three-tails-occur-97131
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch0.db
question:
<div class="question">A fair coin is tossed four times. What is the probability that at most three tails occur ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$7 / 9$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$15 / 16$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$13 / 16$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$3 / 4$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$15 / 16$</span> </div>

answer:
<div class="solution">$\begin{array}{l}\quad \mathrm{n}(\mathrm{S})=2^{4}=16 \\ \text { and } \mathrm{n}(\mathrm{E})={ }^{4} \mathrm{C}_{0}+{ }^{4} \mathrm{C}_{1}+{ }^{4} \mathrm{C}_{2}+{ }^{4} \mathrm{C}_{3} \\ &amp; =1+4+\frac{4 \times 3}{2 \times 1}+4=1+4+6+4=15 \\ &amp; \therefore \mathrm{P}(\mathrm{E})=\frac{\mathrm{n}(\mathrm{E})}{\mathrm{n}(\mathrm{S})}=\frac{15}{16}\end{array}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch0.db
title:
A fair coin is tossed repeatedly. The probability of getting a result in the fifth toss different from those obtained in the first four tosses is:

slug:
a-fair-coin-is-tossed-repeatedly-the-probability-of-getting-a-result-in-the-fifth-toss-different-fr-58789

question:
<div class="question">A fair coin is tossed repeatedly. The probability of getting a result in the fifth toss different from those obtained in the first four tosses is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{32}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{31}{32}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{1}{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{16}$</span> </div>

answer:
<div class="solution">P(HHHHT or TTTTH) = P (HHHHT) + P (TTTTH)<br/>$=\left(\frac{1}{2}\right)^{5}+\left(\frac{1}{2}\right)^{5}=\frac{1}{16}$</div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch0.db
title:
A firm employing 30 workers and paying on an average Rs 500 is combined with another firm employing 20 workers paying on an average Rs 600 . What is the average pay of the workers of the combined firm?

slug:
a-firm-employing-30-workers-and-paying-on-an-average-rs-500-is-combined-with-another-firm-employing-66658

question:
<div class="question">A firm employing 30 workers and paying on an average Rs 500 is combined with another firm employing 20 workers paying on an average Rs 600 . What is the average pay of the workers of the combined firm?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\operatorname{Rs} 540$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\mathrm{Rs} 550$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\operatorname{Rs} 560$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\operatorname{Rs} 580$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\operatorname{Rs} 540$</span> </div>

answer:
<div class="solution">Let $\mathrm{n}$ denote number of workers and $\mathrm{x}$, the pay. Let $\mathrm{n}$ denote number of workers ar<br/>$\mathrm{n}_{1}=30, \mathrm{n}_{2}=20, \mathrm{x}_{1}=â¹ 500, \mathrm{x}_{2}=â¹ 600$<br/>Combined average $=\frac{\mathrm{n}_{1} \mathrm{x}_{1}+\mathrm{n}_{2} \mathrm{x}_{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}$<br/>$=\frac{30 \times 500+20 \times 600}{30+20}=\frac{15000+12000}{50}$<br/>$=\frac{27000}{50}=540$<br/>Combined average pay $=â¹ 540$</div>


tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch0.db
title:
A five-digit number divisible by 3 is to be formed using the digits 0 , 1 , 2 , 3 and 4 without repetition of digits. What is the number of ways this can be done?

slug:
a-five-digit-number-divisible-by-3-is-to-be-formed-using-the-digits-0-1-2-3-and-4-without-repe-58842

question:
<div class="question">A five-digit number divisible by 3 is to be formed using the digits $0,1,2,3$ and 4 without repetition of digits. What is the number of ways this can be done?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">96</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">48</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">32</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Nonumber can be formed</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Nonumber can be formed</span> </div>

answer:
<div class="solution">Since sum of digits $=10$ (which is not divisible by 3 ) $\therefore$ No numbers can be formed.</div>


tags: ['Mathematics', 'Ellipse', 'TS EAMCET', 'TS EAMCET 2021 (04 Aug Shift 1)']
source_db: MarksBatch0.db
title:
A focus of an ellipse having eccentricity 1 2 is at 0 , 0 and a directrix is the line x = 4 . Then the equation of one such ellipse is

slug:
a-focus-of-an-ellipse-having-eccentricity-1-2-is-at-0-0-and-a-directrix-is-the-line-x-4-then-t-65298

question:
<div class="question">A focus of an ellipse having eccentricity <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> is at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> and a directrix is the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn><mo>.</mo></math> Then the equation of one such ellipse 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"><mfrac><mrow><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>64</mn></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><msup><mi>y</mi><mn>2</mn></msup></mrow><mn>16</mn></mfrac><mo>=</mo><mn>1</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>(</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>32</mn></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><mn>16</mn></mfrac><mo>=</mo><mn>1</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"><mfrac><mrow><mo>(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><msup><mo>)</mo><mn>2</mn></msup></mrow><mn>64</mn></mfrac><mo>+</mo><mfrac><msup><mi>y</mi><mn>2</mn></msup><mn>32</mn></mfrac><mo>=</mo><mn>1</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"><mo>(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>64</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"><mo>(</mo><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn><msup><mo>)</mo><mn>2</mn></msup><mo>+</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>64</mn></math></span> </div>

answer:
<div class="solution"><p>A focus of an ellipse is at origin.</p><p>Directrix is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> and eccentricity is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math></p><p>Major axis is along <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis ,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mi>a</mi><mi>e</mi></mfrac><mo>-</mo><mi>a</mi><mi>e</mi><mo>=</mo><mn>4</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mi>a</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mfrac><mo>-</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><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>8</mn><mn>3</mn></mfrac></math></p><p>By the help of options we get the idea that this case is of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>&gt;</mo><mi>b</mi></mrow></mfenced></math></p><p>Then we can say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>=</mo><msqrt><mn>1</mn><mo>-</mo><mfrac><msup><mi>b</mi><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac></msqrt></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mrow><mn>9</mn><msup><mi>b</mi><mn>2</mn></msup></mrow><mn>64</mn></mfrac><mspace linebreak="newline"></mspace><mo>⇒</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>16</mn><mn>3</mn></mfrac></math></p><p>By this time we can say that either the correct option is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>1</mn></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mn>4</mn></mfenced></math></p><p>The standard form of equation of an ellipse is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup></mfrac><mo>=</mo><mn>1</mn></math></p><p>where centre is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow></mfenced></math></p><p>The foci are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>-</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt><mo>,</mo><mi>k</mi></mrow></mfenced><mo> </mo><mo>&amp;</mo><mo> </mo><mfenced><mrow><mi>h</mi><mo>+</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt><mo>,</mo><mi>k</mi></mrow></mfenced></math></p><p>By this we can say that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>+</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt><mo>=</mo><mn>0</mn></math></p><p>Now putting all values, we get</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mo>-</mo><msqrt><mfrac><mn>64</mn><mn>9</mn></mfrac><mo>-</mo><mfrac><mn>16</mn><mn>3</mn></mfrac></msqrt><mspace linebreak="newline"></mspace><mi>h</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>4</mn></mrow><mn>3</mn></mfrac></math></p><p>And <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn></math></p><p>So by this we can say that the required equation of ellipse is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>12</mn><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>64</mn></math></p><p> </p></div>


slug: a-force-f-3-i-2-j-4-k-is-applied-at-the-point-1-1-2-what-is-the-moment-74079
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch0.db
title:
A force F = 3 i ^ + 2 j ^ â â 4 k ^ is applied at the point ( 1 , â 1 , 2 ) What is the moment of the force about the point ( 2 , â 1 , 3 ) ?

question:
<div class="question">A force $\overrightarrow{\mathrm{F}}=3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-4 \hat{\mathrm{k}}$ is applied at the point $(1,-1,2)$<br/>What is the moment of the force about the point $(2,-1,3)$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \hat{\mathrm{i}}-7 \hat{\mathrm{j}}-2 \hat{\mathrm{k}}$</span> </div>

answer:
<div class="solution">Let point $P$ is $(1,-1,2)$<br/>and point $Q$ is $(2,-1,3)$ $\Rightarrow$ Position vector of $P$ w.r.t. $Q$ is<br/>$\vec{r}=(1-2) \hat{i}+(-1+1) \hat{j}+(2-3) \hat{k}$<br/>$\Rightarrow \vec{r}=-\hat{i}+0 \hat{j}-\hat{k}$ and $\vec{F}=3 \hat{i}+2 \hat{j}-4 \hat{k}$<br/>$\Rightarrow$ Moment $=\vec{r} \times \vec{F}=\left|\begin{array}{ccc}\hat{i} &amp; \hat{j} &amp; \hat{k} \\ -1 &amp; 0 &amp; -1 \\ 3 &amp; 2 &amp; -4\end{array}\right|$<br/>$=\hat{i}(0+2)-\hat{j}(4+3)+\hat{k}(-2+0)=2 \hat{i}-7 \hat{j}-2 \hat{k}$</div>


slug: a-force-f-3-i-4-j-3-k-is-applied-at-the-point-p-whose-position-vector-is-r-2-i-74003
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch0.db
title:
A force F = 3 i ^ + 4 j ^ â â 3 k ^ is applied at the point P , whose position vector is r = 2 i â 2 j ^ â â 3 k ^ . What is the magnitude of the moment of the force about the origin?

question:
<div class="question">A force $\vec{F}=3 \hat{i}+4 \hat{j}-3 \hat{k}$ is applied at the point $\mathrm{P}$, whose<br/>position vector is $\vec{r}=\widehat{2} i-2 \hat{j}-3 \hat{k}$. What is the magnitude of the moment of the force about the origin?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">23 units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">19 units</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">18 units</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">21 units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">23 units</span> </div>

answer:
<div class="solution">Moment of force, $\mathrm{m}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{F}}$<br/>$\mathrm{m}=\left|\begin{array}{ccc}\hat{\mathrm{i}} &amp; \hat{\mathrm{j}} &amp; \hat{\mathrm{k}} \\ 2 &amp; -2 &amp; -3 \\ 3 &amp; 4 &amp; -3\end{array}\right|$<br/>$=\hat{\mathrm{i}}(6+12)-\hat{\mathrm{j}}(-6+9)+\hat{\mathrm{k}}(8+6)$<br/>$=18 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+14 \hat{\mathrm{k}}$<br/>$=\sqrt{(18)^{2}+(-3)^{2}+(14)^{2}}$<br/>$=\sqrt{529}=23$ units.</div>


slug: a-force-f-i-3-j-2-k-acts-on-a-particle-to-displace-it-from-the-point-a-i-2-j-84075
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch0.db
title:
A force F = i ^ + 3 j ^ â + 2 k ^ acts on a particle to displace it from the point A ( i ^ + 2 j ^ â â 3 k ^ ) to the point B ( 3 i ^ â j ^ â + 5 k ^ ) . The work done by the force will be

question:
<div class="question">A force $\overrightarrow{\mathrm{F}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ acts on a particle to displace it from<br/>the point $\mathrm{A}(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})$ to the point $\mathrm{B}(3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}})$. The<br/>work done by the force will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5 units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">7 units</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">9 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">D</span> <span class="option-data">10 units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">9 units</span> </div>

answer:
<div class="solution">$\overrightarrow{\mathrm{F}}=\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$<br/>$\overrightarrow{\mathrm{AB}}=(3-1) \hat{\mathrm{i}}+(-1-2) \hat{\mathrm{j}}+(5-(-3)) \hat{\mathrm{k}}$<br/>$=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+8 \hat{\mathrm{k}}$<br/>Work done $=\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{AB}}=(\hat{\mathrm{i}}+3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}) \cdot(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+8 \hat{\mathrm{k}})$<br/>$=(1 \times 2)+(3 \times(-3))+(2 \times 8)$<br/>$=2-9+16=9$ units.</div>


slug: a-force-m-i-3-j-k-acts-on-a-point-and-so-the-point-moves-from-20-3-m-0-to-77495
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2006 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: MarksBatch0.db
title:
A force m i ^ â 3 j ^ â + k ^ acts on a point and so the point moves from ( 20 , 3 m , 0 ) to ( 0 , 0 , 7 ) . If the work done by the force is â 48 unit, what is the value of m ?

question:
<div class="question">A force $m \hat{i}-3 \hat{j}+\hat{k}$ acts on a point and so the point moves<br/>from $(20,3 \mathrm{~m}, 0)$ to $(0,0,7)$. If the work done by the force is $-48$ unit, what is the value of $\mathrm{m}$ ?</div>

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

answer:
<div class="solution">Force, $\overrightarrow{\mathrm{F}}=\mathrm{m} \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+\hat{\mathrm{k}}$<br/>Due to this force, point moves from $\mathrm{A}(20,3 \mathrm{~m}, 0)$ to $\mathrm{B}(0,0,$,7). So, the displacement vector $\overrightarrow{\mathrm{AB}}$ is given by $\overrightarrow{\mathrm{AB}}=-20 \hat{\mathrm{i}}-3 \mathrm{~m} \hat{\mathrm{j}}+7 \hat{\mathrm{k}}$<br/>Work done $=\overrightarrow{\mathrm{F}} \cdot \overrightarrow{\mathrm{AB}}$<br/>$=(\mathrm{m} \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+\hat{\mathrm{k}}) \cdot(-20 \hat{\mathrm{i}}-3 \mathrm{~m} \hat{\mathrm{j}}+7 \hat{\mathrm{k}})$<br/>$=(-20 \mathrm{~m}+9 \mathrm{~m}+7)$ unit<br/>But work done $=-48$ unit, as given $\Rightarrow-11 \mathrm{~m}+7=-48$<br/>$\Rightarrow-11 \mathrm{~m}=-55$<br/>$\Rightarrow \mathrm{m}=5$</div>


tags: ['Biology', 'Ecosystem', 'NEET']
source_db: MarksBatch0.db
title:
A fowl consumes grains. What will we call the rate at which new organic matter is produced by this fowl?

slug:
a-fowl-consumes-grains-what-will-we-call-the-rate-at-which-new-organic-matter-is-produced-by-this-f-20481

question:
<div class="question">A fowl consumes grains. What will we call the rate at which new organic matter is produced by this fowl?</div>

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

answer:
<div class="solution">The amount of biomass or organic matter produced by plants by photosynthesis per unit area and over a period of time is called primary productivity. It is divided into gross and net primary productivity. The rate of production of organic matter by plants through photosynthesis is called gross primary productivity and the net primary productivity is the availability of biomass for consumption by heterotrophs. The rate at which the new organic matter is produced by consumers is called secondary productivity. Here fowl or chicken is a consumer.</div>


tags: ['Physics', 'Wave Optics', 'AP EAMCET', 'AP EAMCET 2020 (18 Sep Shift 2)']
source_db: MarksBatch0.db
title:
A Fraunhofer diffraction pattern due to a narrow slit is obtained on a screen placed at a distance \(D\) from the slit whose slit width is \(a\). The distance of first secondary maximum from the central maximum is

slug:
a-fraunhofer-diffraction-pattern-due-to-a-narrow-slit-is-obtained-on-a-screen-placed-at-a-distance-45515

question:
<div class="question">A Fraunhofer diffraction pattern due to a narrow slit is obtained on a screen placed at a distance \(D\) from the slit whose slit width is \(a\). The distance of first secondary maximum from the central maximum is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{3 D \lambda}{2 a}\)</span> </div>

answer:
<div class="solution">In Fraunhofer diffraction pattern, the direction of secondary maximum is given as<br/>\(\begin{aligned}<br/>\theta &amp; =(2 n+1) \frac{\lambda}{2 a}=(2 \times 1+1) \frac{\lambda}{2 a} \\<br/>\Rightarrow \theta &amp; =\frac{3 \lambda}{2 a}<br/>\end{aligned}\)<br/>\(\therefore\) Distance of first secondary maximum from the central maximum is given by<br/>\(x=\theta D=\frac{3 \lambda}{2 a} \cdot D=\frac{3 D \lambda}{2 a}\)</div>


slug: a-function-f-a-r-is-defined-by-the-equation-f-x-x-2-4-x-5-where-a-1-4-wh-85491
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch0.db
title:
A function f : A â R is defined by the equation f ( x ) = x 2 â 4 x + 5 where A = ( 1 , 4 ) . What is the range of the function?

question:
<div class="question">A function $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{R}$ is defined by the equation $\mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-4 \mathrm{x}+5$<br/>where $\mathrm{A}=(1,4)$. What is the range of the function?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$[1,5)$</span> </div>

answer:
<div class="solution">$\begin{aligned} f(x) &amp;=x^{2}-4 x+5=(x-2)^{2}+1 \\ f(1) &amp;=2, f(4)=5 \end{aligned}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/2E-vdG06ksEwJCocguGhkQMFkbjTJXajS331u7O3Krs.original.fullsize.png"/><br/><br/>$\therefore$ Range $f(x)=[1,5)$</div>


title: A function f defined by f ( x ) = ln ( x 2 + 1 â â x ) is
slug: a-function-f-defined-by-f-x-ln-x-2-1-x-is-65383
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch0.db
question:
<div class="question">A function $\mathrm{f}$ defined by $\mathrm{f}(\mathrm{x})=\ln \left(\sqrt{x^{2}+1}-x\right)$ is</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; f(x)=\ln \left(\sqrt{x^{2}+1}-x\right) \\ &amp; f(-x)=\ln \left(\sqrt{(-x)^{2}+1}-(-x)\right)=\ln \left(\sqrt{x^{2}+1}+x\right) \\ &amp;=\ln \left(\frac{\left(\sqrt{x^{2}+1}+x\right)\left(\sqrt{x^{2}+1}-x\right)}{\sqrt{x^{2}+1}-x}\right) \\ &amp;=\ln \left(\frac{x^{2}+1-x^{2}}{\sqrt{x^{2}+1}-x}\right)=\ln \left(\frac{1}{\sqrt{x^{2}+1}-x}\right) \\ &amp;=-\ln \left(\sqrt{x^{2}+1}-x\right)=-f(x) \end{aligned}$<br/>So, $\mathrm{f}(\mathrm{x})$ is odd function.</div>


slug: a-function-f-is-defined-as-follows-f-x-x-p-cos-x-1-x-0-f-0-0-what-condit-49917
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch0.db
title:
A function f is defined as follows f ( x ) = x p cos ( x 1 â ) , x î  = 0 f ( 0 ) = 0 What conditions should be imposed on p so that f may be continuous at x = 0 ?

question:
<div class="question">A function $\mathrm{f}$ is defined as follows<br/>$\mathrm{f}(\mathrm{x})=\mathrm{x}^{\mathrm{p}} \cos \left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \neq 0$<br/>$f(0)=0$<br/>What conditions should be imposed on p so that $\mathrm{f}$ may be continuous at $x=0$ ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{p}=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{p}&gt;0$</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">C</span> <span class="option-data">$\mathrm{p}   &lt; 0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">No value of p</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{p}&gt;0$</span> </div>

answer:
<div class="solution">Given function is defined as:<br/>$f(x)=\left\{\begin{array}{ll}x^{p} \cos \left(\frac{1}{x}\right) &amp; x \neq 0 \\ 0, &amp; x=0\end{array}\right.$<br/>For continuity:<br/>LHS $: \lim _{x \rightarrow 0} f(x)=$ RHS $\lim _{x \rightarrow 0} f(x)=f(0)$<br/>$\Rightarrow \lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0} x^{p} \cos \left(\frac{1}{x}\right)=0$<br/>$\Rightarrow \lim _{x \rightarrow 0} x^{p} \cos \left(\frac{1}{x}\right)=0$<br/>$\cos \left(\frac{1}{\mathrm{x}}\right)$ is always a finite quantity if $\mathrm{x} \rightarrow 0$<br/>$\Rightarrow \mathrm{x}^{\mathrm{p}}=0$<br/>which is possible only if $\mathrm{p}&gt;0$.</div>


slug: a-function-f-is-defined-by-f-x-x-x-1-consider-the-following-1-f-x-2-f-x-24188
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch0.db
title:
A function f is defined by f ( x ) = x + x 1 â . Consider the following. (1) ( f ( x ) ) 2 = f ( x 2 ) + 2 (2) ( f ( x ) ) 3 = f ( x 3 ) + 3 f ( x ) Which of the above is/are correct?

question:
<div class="question">A function $f$ is defined by $f(x)=x+\frac{1}{x}$. Consider the<br/>following. <br/>(1) $\quad(f(x))^{2}=f\left(x^{2}\right)+2$<br/>(2) $(f(x))^{3}=f\left(x^{3}\right)+3 f(x)$<br/>Which of the above is/are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1 only</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Both 1 and 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">Neither 1 nor 2</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Both 1 and 2</span> </div>

answer:
<div class="solution">$f\left(x^{2}\right)+2=x^{2}+\frac{1}{x^{2}}+2$<br/>$=\left(x+\frac{1}{x}\right)^{2}=\{f(x)\}^{2}$<br/>and $f\left(x^{3}\right)+3 f(x)$<br/>$=x^{3}+\frac{1}{x^{3}}+3\left(x+\frac{1}{x}\right)=\left(x+\frac{1}{x}\right)^{3}=\{f(x)\}^{3}$<br/>Thus, both 1 and 2 are correct.</div>


title: A function f is such that f â² ( x ) = 6 â 4 sin 2 x and f ( 0 ) = 3 . What is f ( x ) equal to?
slug: a-function-f-is-such-that-f-x-6-4-sin-2-x-and-f-0-3-what-is-f-x-equal-to-82089
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch0.db
question:
<div class="question">A function $f$ is such that $f^{\prime}(x)=6-4 \sin 2 x$ and $f(0)=3$. What is $f(x)$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$6 x+2 \cos 2 x+1$</span> </div>

answer:
<div class="solution">Given,<br/>$f^{\prime}(x)=6-4 \sin 2 x$ and $f(0)=3$<br/>Consider $f^{\prime}(x)=6-4 \sin 2 x$<br/>Integrate both sides w.r.t $x$<br/>$\int f^{\prime}(x) d x=\int(6-4 \sin 2 x) d x$<br/>$f(x)=6 x-\frac{4(-\cos 2 x)}{2}+c$<br/>Where ' $c$ ' is constant of integration $\Rightarrow f(x)=6 x+2 \cos 2 x+c$<br/>By using $f(0)=3$, we have $3=f(0)=6.0+2 \cos 0+c$<br/>$\Rightarrow 3=2+c \Rightarrow c=1$<br/>Hence, $f(x)=6 x+2 \cos 2 x+1$</div>


slug: a-function-f-r-r-is-defined-as-f-x-x-2-for-x-0-and-f-x-x-for-x-0-consider-10605
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch0.db
title:
A function f : R â R is defined as f ( x ) = x 2 for x â¥ 0 and f ( x ) = â x for x 0. Consider the following statements in respect of the above function: 1- The function is continuous at x = 0 . 2- The function is differentiable at x = 0 . Which of the above statements is/are correct?

question:
<div class="question">A function $f: R \rightarrow R$ is defined as $f(x)=x^{2}$ for $x \geq 0$ and $f(x)=-x$ for $x   &lt; 0 .$<br/>Consider the following statements in respect of the above function:<br/>1- The function is continuous at $x=0$.<br/>2- The function is differentiable at $x=0$.<br/>Which of the above statements is/are correct?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">1 only</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2 only</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Both 1 and 2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Neither 1 nor 2</span> </li> </ul>

answer:
<div class="solution">$\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}, \mathrm{f}(\mathrm{x})=\left\{\begin{array}{ll}\mathrm{x}^{2} &amp; , \quad \mathrm{x} \geq 0 \\ -\mathrm{x} &amp; , \quad \mathrm{x}   &lt; 0\end{array}\right.$<br/>For continuity at $\mathrm{x}=0$ $\mathrm{f}(0-0)=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}(0-\mathrm{h})$<br/>$=\lim _{\mathrm{h} \rightarrow 0}[(0-\mathrm{h})]=\lim _{\mathrm{h} \rightarrow 0} \mathrm{~h}=0$<br/>$\mathrm{f}(0+0)=\lim _{\mathrm{h} \rightarrow 0} \mathrm{f}(0+\mathrm{h})=\lim _{\mathrm{h} \rightarrow 0}(0+\mathrm{h})^{2}=0$ and $\mathrm{f}(0)=0$<br/>$\mathrm{f}(\mathrm{x})$ is continuous at $\mathrm{x}=0$<br/>For differentiability at $\mathrm{x}=0$ $\lim _{h \rightarrow 0} \frac{-(-h)-0}{-h}=\lim _{h \rightarrow 0}=\frac{h}{-h}=-1$<br/>and $\lim _{h \rightarrow 0} \frac{f(0+h)-f(0)}{h}=\lim _{h \rightarrow 0} h=0$<br/>$\mathrm{f}(\mathrm{x})$ is not differentiable at $\mathrm{x}-0$</div>


slug: a-function-is-defined-as-follows-f-x-x-2-x-0-x-0-x-0-which-59098
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch0.db
title:
A function is defined as follows: f ( x ) = { â x 2 â x â 0 , â , x î  = 0 x = 0 â Which one of the following is correct in respect of the above function?

question:
<div class="question">A function is defined as follows: <br/>$f(\mathrm{x})=\left\{\begin{array}{cl}-\frac{\mathrm{x}}{\sqrt{\mathrm{x}^{2}}} &amp; , \mathrm{x} \neq 0 \\ 0, &amp; \mathrm{x}=0\end{array}\right.$<br/>Which one of the following is correct in respect of the above function?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f(\mathrm{x})$ is discontinuous at $\mathrm{x}=0$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}\frac{-\mathrm{x}}{\sqrt{\mathrm{x}^{2}}}, \mathrm{x} \neq 0 \\ 0, \mathrm{x}=0\end{array}\right.$<br/>$=\left\{\begin{array}{l}\frac{-\mathrm{x}}{|\mathrm{x}|}, \mathrm{x} \neq 0 \\ 0, \mathrm{x}=0\end{array}\right.$<br/>$\mathrm{f}(0+\mathrm{h})=\frac{-\mathrm{h}}{\mathrm{h}}=-1 ; \mathrm{f}(0-\mathrm{h})=\frac{\mathrm{h}}{\mathrm{h}}=1$<br/>So, $\mathrm{f}(\mathrm{x})$ is discontinuous at $\mathrm{x}=0$.</div>


slug: a-function-is-defined-in-0-by-2017-ii-f-x-1-x-2-in-in-2-23260
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch0.db
title:
A function is defined in ( 0 , â ) by [ 2017 â II ] f ( x ) = â© â¨ â§ â 1 â x 2 In In 2 â 1 + 0.5 x â for x for â 0 x â¤ 1 for 1 x â¤ 2 2 x â â Which one of the following is correct in respect of the derivative of the function, i.e., f â² ( x )?

question:
<div class="question">A function is defined in $(0, \infty)$ by $\quad[2017-I I]$ $f(x)=\left\{\begin{array}{ccc}1-x^{2} &amp; \text { for } &amp; 0   &lt; x \leq 1 \\ \operatorname{In} &amp; x &amp; \text { for } 1 &lt; x \leq 2 \\ \text { In } 2-1+0.5 x &amp; \text { for } &amp; 2 &lt; x &lt; \infty\end{array}\right.$<br/>Which one of the following is correct in respect of the derivative of the function, i.e., $f^{\prime}(\mathrm{x}) ?$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$f^{\prime}(x)=-2 x$ for $0   &lt; x &lt; 1$</span> </div>

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}1-\mathrm{x}^{2} \text { for } 0   &lt; \mathrm{x} \leq 1 \\ \ln \mathrm{x} \text { for } 1 &lt; \mathrm{x} \leq 2 \\ \ell \mathrm{n} 2-1+0.5 \mathrm{x} \text { for } 2 &lt; \mathrm{x} &lt; \infty\end{array}\right.$<br/>$\mathrm{f}^{\prime}(\mathrm{x})=-2 \mathrm{x}$ for $0 &lt; \mathrm{x} &lt; 1$</div>


tags: ['Biology', 'Biological Classification', 'NEET']
source_db: MarksBatch0.db
title:
A fungus with hyphae containing nuclei from different genomes, the nuclei do not fuse and are without sex organ but produces sexual spore belongs to

slug:
a-fungus-with-hyphae-containing-nuclei-from-different-genomes-the-nuclei-do-not-fuse-and-are-withou-45033

question:
<div class="question">A fungus with hyphae containing nuclei from different genomes, the nuclei do not fuse and are without sex organ but produces sexual spore belongs to</div>

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

answer:
<div class="solution">The intermediate phase between two stages (plasmogamy and karyogamy) is known as dikaryophase.<br/><br/>It gives rise to secondary mycelium. It has multiple septate hyphae. The septa have spores and hook shaped outgrowths. They are known as clamp connections. Septa mycelia multiply by formation of many spores including uredospores and chlamydospores. In basidia, karyogamy and meiosis occur which leads to the production of basidiospores.</div>


slug: a-galvanic-cell-is-set-up-from-electrodes-a-and-b-electrode-a-cr-2-o-7-2-cr-3-e-90068
tags: ['Chemistry', 'Electrochemistry', 'VITEEE', 'VITEEE 2021']
answer: <div class="solution">No Solution Available</div>
source_db: MarksBatch0.db
title:
A galvanic cell is set up from electrodes A and B Electrode A : Cr 2 â O 7 2 â â / Cr 3 + , E red â â = + 1.33 V Electrode B : Fe 3 + / Fe 2 + , E red â â = 0.77 V Which of the following statements is false ?

question:
<div class="question">A galvanic cell is set up from electrodes $A$ and $B$ Electrode $A: \mathrm{Cr}_2 \mathrm{O}_7^{2-} / \mathrm{Cr}^{3+}, E_{\text {red }}^{\circ}=+1.33 \mathrm{~V}$ Electrode $B: \mathrm{Fe}^{3+} / \mathrm{Fe}^{2+}, \quad E_{\text {red }}^{\circ}=0.77 \mathrm{~V}$ Which of the following statements is false ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Standard e.m.f of the cell is $0.56 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Current will flow from electrode $A$ to $B$ in the external circuit</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$A$ will act as cathode and have positive polarity</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of these</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">None of these</span> </div>


title: A gas deviates from ideal behaviour at a high pressure because its molecules
slug: a-gas-deviates-from-ideal-behaviour-at-a-high-pressure-because-its-molecules-94174
tags: ['Chemistry', 'States of Matter', 'KCET', 'KCET 2008']
source_db: MarksBatch0.db
question:
<div class="question">A gas deviates from ideal behaviour at a high pressure because its molecules</div>

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

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

answer:
<div class="solution">At high pressure, the volume is decreased appreciably, so the attractive forces become large and the molecules are crowded together. Thus, pressure correction is necessary and the gas deviates more from ideal behaviour.</div>


title: A gas deviates from ideal behaviour at a high pressure because its molecules
slug: a-gas-deviates-from-ideal-behaviour-at-a-high-pressure-because-its-molecules-27982
tags: ['Chemistry', 'States of Matter', 'COMEDK', 'COMEDK 2017']
source_db: MarksBatch0.db
question:
<div class="question">A gas deviates from ideal behaviour at a high pressure because its molecules</div>

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

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

answer:
<div class="solution">A gas deviates from ideal behaviour at high pressure because its molecules attract one another as their intermolecular distance decreases.</div>


tags: ['Biology', 'Biological Classification', 'NEET', 'NEET 2007']
source_db: MarksBatch0.db
title:
A genetically engineered micro-organism used successfully in bioremediation of oil spills is a species of:

slug:
a-genetically-engineered-micro-organism-used-successfully-in-bioremediation-of-oil-spills-is-a-speci-47471

question:
<div class="question">A genetically engineered micro-organism used successfully in bioremediation of oil spills is a species of:</div>

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

answer:
<div class="solution">A genetically engineered microorganism called Pseudomonas putida is used in thebioremediation of oil spills. It is also known as super bug and was discovered by Ananda Mohan Chakrabarty. Cyclosporin is obtained from Trichoderma polysporum which is used as immunosuppressive agent. Xanthomonas is a disease causing gram negative bacteria.<br/>Related Theory<br/>Bioremediation is a biotechnical process which involves the removal of complex material or contamination by degrading environmental pollutants using living microorganisms. The ability of micro-organisms could be enhanced by providing optimum growth conditions to micro-organism or by employing the genetically modified microorganisms (GMMs). This technology has been implemented to eradicate environmentally hazardous chemicals and detoxify them into nontoxic forms.<br/>Caution<br/>Bioaugmentation involves the injection of a small amount of oil-degrading microbes into an affected area. Biostimulation is the addition of nutrients to stimulate the growth of innate oil-degrading microbes to increase the rate of remediation.</div>


tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch0.db
title:
A given quantity of metal is to be cast into a half cylinder (i,e, with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends is

slug:
a-given-quantity-of-metal-is-to-be-cast-into-a-half-cylinder-i-e-with-a-rectangular-base-and-semic-85816

question:
<div class="question">A given quantity of metal is to be cast into a half cylinder (i,e, with a rectangular base and semicircular ends). If the total surface area is to be minimum, then the ratio of the height of the half cylinder to the diameter of the semicircular ends is</div>

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

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

answer:
<div class="solution">Volume of half cylinder with height ${ }^{\prime} 1^{\prime}$ and radius ' $\mathrm{r}$ '<br/>$=\frac{1}{2} \pi r^{2} l$<br/>Surface area of half cylinder.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/GXBavCo3HyN3l9KsZ5k9bWrT1UlXsDKmF9ux5G-CBaM.original.fullsize.png"/><br/><br/>$S=l \times 2 r+2\left(\frac{1}{2} \pi r^{2}\right)+2\left(\frac{1}{2} \pi r l\right)$<br/>$S=2 r l+\pi r^{2}+\pi r l$<br/>$S=\frac{4 V}{\pi r}+\frac{\pi \cdot 2 V}{\pi r}+\pi r^{2} \quad($ from $(1))$<br/>$\frac{d s}{d r} \Rightarrow \frac{-4 V}{\pi r^{2}}-\frac{2 V}{r^{2}}+2 \pi r=0$<br/>$\Rightarrow 2 \pi r=\frac{4 V}{\pi r^{2}}+\frac{2 V}{r^{2}}$<br/>$\Rightarrow 2 \pi r=\frac{4 . \pi r^{2} l}{2 \pi r^{2}}+\frac{2}{r^{2}} \cdot \frac{1}{2} \pi r^{2} l$<br/>$\Rightarrow \frac{l}{2 r}=\frac{\pi}{\pi+2}$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch0.db
title:
A glass tube is open at both the ends. A tuning fork of frequency f resonates with the air column inside the tube. Now the tube is placed vertically inside water so that half the length of the tube is filled with water. Now the air column inside the tube is in unison with another fork of frequency f â² . Then

slug:
a-glass-tube-is-open-at-both-the-ends-a-tuning-fork-of-frequency-f-resonates-with-the-air-column-in-11830

question:
<div class="question">A glass tube is open at both the ends. A tuning fork of frequency $f$ resonates with the air column inside the tube. Now the tube is placed vertically inside water so that half the length of the tube is filled with water. Now the air column inside the tube is in unison with another fork of frequency $f^{\prime}$. Then</div>

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

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

answer:
<div class="solution">For fundamental frequency of vibration of air column<br/>$$<br/>l=\frac{\lambda}{2} \Rightarrow \lambda=2 l<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/I7g4mxQi6yu9o20YKdj1gqtH7fDccidkmHhB3vEMndU.original.fullsize.png"/><br/><br/>Frequency, $f=\frac{v}{\lambda}$<br/>$\Rightarrow \quad f=\frac{v}{2 l}$<br/>When tube is placed inside water with half length immersed in water then<br/>Frequency, $f^{\prime}=\frac{v}{\lambda^{\prime}}=\frac{v}{2 l}$<br/>From Eqs. (i) and (ii), we get<br/>$$<br/>f^{\prime}=f<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'KCET', 'KCET 2007']
source_db: MarksBatch0.db
title:
A glass tube is open at both the ends. A tuning fork of frequency f resonates with the air column inside the tube. Now the tube is placed vertically inside water so that half the length of the tube is filled with water. Now the air column inside the tube is in unison with another fork of frequency f â² . Then

slug:
a-glass-tube-is-open-at-both-the-ends-a-tuning-fork-of-frequency-f-resonates-with-the-air-column-in-30999

question:
<div class="question">A glass tube is open at both the ends. A tuning fork of frequency $f$ resonates with the air column inside the tube. Now the tube is placed vertically inside water so that half the length of the tube is filled with water. Now the air column inside the tube is in unison with another fork of frequency $f^{\prime}$. Then</div>

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

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

answer:
<div class="solution">For fundamental frequency of vibration of air column<br/>$$<br/>l=\frac{\lambda}{2} \Rightarrow \lambda=2 l<br/>$$<br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/kcet/I7g4mxQi6yu9o20YKdj1gqtH7fDccidkmHhB3vEMndU.original.fullsize.png"/><br/><br/>Frequency, $f=\frac{v}{\lambda}$<br/>$\Rightarrow \quad f=\frac{v}{2 l}$<br/>When tube is placed inside water with half length immersed in water then<br/>Frequency, $f^{\prime}=\frac{v}{\lambda^{\prime}}=\frac{v}{2 l}$<br/>From Eqs. (i) and (ii), we get<br/>$$<br/>f^{\prime}=f<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2018 (08 Apr)']
source_db: MarksBatch0.db
title:
A granite rod of 60 cm length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is 2.7 Ã 10 3 kg m - 3 and its Young's modulus is 9.27 Ã 10 10 Pa . What will be the fundamental frequency of the longitudinal vibrations?

slug:
a-granite-rod-of-60-cm-length-is-clamped-at-its-middle-point-and-is-set-into-longitudinal-vibrations-81164

question:
<div class="question">A granite rod of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><mi>cm</mi></math> length is clamped at its middle point and is set into longitudinal vibrations. The density of granite is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2.7</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mi> </mi><mi>kg</mi><mo> </mo><msup><mi mathvariant="normal">m</mi><mrow><mo>-</mo><mn>3</mn></mrow></msup></math> and its Young's modulus is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9.27</mn><mo>×</mo><msup><mn>10</mn><mn>10</mn></msup><mi> </mi><mi>Pa</mi></math>. What will be the fundamental frequency of the longitudinal vibrations?</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>7</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>kHz</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>5</mn><mo> </mo><mi>kHz</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>2</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>kHz</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>10</mn><mo> </mo><mi>kHz</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>5</mn><mo> </mo><mi>kHz</mi></math></span> </div>

answer:
<div class="solution"><img alt="" src="https://cdn-question-pool.getmarks.app/pyq/jee_main/bbbi.JPG" style="width: 250px; height: 134px;"/><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>=</mo><mfrac><mo>⋋</mo><mn>2</mn></mfrac><mo>⇒</mo><mo>⋋</mo><mo>=</mo><mn>2</mn><mi>L</mi><mo>=</mo><mn>1.2</mn><mo> </mo><mi mathvariant="normal">m</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msqrt><mfrac><mi>y</mi><mi>ρ</mi></mfrac></msqrt><mo>=</mo><mn>5.86</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mi> </mi><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mo> </mo><mi>Fundamental</mi><mo> </mo><mo>frequency</mo><mo>=</mo><mfrac><mi>v</mi><mo>⋋</mo></mfrac><mo>=</mo><mn>4.88</mn><mi> </mi><mi>kHz</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mn>5</mn><mo> </mo><mi>kHz</mi></math></div>


tags: ['Physics', 'Units and Dimensions']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
A great physicist of this century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics ( c , e , mass of electron, mass of proton) and the gravitational constant G , he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe ( â¼ 15 /billion years). From the table of fundamental constants in this book, try to see if you too can construct this number (or any other interesting number you can think of ). If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants?

slug:
a-great-physicist-of-this-century-p-a-m-dirac-loved-playing-with-numerical-values-of-fundamental-60890

question:
<div class="question">A great physicist of this century (P.A.M. Dirac) loved playing with numerical values of Fundamental constants of nature. This led him to an interesting observation. Dirac found that from the basic constants of atomic physics $(c, e$, mass of electron, mass of proton) and the gravitational constant $\mathrm{G}$, he could arrive at a number with the dimension of time. Further, it was a very large number, its magnitude being close to the present estimate on the age of the universe ( $\sim 15$ /billion years). From the table of fundamental constants in this book, try to see if you too can construct this number (or any other interesting number you can think of ). If its coincidence with the age of the universe were significant, what would this imply for the constancy of fundamental constants?</div>

answer:
<div class="solution">Using basic constants such as speed of light (c), charge on electron $(e)$, mass of electron $\left(\mathrm{m}_e\right)$, mass of proton $\left(\mathrm{m}_v\right)$ and gravitational constant $(\mathrm{G})$, we can construct the quantity,<br/>$$<br/>\tau=\left(\frac{e^2}{4 \pi \varepsilon_0}\right)^2 \times \frac{1}{m_p m_e^2 c^3 \mathrm{G}}<br/>$$<br/>$$<br/>\begin{aligned}<br/>\text { Now }\left[\frac{e^2}{4 \pi \varepsilon_0}\right] &amp;=\left[\frac{1}{4 \pi \varepsilon_0} \frac{e^2}{r^2} r^2\right]=\left[\mathrm{Fr}^2\right] \\<br/>&amp;=\left[\mathrm{MLT}^{-2} \cdot \mathrm{L}^2\right]=\left[\mathrm{ML}^3 \mathrm{~T}^{-2}\right] \\<br/>\therefore[\tau] &amp;=\frac{\left[\mathrm{ML}^3 \mathrm{~T}^{-2}\right]^2}{[\mathrm{M}][\mathrm{M}]^2\left[\mathrm{LT}^{-1}\right]^3\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]}=[\mathrm{T}]<br/>\end{aligned}<br/>$$<br/>Clearly, the quantity $\tau$ has the dimensions of time.<br/>Put $\mathrm{G}=6.67 \times 10^{-11} \mathrm{Nm}^2 \mathrm{~kg}^{-2}$,<br/>$$<br/>\begin{aligned}<br/>&amp;c=3 \times 10^8 \mathrm{~m} / \mathrm{s} \\<br/>&amp;e=1.6 \times 10^{-19} \mathrm{C}, \mathrm{m}_{\mathrm{e}}=9.1 \times 10^{-31} \mathrm{~kg} \\<br/>&amp;\mathrm{~m}=1.67 \times 10^{-27} \mathrm{~kg} \\<br/>&amp;\text { and } \frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^2 \\<br/>&amp;\therefore \tau=\frac{\left[9 \times 10^9 \times\left(1.6 \times 10^{-19}\right)^2\right]^2}{1.67 \times 10^{-27} \times\left(9.1 \times 10^{-31}\right)^2 \times\left(3 \times 10^8\right)^3 \times 6.67 \times 10^{-11}}<br/>\end{aligned}<br/>$$<br/>$$<br/>\begin{aligned}<br/>&amp;=2.13 \times 10^{16} \mathrm{~s} \\<br/>&amp;=\frac{2.13 \times 10^{16}}{3.156 \times 10^7} \text { years }=0.667 \times 10^9 \text { years. } \\<br/>&amp;=0.667 \text { billion years. }<br/>\end{aligned}<br/>$$<br/>This time is slightly less than the age of the universe ( $\approx 15$ billion years). It implies that the values of the basic constants of physics should change with time because the age of the universe increases with time.</div>


tags: ['Mathematics', 'Permutation Combination', 'NDA', 'NDA 2008 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 or 3</span> </div>
source_db: MarksBatch0.db
title:
A group consists of 5 men and 5 women. If the number of different five-person committees containing k men and ( 5 â k ) women is 100, what is the value of k ?

slug:
a-group-consists-of-5-men-and-5-women-if-the-number-of-different-five-person-committees-containing-84375

question:
<div class="question">A group consists of 5 men and 5 women. If the number of different five-person committees containing $\mathrm{k}$ men and $(5-\mathrm{k})$ women is 100, what is the value of $\mathrm{k}$ ?</div>

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

answer:
<div class="solution">$\mathrm{K}$ men selected out of 5 and $5-\mathrm{k}$ women out of 5 . These are ${ }^{5} \mathrm{C}_{\mathrm{k}}$ and ${ }^{5} \mathrm{C}_{5-\mathrm{k}}$<br/>According to problem:<br/>${ }^{5} \mathrm{C}_{\mathrm{k}} \times{ }^{5} \mathrm{C}_{5-\mathrm{k}}=100$<br/>$\Rightarrow \frac{5 !}{\mathrm{k} !(5-\mathrm{k}) !} \times \frac{5 !}{(5-\mathrm{k}) ! 5 !}=100$<br/>$\Rightarrow\left(\frac{5}{k !(5-k) !}\right)^{2}=100$<br/>$\Rightarrow \frac{5 !}{\mathrm{k} !(5-\mathrm{k}) !}=10$<br/>This is true for $\mathrm{k}=2$ or 3 .</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2023 (08 Apr Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">60</span> </div>
source_db: MarksBatch0.db
title:
A guitar string of length 90 cm vibrates with a fundamental frequency of 120 Hz . The length of the string producing a fundamental of 180 Hz will be _____ cm

slug:
a-guitar-string-of-length-90-cm-vibrates-with-a-fundamental-frequency-of-120-hz-the-length-of-the-63775

question:
<div class="question">A guitar string of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo> </mo><mi>cm</mi></math> vibrates with a fundamental frequency of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>120</mn><mo> </mo><mi>Hz</mi></math>. The length of the string producing a fundamental of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo> </mo><mi>Hz</mi></math> will be _____ <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cm</mi></math></div>

answer:
<div class="solution"><blockquote><p>The formula to calculate the fundamental frequency produced in a vibrating string is given by</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>0</mn></msub><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mi>L</mi></mrow></mfrac><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>1</mn></mfenced></math></p><p>For the first case, it can be written that</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>120</mn><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mo>(</mo><mn>90</mn><mo>)</mo></mrow></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>2</mn></mfenced></math></p><p>Similarly, for the second case, it can be written that</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo>=</mo><mfrac><mi>v</mi><mrow><mn>2</mn><mo>(</mo><mi>L</mi><mo>)</mo></mrow></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo>.</mo><mo>.</mo><mo>.</mo><mfenced><mn>3</mn></mfenced></math></p><p>Divide equation (3) by equation (2) and solve to obtain the required length of the string.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="right center left" columnspacing="0px"><mtr><mtd><mfrac><mn>180</mn><mn>120</mn></mfrac></mtd><mtd><mo>=</mo></mtd><mtd><mfrac><mstyle displaystyle="true"><mfrac><mi>v</mi><mrow><mn>2</mn><mfenced><mi>L</mi></mfenced></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><mi>v</mi><mrow><mn>2</mn><mfenced><mn>90</mn></mfenced></mrow></mfrac></mstyle></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd><mo> </mo><mfrac><mn>90</mn><mi>L</mi></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo></mtd><mtd><mi>L</mi><mo>=</mo><mo> </mo><mn>90</mn><mo>×</mo><mfrac><mn>120</mn><mn>180</mn></mfrac></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd><mn>60</mn></mtd></mtr></mtable></math> </p></blockquote></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch0.db
title:
A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 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 N and the speed of sound is 320 ms â 1 , the mass of the string is

slug:
a-hollow-pipe-of-length-0-8-m-is-closed-at-one-end-at-its-open-end-a-0-5-m-long-uniform-string-is-v-81941

question:
<div class="question">A hollow pipe of length $0.8 \mathrm{~m}$ is closed at one end. 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 $320 \mathrm{~ms}^{-1}$, the mass of the string is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$5 \mathrm{~g}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$10 \mathrm{~g}$<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/>$20 \mathrm{~g}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$40 \mathrm{~g}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$10 \mathrm{~g}$<br/></span> </div>

answer:
<div class="solution">$2\left(\frac{v_1}{2 l_1}\right)=\frac{v_2}{4 l_2}$ $\therefore \quad \frac{\sqrt{T / \mu}}{l_1}=\frac{320}{4 l_2}$<br/>( $\mu=$ mass per unit length of wire)<br/>or $\frac{\sqrt{50 / \mu}}{0.5}=\frac{320}{4 \times 0.8}$<br/>Solving we get $\mu=0.02 \frac{\mathrm{kg}}{\mathrm{m}}=20 \frac{\mathrm{g}}{\mathrm{m}}$<br/>$\therefore$ Mass of string<br/>$$<br/>=\left(20 \frac{\mathrm{g}}{\mathrm{m}}\right)(0.5 \mathrm{~m})=10 \mathrm{~g}<br/>$$<br/>$\therefore$ The correct option is (b).</div>


tags: ['Physics', 'Waves and Sound', 'JEE Advanced', 'JEE Advanced 2010 (Paper 2)']
source_db: MarksBatch0.db
title:
A hollow pipe of length 0.8 m is closed at one end. At its open end a 0.5 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 N and the speed of sound is 320 ms â 1 , the mass of the string is

slug:
a-hollow-pipe-of-length-0-8-m-is-closed-at-one-end-at-its-open-end-a-0-5-m-long-uniform-string-is-v-22916

question:
<div class="question">A hollow pipe of length $0.8 \mathrm{~m}$ is closed at one end. 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 $320 \mathrm{~ms}^{-1}$, the mass of the string is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>$5 \mathrm{~g}$<br/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><br/>$10 \mathrm{~g}$<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/>$20 \mathrm{~g}$<br/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><br/>$40 \mathrm{~g}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><br/>$10 \mathrm{~g}$<br/></span> </div>

answer:
<div class="solution">$2\left(\frac{v_1}{2 l_1}\right)=\frac{v_2}{4 l_2}$ $\therefore \quad \frac{\sqrt{T / \mu}}{l_1}=\frac{320}{4 l_2}$<br/>( $\mu=$ mass per unit length of wire)<br/>or $\frac{\sqrt{50 / \mu}}{0.5}=\frac{320}{4 \times 0.8}$<br/>Solving we get $\mu=0.02 \frac{\mathrm{kg}}{\mathrm{m}}=20 \frac{\mathrm{g}}{\mathrm{m}}$<br/>$\therefore$ Mass of string<br/>$$<br/>=\left(20 \frac{\mathrm{g}}{\mathrm{m}}\right)(0.5 \mathrm{~m})=10 \mathrm{~g}<br/>$$<br/>$\therefore$ The correct option is (b).</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2022 (10 Aug Shift 2)']
source_db: MarksBatch0.db
title:
A hollow pipe of length 0.8 m is closed at one end. At its open end, a 0.8 m long uniform string is vibrating in its second harmonic and it resonates with the fundamental frequency of pipe. If the tension in the sting is 50 N and speed of sound in air is 320 m / s , the mass of the strings is

slug:
a-hollow-pipe-of-length-0-8-m-is-closed-at-one-end-at-its-open-end-a-0-8-m-long-uniform-string-is-75656

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

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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><li class=""> <span class="option-label">B</span> <span class="option-data">$20 \mathrm{~g}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$5 \mathrm{~g}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$40 \mathrm{~g}$</span> </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 a closed organ pipe, the fundamental frequency is given $f=\frac{v}{4 L}$, where $v=320 \mathrm{~m} / \mathrm{s}$ is the velocity of sound in the medium of organ pipe and $L=0.8 \mathrm{~m}$ being the length of pipe.<br/>Now, we are given second harmonic frequency of wire is equal and in resonance with fundamental frequency of pipe, thus;<br/>$f=\frac{v}{4 L}=\frac{1}{l} \sqrt{\frac{T}{\mu}}$<br/>Putting, $T=50 \mathrm{~N}$ and $l=0.8 \mathrm{~m}$<br/>$\begin{aligned} &amp; \frac{320}{4(0.8)}=\frac{1}{0.5} \sqrt{\frac{50}{\mu}} \\ &amp; \Rightarrow \sqrt{\frac{\mu}{50}}=\frac{1}{50}\end{aligned}$<br/>The length of string $=0.5 \mathrm{~m}$<br/>Thus, mass of string $=0.02 \times 0.5=0.01 \mathrm{~kg}=10 \mathrm{~g}$</div>


tags: ['Physics', 'Laws of Motion', 'JEE Main']
source_db: MarksBatch0.db
title:
A horizontal force F is applied at the top of an equilateral triangular block having mass m . The minimum coefficient of friction required to topple the block before translation will be

slug:
a-horizontal-force-f-is-applied-at-the-top-of-an-equilateral-triangular-block-having-mass-m-the-mi-94533

question:
<div class="question">A horizontal force <math><mi>F</mi></math> is applied at the top of an equilateral triangular block having mass <math><mi>m</mi><mo>.</mo></math> The minimum coefficient of friction required to topple the block before translation will be<br/> <br/> <img src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/cjeojglikbmefxdjgldtmshjackstbtyxbyfucyplfyyfbqwcn.png" style="max-width:100%; object-fit:contain"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mfrac><mrow><mn>2</mn></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mn>3</mn></msqrt></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">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><math><mi>I</mi><mo>=</mo><mfrac><mrow><mi>m</mi><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>6</mn></mrow></mfrac><mo>+</mo><mi>m</mi><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mi>a</mi></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>â</mo><mi>I</mi><mo>=</mo><mfenced separators="|"><mrow><mfrac><mrow><mi>m</mi><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced></math> <br/><br/> <math><mo>â</mo><mi>F</mi><mo>Ã</mo><mfrac><mrow><mi>a</mi><msqrt><mn>3</mn></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mo>=</mo><mfenced separators="|"><mrow><mi>m</mi><mi>g</mi></mrow></mfenced><mfrac><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>â</mo><mi>F</mi><mo>=</mo><mfenced separators="|"><mrow><mfrac><mrow><mi>m</mi><mi>g</mi></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></mfenced></math> <br/><br/> <math><mi>F</mi><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi></mrow></msub><mo>â</mo><mfrac><mrow><mi>m</mi><mi>g</mi></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac><mi>Î¼</mi><mi>m</mi><mi>g</mi><mo>â</mo><mi>Î¼</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></math> <br/><br/> <img call-to-action="" data-pagefind-ignore="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/fxwdkvrxzdgzjfprjtxmcdgtzcowlgavwqssuonwqwgpsigsro.png" style="max-width:100%; object-fit:contain /&gt;&lt;/div&gt; &lt;/section&gt; &lt;!-- /Solution --&gt; &lt;!-- Call to Action --&gt; &lt;section class="/> <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: ['Mathematics', 'Probability', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch0.db
title:
A husband and wife appear in an interview for two vacancies in the same post. The probability of husband's selection is 1/5 and that of wife's selection is 1/3 . What is the probability that only one of them will be selected?

slug:
a-husband-and-wife-appear-in-an-interview-for-two-vacancies-in-the-same-post-the-probability-of-hus-90548

question:
<div class="question">A husband and wife appear in an interview for two vacancies in the same post. The probability of husband's selection is $1 / 5$ and that of wife's selection is $1 / 3$. What is the probability that only one of them will be selected?</div>

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

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

answer:
<div class="solution">Let $\mathrm{A}=$ Event of husband's selection and $\mathrm{B}=$ Event of wife's selection<br/>Given : $P(A)=\frac{1}{5} \Rightarrow P(\bar{A})=1-\frac{1}{5}=\frac{4}{5}$<br/>$P(B)=\frac{1}{3} \Rightarrow P(\bar{B})=1-\frac{1}{3}=\frac{2}{3}$<br/>$P($ Only one of them selected $)$ $=P(A) \times P(\bar{B})+P(\bar{A}) \times P(B)$<br/>$=\frac{1}{5} \times \frac{2}{3}+\frac{4}{5} \times \frac{1}{3}=\frac{2}{15}+\frac{4}{15}=\frac{6}{15}=\frac{2}{5}$</div>


slug: a-hyperbola-passes-through-the-point-p-2-3-and-has-foci-at-2-0-then-the-tangent-to-this-hyp-89730
tags: ['Mathematics', 'Hyperbola', 'JEE Main', 'JEE Main 2017 (02 Apr)']
source_db: MarksBatch0.db
title:
A hyperbola passes through the point P 2 , 3 and has foci at Â± 2 , 0 . Then the tangent to this hyperbola at P also passes through the point

question:
<div class="question">A hyperbola passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced separators="|"><mrow><msqrt><mn>2</mn></msqrt><mo>,</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced></math> and has foci at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="|"><mrow><mo>±</mo><mi> </mi><mn>2</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math>. Then the tangent to this hyperbola at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> also passes through the point</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced separators="|"><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>,</mo><mn>3</mn><msqrt><mn>3</mn></msqrt></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>From the standard equation of hyperbola</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>±</mo><mi>a</mi><mi>e</mi><mo>=</mo><mo>±</mo><mi> </mi><mn>2</mn></math></p><p>As we know, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>a</mi><mn>2</mn></msup><mfenced separators="|"><mrow><msup><mi>e</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p><p><math><mo>⇒</mo><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>4</mn><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi> </mi></math>Equation of hyperbola is <math><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∵</mo><mo> </mo></math>It passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msqrt><mn>2</mn></msqrt><mo>,</mo><msqrt><mn>3</mn></msqrt><mo> </mo></mrow></mfenced></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup></mfrac><mo>-</mo><mfrac><mn>3</mn><mrow><mn>4</mn><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mi>t</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mfenced separators="|"><mrow><mn>4</mn><mo>-</mo><mi>t</mi></mrow></mfenced><mo>-</mo><mn>3</mn><mi>t</mi><mo>-</mo><mi>t</mi><mfenced separators="|"><mrow><mn>4</mn><mo>-</mo><mi>t</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>8</mn><mo>-</mo><mn>2</mn><mi>t</mi><mo>-</mo><mn>3</mn><mi>t</mi><mo>-</mo><mn>4</mn><mi>t</mi><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>t</mi><mo>-</mo><mi>t</mi><mo>+</mo><mn>8</mn><mo>=</mo><mn>0</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi>t</mi><mo>=</mo><mi> </mi><mn>8</mn><mo>,</mo><mi> </mi><mn>1</mn></math></p><p>So, <math><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>8</mn><mo>,</mo><mi> </mi><mn>1</mn></math>.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>,</mo><mi> </mi><mn>1</mn></math></p><p>But for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>,</mo><mo> </mo><mi>b</mi></math> becomes imaginary. So this case is rejected.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo></math> Hyperbola is <math><mfrac><mrow><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>1</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mfrac><mo>=</mo><mn>1</mn></math>.</p><p>The equation of the tangent at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>2</mn></msqrt><mi>x</mi><mo>-</mo><mfrac><mi>y</mi><msqrt><mn>3</mn></msqrt></mfrac><mo>=</mo><mn>1</mn></math>. </p><p>Now checking each points.</p></div>


tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch0.db
title:
a. In the young dicot stem, cambium present in patches as a single layer between xylem and phloem. b. Vascular and cork cambium both are from the lateral meristem. c. The medullary cells are derived from the interfascicular cambium. In given statements, which of the following are correct?

slug:
a-in-the-young-dicot-stem-cambium-present-in-patches-as-a-single-layer-between-xylem-and-phloem-b-28276

question:
<div class="question"><math><mfenced separators="|"><mrow><mi mathvariant="normal">a.</mi></mrow></mfenced></math> In the young dicot stem, cambium present in patches as a single layer between xylem and phloem.<br/> <math><mfenced separators="|"><mrow><mi mathvariant="normal">b.</mi></mrow></mfenced></math> Vascular and cork cambium both are from the lateral meristem.<br/> <math><mfenced separators="|"><mrow><mi mathvariant="normal">c.</mi></mrow></mfenced></math> The medullary cells are derived from the interfascicular cambium.<br/> <br/>
In given statements, which of the following are correct?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a and c</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a and b</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">bâand c</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">a,b 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">a,b and c</span> </div>

answer:
<div class="solution">Intrafascicular cambium is present in between xylem and phloem of vascular bundle. It combines with interfascicular cambium to form complete vascular cambium ring. Both vascular cambium and cork cambium are example of lateral meristem and contribute to secondary growth.</div>


tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch0.db
title:
A is a certain positive acute angle which satisfies the following equation: Number of degrees in A + Number of radians in A = ( 180 + Ï ) /3 What is the angle A?

slug:
a-is-a-certain-positive-acute-angle-which-satisfies-the-following-equation-number-of-degrees-in-a-92725

question:
<div class="question">A is a certain positive acute angle which satisfies the following equation:<br/>Number of degrees in A + Number of radians in $\mathrm{A}=(180+\pi) / 3$<br/>What is the angle A?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$40^{\circ}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$60^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$80^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$60^{\circ}$</span> </div>

answer:
<div class="solution">Given that number of degrees in A+ Number of radians<br/>in $\mathrm{A}=\frac{180^{\circ}+\pi}{3}=\frac{180^{\circ}}{3}+\frac{\pi}{3}=60^{\circ}+\frac{\pi}{3}$<br/>$\Rightarrow$ Angle $\mathrm{A}=60^{\circ}$</div>


tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: MarksBatch0.db
title:
A is a square matrix of order 3 such that its determinant is 4. What is the determinant of its transpose?

slug:
a-is-a-square-matrix-of-order-3-such-that-its-determinant-is-4-what-is-the-determinant-of-its-trans-76408

question:
<div class="question">A is a square matrix of order 3 such that its determinant is 4. What is the determinant of its transpose?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">64</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">36</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">32</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">4</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">The determinant of transpose will not change. So, the determinant is equal to 4.</div>


slug: a-is-an-angle-in-the-fourth-quadrant-if-satisfies-the-trigonometric-equation-3-3-tan-2-a-64107
tags: ['Mathematics', 'Trigonometric Equations', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch0.db
title:
A is an angle in the fourth quadrant. If satisfies the trigonometric equation 3 ( 3 â tan 2 A â cot A ) 2 = 1 . Which one of the following is a value of A?

question:
<div class="question">A is an angle in the fourth quadrant. If satisfies the trigonometric equation $3\left(3-\tan ^{2} \mathrm{~A}-\cot \mathrm{A}\right)^{2}=1$.<br/>Which one of the following is a value of A?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$300^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$315^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$330^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$345^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$300^{\circ}$</span> </div>

answer:
<div class="solution">Checking through options $300^{\circ}=-60^{\circ}$<br/>So, $3\left[3-\tan ^{2}\left(-60^{\circ}\right)-\cot \left(-60^{\circ}\right)\right]^{2}$<br/>$=3\left[3-3+\frac{1}{\sqrt{3}}\right]^{2}=3 \times \frac{1}{3}=1$</div>


title: A is associated with
slug: a-is-associated-with-41956
tags: ['Mathematics', 'Continuity and Differentiability', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch0.db
question:
<div class="question">A is associated with<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/KZ_9YHdRVYib_6RC_uIf8TdDaNVmMVwucfK0_fpsgIw.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$1,3,5,8,9,12$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2,4,6,8,10,11$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1,3,5,7,10,11$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1,3,5,8,9,12$</span> </div>

answer:
<div class="solution">$\sin x$ is periodic, continuous at every point on $(-\infty, \infty)$, differentiable at every point on $(-\infty, \infty)$, has a period $2 \pi, \sin x$ increases on $\left(0, \frac{\pi}{2}\right)$ and decreases on $\left(\frac{\pi}{2}, \pi\right)$.</div>


tags: ['Chemistry', 'Redox Reactions', 'NEET']
source_db: MarksBatch0.db
title:
A K M n O 4 solution can be standardised by titration against A s 2 O 3 ( s ) . A 0.1156 g sample of A s 2 O 3 requires 27.08 mL of the K M n O 4 (aq) for its titration. What is the molarity of the K M n O 4 ( a q ) [As = 75].

slug:
a-k-m-n-o-4-solution-can-be-standardised-by-titration-against-a-s-2-o-3-s-a-0-1156-g-sample-of-13766

question:
<div class="question">A <math><msub><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">O</mi></mrow><mrow><mn>4</mn></mrow></msub></math> solution can be standardised by titration against <math><msub><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">s</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi mathvariant="normal">O</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo><mo>.</mo><mi mathvariant="normal"> </mi></math> A 0.1156 g sample of <math><msub><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">s</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi mathvariant="normal">O</mi></mrow><mrow><mn>3</mn></mrow></msub></math> requires 27.08 mL of the <math><mi mathvariant="normal">K</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">n</mi><msub><mrow><mi mathvariant="normal">O</mi></mrow><mrow><mn>4</mn></mrow></msub></math> (aq) for its titration. What is the molarity of the <math><msub><mrow><mi mathvariant="normal">K</mi><mi mathvariant="normal">M</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">O</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo></math> [As = 75]. </div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.0172 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">1.0172 M</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.172 M</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.9172 M</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">0.0172 M</span> </div>

answer:
<div class="solution">Meq. of <math><msub><mrow><mi mathvariant="normal">A</mi><mi mathvariant="normal">s</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi mathvariant="normal">O</mi></mrow><mrow><mn>3</mn></mrow></msub></math> = Meq of <math> <mrow> <msub> <mrow> <mtext>KMnO</mtext></mrow> <mn>4</mn> </msub> <mfenced> <mrow> <msubsup> <mrow> <mtext>As</mtext></mrow> <mn>2</mn> <mrow> <mn>3</mn><mo>+</mo></mrow> </msubsup> <mo>â</mo><mn>2</mn><msup> <mrow> <mtext>As</mtext></mrow> <mrow> <mn>5</mn><mo>+</mo></mrow> </msup> <mo>+</mo><mn>4</mn><msup> <mtext>e</mtext> <mo>â</mo> </msup> </mrow> </mfenced></mrow> </math><br/><br/><math> <mrow> <mfrac> <mrow> <mn>0.1156</mn></mrow> <mrow> <mn>198</mn><mo>/</mo><mn>4</mn></mrow> </mfrac> <mo>×</mo><mn>1000</mn><mtext> </mtext><mo>=</mo><mtext> M</mtext><mo>×</mo><mn>5</mn><mo>×</mo><mn>27.08</mn><mfenced> <mrow> <msup> <mrow> <mtext>Mn</mtext></mrow> <mrow> <mn>7</mn><mo>+</mo></mrow> </msup> <mo>+</mo><mn>5</mn><msup> <mtext>e</mtext> <mo>−</mo> </msup> <mo>→</mo><msup> <mrow> <mtext>Mn</mtext></mrow> <mrow> <mn>2</mn><mo>+</mo></mrow> </msup> </mrow> </mfenced></mrow></math><br/><br/><math><mo>â´</mo></math> M = 0.0172<br/><br/><math><mo>â´</mo></math> <math><msub><mrow><mi mathvariant="normal">M</mi></mrow><mrow><mi mathvariant="normal">M</mi><mi mathvariant="normal">n</mi><msubsup><mrow><mi mathvariant="normal">O</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>-</mo></mrow></msubsup></mrow></msub><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mo>=</mo><mi mathvariant="normal"> </mi><mn>0.0172</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">M</mi></math></div>


tags: ['Biology', 'Principles of Inheritance and Variation', 'NEET']
source_db: MarksBatch0.db
title:
A karyotype is used to study I) Number of chromosomes II) Position of centromere III) Position of chromosome IV) Length of the arms V) Secondary constriction

slug:
a-karyotype-is-used-to-study-i-number-of-chromosomes-ii-position-of-centromere-iii-position-of-ch-51000

question:
<div class="question">A karyotype is used to study<br/> <br/>
I) Number of chromosomes<br/>
II) Position of centromere<br/>
III) Position of chromosome<br/>
IV) Length of the arms<br/>
V) Secondary constriction</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">I, II and III</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">I, II, IV, 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">All of these</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">I, IV, V</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">I, II, IV, V</span> </div>

answer:
<div class="solution">Karyotyping is the process by which photographs of chromosomes are taken to determine the chromosome complement of an individual, including the number of chromosomes and any abnormalities. A karyotype is used to analyze the number of chromosomes to determine the position of the centromere, to measure the length of the arms and to observe the secondary constriction.</div>


slug: a-l-3-a-q-3-e-a-l-s-e-1-66-v-c-u-2-a-q-2-e-c-u-s-e-78669
tags: ['Chemistry', 'Electrochemistry', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2.00 V</span> </div>
source_db: MarksBatch0.db
title:
A l 3 + ( a q ) + 3 e - â A l ( s ) ; E Â° = - 1.66 V C u 2 + ( a q ) + 2 e - â C u ( s ) ; E Â° = + 0.34 V What voltage is produced under standard conditions by combining the half reaction with these standard electrode potentials?

question:
<div class="question"><math><mi mathvariant="normal">A</mi><msup><mrow><mi mathvariant="normal">l</mi></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>+</mo><mn>3</mn><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo></mrow></msup><mo>â</mo><mi mathvariant="normal">A</mi><mi mathvariant="normal">l</mi><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo><mo>;</mo></math> <math><mi mathvariant="normal">E</mi><mo>Â°</mo><mo>=</mo><mo>-</mo><mn>1.66</mn><mi mathvariant="normal">V</mi></math><br/> <math><mi mathvariant="normal">C</mi><msup><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>+</mo><mn>2</mn><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo></mrow></msup><mo>â</mo><mi mathvariant="normal">C</mi><mi mathvariant="normal">u</mi><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo><mo>;</mo></math> <math><mi mathvariant="normal">E</mi><mo>Â°</mo><mo>=</mo><mo>+</mo><mn>0.34</mn><mi mathvariant="normal">V</mi></math><br/>
What voltage is produced under standard conditions by combining the half reaction with these standard electrode potentials?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1.32 V</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2.00 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">2.30 V</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4.34 V</span> </li> </ul>

answer:
<div class="solution">The cell will be<br/><math><mi mathvariant="normal">A</mi><mi mathvariant="normal">l</mi><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo><mo>|</mo><mi mathvariant="normal">A</mi><msup><mrow><mi mathvariant="normal">l</mi></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>|</mo><mo>|</mo><mi mathvariant="normal">C</mi><msup><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>(</mo><mi mathvariant="normal">a</mi><mi mathvariant="normal">q</mi><mo>)</mo><mo>|</mo><mi mathvariant="normal">C</mi><mi mathvariant="normal">u</mi><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></math><br/><math><msubsup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">l</mi><mi mathvariant="normal">l</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msubsup><mo>=</mo><msubsup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">c</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal">e</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msubsup></math><br/><math><mo>=</mo><msubsup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">C</mi><msup><mrow><mi mathvariant="normal">u</mi></mrow><mrow><mn>2</mn><mo>+</mo></mrow></msup><mo>/</mo><mi mathvariant="normal">C</mi><mi mathvariant="normal">u</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msubsup><mo>-</mo><msubsup><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mi mathvariant="normal">A</mi><msup><mrow><mi mathvariant="normal">l</mi></mrow><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>/</mo><mi mathvariant="normal">A</mi><mi mathvariant="normal">l</mi></mrow><mrow><mi mathvariant="normal">o</mi></mrow></msubsup></math><br/><math><mo>=</mo><mo>+</mo><mn>0.34</mn><mo>-</mo><mo>(</mo><mo>-</mo><mn>1.66</mn><mo>)</mo><mo>=</mo><mo>+</mo><mn>2.00</mn><mi mathvariant="normal">V</mi></math> .</div>


tags: ['Mathematics', 'Heights and Distances', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch0.db
title:
A lamp post stands on a horizontal plane. From a point situated at a distance 150 m from its foot, the angle of elevation of the top is 3 0 â . What is the height of the lamp post ?

slug:
a-lamp-post-stands-on-a-horizontal-plane-from-a-point-situated-at-a-distance-150-m-from-its-foot-t-19617

question:
<div class="question">A lamp post stands on a horizontal plane. From a point situated at a distance $150 \mathrm{~m}$ from its foot, the angle of elevation of the top is $30^{\circ}$. What is the height of the lamp post ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$50 \mathrm{~m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$50 \sqrt{3} \mathrm{~m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{50}{\sqrt{3}} \mathrm{~m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$100 \mathrm{~m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$50 \sqrt{3} \mathrm{~m}$</span> </div>

answer:
<div class="solution">In $\Delta \mathrm{ABC}$, we have $\mathrm{BC}=150 \mathrm{~m}, \mathrm{AB}=\mathrm{h}$<br/>$\angle \mathrm{C}=30^{\circ}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/txT3JDe_KqO2SeiAhYibYsVPLm1M53B0BWbLbPtfqM0.original.fullsize.png"/><br/><br/>$\tan 30^{\circ}=\frac{\mathrm{h}}{150}$<br/>$\Rightarrow \frac{1}{\sqrt{3}}=\frac{\mathrm{h}}{150}$<br/>$\therefore \mathrm{h}=\frac{150}{\sqrt{3}} \mathrm{~m}$<br/>$=\frac{150 \times \sqrt{3}}{3}=50 \sqrt{3} \mathrm{~m}$</div>


tags: ['Chemistry', 'Environmental Chemistry']
options: None
correct_option: None
source_db: MarksBatch0.db
title:
A large number of fish are suddenly found floating dead on a lake. There is no evidence of toxic dumping but you find an abundance of phytoplankton. Suggest a reason for the fish kill.

slug:
a-large-number-of-fish-are-suddenly-found-floating-dead-on-a-lake-there-is-no-evidence-of-toxic-dum-82535

question:
<div class="question">A large number of fish are suddenly found floating dead on a lake. There is no evidence of toxic dumping but you find an abundance of phytoplankton. Suggest a reason for the fish kill.</div>

answer:
<div class="solution">Excessive phytoplankton (organic pollutants such as leaves, grass, trash etc. ) present in water is biodegradable. A large population of bacteria decomposes this organic matter in water. During this process, they consume the oxygen dissolved in water. Water has already limited dissolved oxygen ( \(10 \mathrm{ppm})\). Thus, it is further depleted. When the level of dissolved oxygen falls below 6 ppm, the fish cannot survive. Hence, they die and float dead on the lake.</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'NEET']
source_db: MarksBatch0.db
title:
A large number of particles are placed around the origin, each at a distance R from the origin. The distance of the center of mass of the system from the origin is

slug:
a-large-number-of-particles-are-placed-around-the-origin-each-at-a-distance-r-from-the-origin-the-50962

question:
<div class="question">A large number of particles are placed around the origin, each at a distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi></math> from the origin. The distance of the center of mass of the system from the origin is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">equal to <math><mi mathvariant="normal">R</mi></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">less than or equal to <math><mi mathvariant="normal">R</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">greater than  <math><mi mathvariant="normal">R</mi></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">greater than or equal to  <math><mi mathvariant="normal">R</mi></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">less than or equal to <math><mi mathvariant="normal">R</mi></math></span> </div>

answer:
<div class="solution">A large number of particles are situated at a distance <math><mi mathvariant="normal">R</mi></math> from the origin. If particles are uniformly distributed and make a circular boundary around the origin, the centre of mass will lie at the origin.<br/><br/>Whereas, if the particles are not uniformly distributed, then distance can be any value between zero and <math><mi mathvariant="normal">R</mi></math><br/><br/>This implies the distance between the centre of mass and the origin is always less than or equal to <math><mi mathvariant="normal">R</mi></math></div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'JEE Main']
source_db: MarksBatch0.db
title:
A large number of particles are placed around the origin, each at a distance R from the origin. The distance of the centro of mass of the system from the origin is

slug:
a-large-number-of-particles-are-placed-around-the-origin-each-at-a-distance-r-from-the-origin-the-77918

question:
<div class="question">A large number of particles are placed around the origin, each at a distance $R$ from the origin. The distance of the centro of mass of the system from the origin is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">equal to $R$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">less than equal to $R$</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">C</span> <span class="option-data">greater than $R$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">greater than equal to $R$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">less than equal to $R$</span> </div>

answer:
<div class="solution">As large number of particles is situated at a distance $R$ from the ongin If particles are uniformly distubuted and make a circulat boundary atound the ongm, then centre of mass will be at the ongin While if the patticles are not uniformly distnbuted, then centre of mass will lie between particle and origin. This implies distance between centre of mass and ongin is always less than equal to $R$.</div>


tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2015']
source_db: MarksBatch0.db
title:
A large number of particles are placed around the origin, each at a distance R from the origin. The distance of the centro of mass of the system from the origin is

slug:
a-large-number-of-particles-are-placed-around-the-origin-each-at-a-distance-r-from-the-origin-the-28871

question:
<div class="question">A large number of particles are placed around the origin, each at a distance $R$ from the origin. The distance of the centro of mass of the system from the origin is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">equal to $R$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">less than equal to $R$</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">C</span> <span class="option-data">greater than $R$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">greater than equal to $R$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">less than equal to $R$</span> </div>

answer:
<div class="solution">As large number of particles is situated at a distance $R$ from the ongin If particles are uniformly distubuted and make a circulat boundary atound the ongm, then centre of mass will be at the ongin While if the patticles are not uniformly distnbuted, then centre of mass will lie between particle and origin. This implies distance between centre of mass and ongin is always less than equal to $R$.</div>


tags: ['Physics', 'Ray Optics', 'JEE Main']
source_db: MarksBatch0.db
title:
A light ray is incident on the lower medium boundary at an angle 30 Â° with the normal. Which of the following statements is true?

slug:
a-light-ray-is-incident-on-the-lower-medium-boundary-at-an-angle-30-with-the-normal-which-of-the-42504

question:
<div class="question">A light ray is incident on the lower medium boundary at an angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>Â°</mo></math> with the normal. Which of the following statements is true?<br/> <br/> <img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/EM0029938_Question.jpg" style="width: 170px; height: 131px;"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">If <math> <mrow> <msub> <mi>Î¼</mi> <mn>2</mn> </msub> <mo>&gt;</mo> <mn>2</mn> </mrow> </math> ,then total deviation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo>Â°</mo></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">If <math> <mrow> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>2</mn> </mrow> </math> , then total deviation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>°</mo></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">If <math> <mrow> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mo>&gt;</mo> <mn>2</mn> </mrow> </math> ,then total deviation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>120</mn><mo>°</mo></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">If <math> <mrow> <msub> <mi>μ</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>2</mn> </mrow> </math> ,then total deviation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo>°</mo></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">If <math> <mrow> <msub> <mi>Î¼</mi> <mn>2</mn> </msub> <mo>&gt;</mo> <mn>2</mn> </mrow> </math> ,then total deviation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo>Â°</mo></math></span> </div>

answer:
<div class="solution">For    T.I.R. at A<br/><br/><math> <mrow> <mn>4</mn> <mo>sin</mo> <mn>3</mn> <msup> <mn>0</mn> <mo>â</mo> </msup> <mo>=</mo> <msub> <mi>Î¼</mi> <mn>2</mn> </msub> <mo>sin</mo> <mn>9</mn> <msup> <mn>0</mn> <mo>â</mo> </msup> </mrow> </math><br/><br/><math> <mrow> <msub> <mi>Î¼</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>2</mn> </mrow> </math><br/><br/><strong>Case -I</strong><br/><br/>If <math> <mrow> <msub> <mi>Î¼</mi> <mn>2</mn> </msub> <mo>&lt;</mo> <mn>2</mn> </mrow> </math>, then always T.I.R. takes place and in this situation angle of deviation is 120<sup>o</sup>.<br/><br/><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/EM0029938_Ans_1.jpg" style="width: 190px; height: 149px;"/><br/><br/><strong>Case-II</strong><br/><br/><math> <mrow> <msub> <mi>Î¼</mi> <mn>2</mn> </msub> <mo>&gt;</mo> <mn>2</mn> </mrow> </math>, then for the given angle of incidence no. T.I.R takes place at A, so light strikes on interface B for T.I.R.<br/><br/><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/EM0029938_Ans_2.jpg" style="width: 206px; height: 180px;"/><br/><br/><math> <mrow> <msub> <mi>Î¼</mi> <mn>2</mn> </msub> <mo>sin</mo> <mi>Î±</mi> <mo>=</mo> <mn>2</mn> <mo>sin</mo> <mn>9</mn> <msup> <mn>0</mn> <mo>â</mo> </msup> </mrow> </math><br/><br/>or     <math> <mrow> <msub> <mi>Î¼</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> </mrow> <mrow> <mo>sin</mo> <mi>Î±</mi> </mrow> </mfrac> </mrow> </math><br/><br/><math> <mrow> <mo>sin</mo> <mi>Î±</mi> <mo>&lt;</mo> <mn>1</mn> </mrow> </math><br/><br/><math> <mrow> <msub> <mi>Î¼</mi> <mn>2</mn> </msub> <mo>&gt;</mo> <mn>2</mn> </mrow> </math></div>


slug: a-line-l-passes-through-the-point-p-5-6-7-and-is-parallel-to-the-planes-x-y-z-1-an-14261
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch0.db
title:
A line L passes through the point P ( 5 , â 6 , 7 ) and is parallel to the planes x + y + z = 1 and 2 x â y â 2 z = 3 . What are the direction ratios of the line of intersection of the given planes?

question:
<div class="question">A line L passes through the point $\mathrm{P}(5,-6,7)$ and is parallel to the planes $x+y+z=1$ and $2 x-y-2 z=3$. What are the direction ratios of the line of intersection of the given planes?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$   &lt; 1,-4,3&gt;$</span> </div>

answer:
<div class="solution">Let $a, b, c$ be the direction ratios of the line Then its equation is<br/>$\frac{x-5}{a}=\frac{y+6}{b}=\frac{z-7}{c}$<br/>Since (i) is parallel to the planes $x+y+z=1$ and<br/>$2 x-v-2 z=3$ then<br/>$a(1)+b(1)+c(1)=0$ and $a(2)+b(-1)+c(-2)=0$<br/>Bycross multiplication<br/>$\frac{a}{-1}=\frac{b}{4}=\frac{c}{-3}=\lambda$<br/>$\Rightarrow a=-\lambda, b=4 \lambda, c=-3 \lambda$<br/>$\Rightarrow$ Direction ratios of the line are $   &lt; -1,4,-3&gt;= &lt; 1,-4,3&gt;$</div>


slug: a-line-l-passes-through-the-point-p-5-6-7-and-is-parallel-to-the-planes-x-y-z-1-an-91291
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch0.db
title:
A line L passes through the point P ( 5 , â 6 , 7 ) and is parallel to the planes x + y + z = 1 and 2 x â y â 2 z = 3 . What is the equation of the line L ?

question:
<div class="question">A line L passes through the point $\mathrm{P}(5,-6,7)$ and is parallel to the planes $x+y+z=1$ and $2 x-y-2 z=3$. What is the equation of the line $\mathrm{L}$ ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{x-5}{-1}=\frac{y+6}{4}=\frac{z-7}{-3}$</span> </div>

answer:
<div class="solution">Substituting $\mathrm{a}, \mathrm{b}, \mathrm{c}$ in (i), we get<br/>$\frac{x-5}{-1}=\frac{y+6}{4}=\frac{z-7}{-3}$<br/>Hence, equation of the line is<br/>$\frac{x-5}{-1}=\frac{y+6}{4}=\frac{z-7}{-3}$</div>


slug: a-line-makes-4-5-with-positive-x-axis-and-makes-equal-angles-with-positive-y-z-axes-respecti-65065
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch0.db
title:
A line makes 4 5 â with positive x -axis and makes equal angles with positive y , z axes, respectively. What is the sum of the three angles which the line makes with positive x , y and z axes?

question:
<div class="question">A line makes $45^{\circ}$ with positive $\mathrm{x}$ -axis and makes equal angles with positive $y, z$ axes, respectively. What is the sum of the three angles which the line makes with positive $x, y$ and $z$ axes?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$180^{\circ}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$165^{\circ}$</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">$150^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$135^{\circ}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$165^{\circ}$</span> </div>

answer:
<div class="solution">We know that sum of square of direction cosines $=1$ i.e. $\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1$<br/>$\Rightarrow \quad \cos ^{2} 45^{\circ}+\cos ^{2} \beta+\cos ^{2} \beta=1$<br/>(As given $\alpha=45^{\circ}$ and $\beta=\gamma$ )<br/>$\Rightarrow \quad \frac{1}{2}+2 \cos ^{2} \beta=1$<br/>$\Rightarrow \cos ^{2} \beta=\frac{1}{4}$<br/>$\Rightarrow \cos \beta=\pm \frac{1}{2}$, Negative value is discarded, since the line<br/>makes angle with positive axes.<br/>Hence, $\cos \beta=\frac{1}{2}$<br/>$\Rightarrow \cos \beta=\cos 60^{\circ}$<br/>$\Rightarrow \quad \beta=60^{\circ}$<br/>Required sum $=\alpha+\beta+\gamma=45^{\circ}+60^{\circ}+60^{\circ}=165^{\circ}$</div>


slug: a-line-makes-angles-and-with-x-y-z-axes-respectively-consider-the-following-1-sin-51081
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2007 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2 only</span> </div>
source_db: MarksBatch0.db
title:
A line makes angles Î¸ , Ï and Ï with x , y , z axes respectively. Consider the following 1 . sin 2 Î¸ + sin 2 Ï = cos 2 Ï 2 . cos 2 Î¸ + cos 2 Ï = sin 2 Ï 3 . sin 2 Î¸ + cos 2 Ï = cos 2 Ï Which of the above is/are correct?

question:
<div class="question">A line makes angles $\theta, \phi$ and $\psi$ with $\mathrm{x}, \mathrm{y}, \mathrm{z}$ axes respectively.<br/>Consider the following<br/> 1 . $\sin ^{2} \theta+\sin ^{2} \phi=\cos ^{2} \psi$<br/> 2 . $\quad \cos ^{2} \theta+\cos ^{2} \phi=\sin ^{2} \psi$<br/> 3 . $\sin ^{2} \theta+\cos ^{2} \phi=\cos ^{2} \psi$<br/>Which of the above is/are correct?</div>

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

answer:
<div class="solution">If a line makes angle $\theta, \phi$ and $\psi$ with $\mathrm{x}, \mathrm{y}, \mathrm{z}$ axes respectively, then $\cos ^{2} \theta+\cos ^{2} \phi+\cos ^{2} \psi=1$<br/>$\Rightarrow \cos ^{2} \theta+\cos ^{2} \phi=1-\cos ^{2} \psi \Rightarrow \cos ^{2} \theta+\cos ^{2} \phi=\sin ^{2} \psi$<br/>Statement $(2)$ is correct</div>


slug: a-line-makes-the-same-angle-with-each-of-the-x-and-y-axes-if-the-angle-which-it-makes-with-16011
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch0.db
title:
A line makes the same angle Î± with each of the x and y axes. If the angle Î¸ , which it makes with the z -axis, is such that sin 2 Î¸ = 2 sin 2 Î± , then what is the value of Î± ?

question:
<div class="question">A line makes the same angle $\alpha$ with each of the $x$ and $y$ axes. If the angle $\theta$, which it makes with the $z$ -axis, is such that $\sin ^{2} \theta=2 \sin ^{2} \alpha$, then what is the value of $\alpha$ ?</div>

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

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

answer:
<div class="solution">Since $l^{2}+m^{2}+n^{2}=1$<br/>$\therefore \cos ^{2} \alpha+\cos ^{2} \alpha+\cos ^{2} \theta=1$<br/>$(\because A$ line makes the same angle $\alpha$ with $x$ and $y$ -axes and $\theta$ with z-axis) Also, $\sin ^{2} \theta=2 \sin ^{2} \alpha$<br/>$\Rightarrow 1-\cos ^{2} \theta=2\left(1-\cos ^{2} \alpha\right)\left(\because \sin ^{2} A+\cos ^{2} A=1\right)$<br/>$\Rightarrow \cos ^{2} \theta=2 \cos ^{2} \alpha-1$<br/>From Eq. (i) and (ii) $2 \cos ^{2} \alpha+2 \cos ^{2} \alpha-1=1$<br/>$\Rightarrow 4 \cos ^{2} \alpha=2 \Rightarrow \cos ^{2} \alpha=\frac{1}{2}$<br/>$\Rightarrow \cos \alpha=\pm \frac{1}{\sqrt{2}}$<br/>$\Rightarrow \alpha=\frac{\pi}{4}, \frac{3 \pi}{4}$</div>


title: A line passes through ( 2 , 2 ) and is perpendicular to the line 3 x + y = 3 . Its y -intercept is
slug: a-line-passes-through-2-2-and-is-perpendicular-to-the-line-3-x-y-3-its-y-intercept-is-49767
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch0.db
question:
<div class="question">A line passes through $(2,2)$ and is perpendicular to the line $3 x+y=3$. Its $y$ -intercept is</div>

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

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

answer:
<div class="solution">A line passes through $(2,2)$ and is perpendicular to the line $3 x+y=3$ Slope of line $3 \mathrm{x}+\mathrm{y}=3$ is $-3$<br/>Slope of line which passes through $(2,2)$ is $\frac{1}{3}$<br/>$\therefore$ Equation of line passes through $(2,2)$ and having<br/>slope $\left(\frac{1}{3}\right)$ is<br/>$(\mathrm{y}-2)=\frac{1}{3}(\mathrm{x}-2)$<br/>$\therefore \quad 3 y-6=x-2$<br/>$\therefore \quad x-3 y+4=0$<br/>In order to find y-intercept of the line Put $x=0$ in $x-3 y+4=0$ $\therefore \quad-3 \mathrm{y}=-4$<br/>$\mathrm{y}=\frac{4}{3}$<br/>$\therefore$ Option (b) is correct.</div>


slug: a-line-passes-through-the-points-6-7-1-and-2-3-1-what-are-the-directio-95303
tags: ['Mathematics', 'Three Dimensional Geometry', 'NDA', 'NDA 2014 (Phase 1)']
answer: <div class="solution">Direction ratios $ &lt; (2-6),(-3+7),(1+1)&gt;$ $=\langle-4,4,2&gt;$</div>
source_db: MarksBatch0.db
title:
A line passes through the points ( 6 , â 7 , â 1 ) and ( 2 , â 3 , 1 ) . What are the direction ratios of the line?

question:
<div class="question">A line passes through the points $(6,-7,-1)$ and $(2,-3,1)$. What are the direction ratios of the line?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$   &lt; -4,4,2&gt;$</span> </div>


tags: ['Chemistry', 'Electrochemistry', 'KCET', 'KCET 2012']
source_db: MarksBatch1_P1.db
title:
100-ml-of-01-m-acetic-acid-is-completely-neutralized-using-a-standard-solution-of-naoh-the-volume-of-ethane-obtained-at-stp-after-the-complete-electro

slug:
100-ml-of-01-m-acetic-acid-is-completely-neutralized-using-a-standard-solution-of-naoh-the-volume-of-ethane-obtained-at-stp-after-the-complete-electro-73231

question:
<div class="question">$100 \mathrm{~mL}$ of $0.1 \mathrm{M}$ acetic acid is completely neutralized using a standard solution of $\mathrm{NaOH}$. The volume of ethane obtained at STP after the complete electrolysis of the resulting solution is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$56 \mathrm{~mL}$.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$224 \mathrm{~mL}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$560 \mathrm{~mL}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$112 \mathrm{~mL}$.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$112 \mathrm{~mL}$.</span> </div>

answer:
<div class="solution">Acetic acid on electrolysis gives ethane as<br/>Given moles of acetic acid,<br/>$$<br/>\mathrm{CH}_{3} \mathrm{COOH}=\frac{0.1 \times 100}{1000}=0.01<br/>$$<br/>$\because 2 \mathrm{~mol} \mathrm{CH}_{3} \mathrm{COOH}$ gives $\mathrm{C}_{2} \mathrm{H}_{6}=22.4 \mathrm{~L}$<br/>$\therefore 0.01 \mathrm{~mol} \mathrm{CH} \mathrm{CHOH}_{3} \mathrm{COOH}$ give<br/>$$<br/>\begin{aligned}<br/>\mathrm{C}_{2} \mathrm{H}_{6} &amp;=\frac{22.4 \times 0.01}{2} \\<br/>&amp;=0.112 \mathrm{~L} \\<br/>&amp;=112 \mathrm{~mL}<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'AP EAMCET', 'AP EAMCET 2021 (23 Aug Shift 1)']
source_db: MarksBatch1_P1.db
title:
100-ml-of-02-m-acetic-acid-is-completely-neutralised-using-a-standard-solution-of-naoh-the-volume-of-ethane-obtained-at-stp-after-complete-electrolysi

slug:
100-ml-of-02-m-acetic-acid-is-completely-neutralised-using-a-standard-solution-of-naoh-the-volume-of-ethane-obtained-at-stp-after-complete-electrolysi-64022

question:
<div class="question">$100 \mathrm{~mL}$ of $0.2 \mathrm{M}$ acetic acid is completely neutralised using a standard solution of $\mathrm{NaOH}$. The volume of ethane obtained at STP after complete electrolysis of the resulting solution is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$11.2 \mathrm{~L}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2.24 \mathrm{~L}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.224 \mathrm{~L}$</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">D</span> <span class="option-data">$22.4 \mathrm{~L}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.224 \mathrm{~L}$</span> </div>

answer:
<div class="solution">Given moles of acetic acid<br/>$$<br/>\begin{array}{r}<br/>=\frac{0.2 \times 100}{1000}=0.02 \text { moles } \\<br/>2 \mathrm{CH}_3 \mathrm{COOH} \longrightarrow \mathrm{C}_2 \mathrm{H}_6+2 \mathrm{CO}_2+\mathrm{H}_2<br/>\end{array}<br/>$$<br/>2 moles of $\mathrm{CH}_3 \mathrm{COOH}$ gives 1 mole of ethane $=22.4 \mathrm{~L}$<br/>0.02 moles of $\mathrm{CH}_3 \mathrm{COOH}$ will give ethane<br/>$$<br/>=\frac{22.4 \times 0.02}{2}=0.224 \mathrm{~L}<br/>$$<br/>$\therefore 0.224 \mathrm{~L}$ of ethane is obtained.</div>


tags: ['Chemistry', 'Thermodynamics (C)']
options: None
correct_option: None
source_db: MarksBatch1_P1.db
title:
180-g-of-water-completely-vaporizes-at-10-0-c-and-1-bar-pressure-and-the-enthalpy-change-in-the-process-is-4079-kj-mol-1-what-will-be-the-enthalpy-cha

slug:
180-g-of-water-completely-vaporizes-at-10-0-c-and-1-bar-pressure-and-the-enthalpy-change-in-the-process-is-4079-kj-mol-1-what-will-be-the-enthalpy-cha-71043

question:
<div class="question">$18.0 \mathrm{~g}$ of water completely vaporizes at $100^{\circ} \mathrm{C}$ and 1 bar pressure and the enthalpy change in the process is $40.79 \mathrm{~kJ} \mathrm{~mol}^{-1}$. What will be the enthalpy change for vaporizing two moles of water under the same conditions? What is the standard enthalpy of vaporization for water?</div>

answer:
<div class="solution">$+81.58 \mathrm{~kJ}, \Delta_{\text {vap }} \mathrm{H}^{\circ}=+40.79 \mathrm{~kJ} \mathrm{~mol}^{-1}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch1_P1.db
title:
180-g-of-water-completely-vapourises-at-100c-and-1-bar-pressure-and-the-enthalpy-change-in-the-process-is-4079-kj-mol-1-what-will-be-the-enthalpy-chan

slug:
180-g-of-water-completely-vapourises-at-100c-and-1-bar-pressure-and-the-enthalpy-change-in-the-process-is-4079-kj-mol-1-what-will-be-the-enthalpy-chan-66285

question:
<div class="question">18.0 g of water completely vapourises at 100Â°C and 1 bar pressure and the enthalpy change in the process is 40.79 kJ mol<sup>â1</sup>. What will be the enthalpy change for vapourising two moles of water under the same conditions? What is the standard enthalphy of vapourisation for water?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Enthalpy change for vapourising two moles of water under the same conditions is +81.58 kJ and â<sub>vap</sub>H = +40.79 kJ mol<sup>â1</sup></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Enthalpy change for vapourising two moles of water under the same conditions is +69.58 kJ and â<sub>vap</sub>H = +40.79 kJ mol<sup>â1</sup></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Enthalpy change for vapourising two moles of water under the same conditions is +81.58 kJ and â<sub>vap</sub>H = +45.79 kJ mol<sup>â1</sup></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Enthalpy change for vapourising two moles of water under the same conditions is +69.58 kJ and â<sub>vap</sub>H = +45.79 kJ mol<sup>â1</sup></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Enthalpy change for vapourising two moles of water under the same conditions is +81.58 kJ and â<sub>vap</sub>H = +40.79 kJ mol<sup>â1</sup></span> </div>

answer:
<div class="solution">Correct Option is : (A)<br/>Enthalpy change for vapourising two moles of water under the same conditions is +81.58 kJ and â<sub>vap</sub>H = +40.79 kJ mol<sup>â1</sup></div>


