# questions

tags: ['Biology', 'Cell - The Unit of Life', 'NEET']
source_db: MarksBatch2_P3.db
title:
to-enter-or-leave-a-cell-the-substance-must-pass-through-and-its-structure-was-explained-by-in-the-year

slug:
to-enter-or-leave-a-cell-the-substance-must-pass-through-and-its-structure-was-explained-by-in-the-year-63907

question:
<div class="question">To enter or leave a cell, the substance must pass through __________ and its structure was explained by ___________ in the year __________.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><table border="1" cellpadding="1" cellspacing="1" style="width:100%;"> <tbody> <tr> <td style="text-align: center;">P</td> <td style="text-align: center;">Q</td> <td style="text-align: center;">R</td> </tr> <tr> <td style="text-align: center;">the cell wall</td> <td style="text-align: center;">the cell theory</td> <td style="text-align: center;">1978</td> </tr> </tbody></table></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><table border="1" cellpadding="1" cellspacing="1" style="width: 368px;"> <tbody> <tr> <td style="text-align: center;">P</td> <td style="text-align: center;">Q</td> <td style="text-align: center;">R</td> </tr> <tr> <td style="text-align: center;">the plasma membrane,</td> <td style="text-align: center;">the cell theory</td> <td style="text-align: center;">1942</td> </tr> </tbody></table></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><table border="1" cellpadding="1" cellspacing="1" style="width: 368px;"> <tbody> <tr> <td style="text-align: center;">P</td> <td style="text-align: center;">Q</td> <td style="text-align: center;">R</td> </tr> <tr> <td style="text-align: center;">the cell wall</td> <td style="text-align: center;">the fluid mosaic model</td> <td style="text-align: center;">1975</td> </tr> </tbody></table></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><table border="1" cellpadding="1" cellspacing="1" style="width:100%;"> <tbody> <tr> <td style="text-align: center;">P</td> <td style="text-align: center;">Q</td> <td style="text-align: center;">R</td> </tr> <tr> <td style="text-align: center;">the plasma membrane,</td> <td style="text-align: center;">the fluid mosaic model</td> <td style="text-align: center;">1972</td> </tr> </tbody></table></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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"><table border="1" cellpadding="1" cellspacing="1" style="width:100%;"> <tbody> <tr> <td style="text-align: center;">P</td> <td style="text-align: center;">Q</td> <td style="text-align: center;">R</td> </tr> <tr> <td style="text-align: center;">the plasma membrane,</td> <td style="text-align: center;">the fluid mosaic model</td> <td style="text-align: center;">1972</td> </tr> </tbody></table></span> </div>

answer:
<div class="solution">The plasma membrane is the outermost boundary of the cell and it is selectively permeable. It regulates the movement of substance within a cell. The most accepted model regarding the structure of the plasma membrane is the fluid mosaic model, which was first proposed in 1972. According to this model, the plasma membrane components are arranged in a mosaic pattern and primarily made up of phospholipids, cholesterol, and proteins that move freely by showing fluidly.</div>


tags: ['Chemistry', 'Electrochemistry', 'JEE Main', 'JEE Main 2017 (09 Apr Online)']
source_db: MarksBatch2_P3.db
title:
to-find-the-standard-potential-of-m-3-m-electrode-the-following-cell-is-constituted-pt-m-m-3-0-001-mol-l-1-ag-0-01-mol-l-1-ag-the-emf-of-the-cell-is-f

slug:
to-find-the-standard-potential-of-m-3-m-electrode-the-following-cell-is-constituted-pt-m-m-3-0-001-mol-l-1-ag-0-01-mol-l-1-ag-the-emf-of-the-cell-is-f-16499

question:
<div class="question">To find the standard potential of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">M</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>/</mo><mi mathvariant="normal">M</mi></math> electrode, the following cell is constituted: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Pt</mi><mo>/</mo><mi mathvariant="normal">M</mi><mo>/</mo><msup><mi mathvariant="normal">M</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mi mathvariant="normal"> </mi><mfenced separators="|"><mrow><mn>0</mn><mo>.001</mo><mi mathvariant="normal"> </mi><mi>mol</mi><mi mathvariant="normal"> </mi><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>/</mo><msup><mi>Ag</mi><mo>+</mo></msup><mi mathvariant="normal"> </mi><mfenced separators="|"><mrow><mn>0</mn><mo>.01</mo><mi mathvariant="normal"> </mi><mi>mol</mi><mi mathvariant="normal"> </mi><msup><mi mathvariant="normal">L</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>/</mo><mi>Ag</mi></math><br/><br/>The emf of the cell is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>421</mn><mo> </mo><mi>volt</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>298</mn><mo> </mo><mi mathvariant="normal">K</mi></math>. The standard potential of half-reaction <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">M</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>+</mo><mn>3</mn><msup><mi mathvariant="normal">e</mi><mo>-</mo></msup><mo>→</mo><mi mathvariant="normal">M</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>298</mn><mo> </mo><mi mathvariant="normal">K</mi></math> will be:<br/><br/>(Given: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi mathvariant="normal">E</mi><mfrac bevelled="true"><msup><mi>Ag</mi><mo>+</mo></msup><mi>Ag</mi></mfrac><mo>⊝</mo></msubsup></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>298</mn><mo> </mo><mi mathvariant="normal">K</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>80</mn><mo> </mo><mi>volt</mi></math>)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>38</mn><mo> </mo><mi>volt</mi></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>28</mn><mo> </mo><mi>volt</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>32</mn><mo> </mo><mi>volt</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>66</mn><mo> </mo><mi>volt</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>32</mn><mo> </mo><mi>volt</mi></math></span> </div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">E</mi><mi>cell</mi></msub><mo>=</mo><msub><msup><mi mathvariant="normal">E</mi><mi mathvariant="normal">o</mi></msup><mi>cell</mi></msub><mo>−</mo><mfenced><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0591</mn></mrow><mi mathvariant="normal">n</mi></mfrac></mfenced><mi>logQ</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mtext> </mtext><mi mathvariant="normal">Q</mi><mo>=</mo><mfrac><mfenced><msup><mi mathvariant="normal">M</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup></mfenced><msup><mfenced><msup><mi>Ag</mi><mo>+</mo></msup></mfenced><mn>3</mn></msup></mfrac><mo> </mo><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.001</mo></mrow><msup><mrow><mo>(</mo><mn>0</mn><mo>.01</mo><mo>)</mo></mrow><mn>3</mn></msup></mfrac><mo>=</mo><msup><mn>10</mn><mn>3</mn></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">E</mi><mi>cell</mi></msub><mo>=</mo><msub><msup><mi mathvariant="normal">E</mi><mi mathvariant="normal">o</mi></msup><mi>cell</mi></msub><mo>−</mo><mfenced><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0591</mn></mrow><mn>3</mn></mfrac></mfenced><mi>log</mi><msup><mn>10</mn><mn>3</mn></msup></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">E</mi><mi>cell</mi></msub><mo>=</mo><msub><msup><mi mathvariant="normal">E</mi><mi mathvariant="normal">o</mi></msup><mi>cell</mi></msub><mo>−</mo><mfenced><mfrac><mrow><mn>0</mn><mo>.</mo><mn>0591</mn></mrow><mn>3</mn></mfrac></mfenced><mo>×</mo><mn>3</mn><mo>=</mo><mn>0</mn><mo>.421</mo><mspace linebreak="newline"></mspace><msub><msup><mi mathvariant="normal">E</mi><mi mathvariant="normal">o</mi></msup><mi>cell</mi></msub><mo>−</mo><mn>0</mn><mo>.0591</mo><mo>=</mo><mn>0</mn><mo>.</mo><mn>421</mn><mspace linebreak="newline"></mspace><msub><msup><mi mathvariant="normal">E</mi><mi mathvariant="normal">o</mi></msup><mi>cell</mi></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>48</mn><mo> </mo><mi mathvariant="normal">V</mi><mo> </mo><mo>=</mo><msub><msup><mi mathvariant="normal">E</mi><mi mathvariant="normal">o</mi></msup><mrow><msup><mi>Ag</mi><mo>+</mo></msup><mo>/</mo><mi>Ag</mi></mrow></msub><mo>-</mo><msub><msup><mi mathvariant="normal">E</mi><mi mathvariant="normal">o</mi></msup><mrow><msup><mi mathvariant="normal">M</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>/</mo><mi mathvariant="normal">M</mi></mrow></msub><mspace linebreak="newline"></mspace><msub><msup><mi mathvariant="normal">E</mi><mi mathvariant="normal">o</mi></msup><mrow><msup><mi mathvariant="normal">M</mi><mrow><mn>3</mn><mo>+</mo></mrow></msup><mo>/</mo><mi mathvariant="normal">M</mi></mrow></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>80</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>48</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>32</mn><mo> </mo><mi mathvariant="normal">V</mi></math></div>


tags: ['Chemistry', 'Surface Chemistry', 'TS EAMCET', 'TS EAMCET 2020 (11 Sep Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.01</span> </div>
source_db: MarksBatch2_P3.db
title:
to-resist-the-coagulation-of-100-cc-gold-sol-1-cc-of-10-nacl-is-added-to-it-in-the-presence-of-1-0-4-g-gelatin-the-gold-number-of-gelatin-is

slug:
to-resist-the-coagulation-of-100-cc-gold-sol-1-cc-of-10-nacl-is-added-to-it-in-the-presence-of-1-0-4-g-gelatin-the-gold-number-of-gelatin-is-60626

question:
<div class="question">To resist the coagulation of $100 \mathrm{cc}$ gold sol; 1 cC of $10 \% \mathrm{NaCl}$ is added to it in the presence of $10^{-4} \mathrm{~g}$ gelatin. The gold number of gelatin is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.00001</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.0001</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.001</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0.01</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Gold number is the amount (in milligrams) of aprotective colloid, which is just sufficient to prevent the coagulation of $10 \mathrm{~mL}$ of standard hydro gold sol, when $1 \mathrm{~mL}$ of $10 \% \mathrm{NaCl}$ solution is added to it.<br/>$\because 10 \mathrm{cc}$ gold sol $=1 \mathrm{cc}$ of $10 \% \mathrm{NaCl}$<br/>$\therefore 100 \mathrm{~mL}$ of gold $\mathrm{sol}=1 \mathrm{cc}$ of gelatin<br/>Therefore, gold number of gelatin is<br/>$\frac{1 \times 10}{100}=0.01$</div>


tags: ['Physics', 'System of Particles and Rotational Motion']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
torques-of-equal-magnitude-are-applied-to-a-hollow-cylinder-and-a-solid-sphere-both-having-the-same-mass-and-radius-the-cylinder-is-free-to-rotate-abo

slug:
torques-of-equal-magnitude-are-applied-to-a-hollow-cylinder-and-a-solid-sphere-both-having-the-same-mass-and-radius-the-cylinder-is-free-to-rotate-abo-97589

question:
<div class="question">Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about its axis passing through its centre. Which of the two will acquire a greater angular speed?</div>

answer:
<div class="solution">$I_1, M . I$ of a hollow cylinder about its axis $=M R^2$<br/>$I_2, M . I$ of a solid sphere about an axis passing through its centre $=\frac{2}{5} M R^2$<br/>$\because \quad$ Applied torque on the cylinder and the sphere is<br/>$$<br/>\begin{array}{ll} <br/>&amp; \text { same } \\<br/>\therefore \quad \tau=I_1 \alpha_1=I_2 \alpha_2 \\<br/>\therefore &amp; \frac{\alpha_2}{\alpha_1}=\frac{I_1}{I_2}=\frac{M R^2}{\frac{2}{5} M R^2}=\frac{5}{2} \\<br/>\Rightarrow &amp; \therefore \alpha_2&gt;\alpha_1\left[\alpha_2=2.5 \alpha_1\right] \\<br/>\because &amp; \omega=\omega_0+\alpha t, \text { where } \omega_0 \text { and } t \text { are constants } \\<br/>\therefore \quad &amp; \omega_2&gt;\omega_1 \\<br/>\end{array}<br/>$$<br/>$\therefore \quad$ Angular speed of the solid sphere is greater than that of the cylinder.</div>


tags: ['Biology', 'Breathing and Exchange of Gases', 'NEET']
source_db: MarksBatch2_P3.db
title:
trachea-is-a-straight-tube-extending-upto-the-mid-thoracic-cavity-which-divides-into-right-and-left-bronchi-at-the-level-of

slug:
trachea-is-a-straight-tube-extending-upto-the-mid-thoracic-cavity-which-divides-into-right-and-left-bronchi-at-the-level-of-41876

question:
<div class="question">Trachea is a straight tube extending upto the mid thoracic cavity, which divides into right and left bronchi at the level of</div>

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

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

answer:
<div class="solution">Human respiratory system consists of external nostrils, nasal chamber, pharynx, trachea, bronchi, bronchioles and Lungs. Pharynx opens into trachea through larynx. Trachea is a straight tube in the thoracic cavity and at the level of fifth thoracic vertebra it divides into right and left bronchi which enters the respective Lung. The region at which the division takes place is known as Carina. Primary bronchi when enters the lung further divides into secondary and tertiary bronchi which then forms bronchioles which further extends as alveoli.</div>


tags: ['Biology', 'Biotechnology and Its Applications']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
transgenic-animals-are-the-animals-in-which-a-foreign-gene-is-expressed-such-animals-can-be-used-to-study-the-fundamental-biological-process-phenomeno

slug:
transgenic-animals-are-the-animals-in-which-a-foreign-gene-is-expressed-such-animals-can-be-used-to-study-the-fundamental-biological-process-phenomeno-39174

question:
<div class="question">Transgenic animals are the animals in which a foreign gene is expressed. Such animals can be used to study the fundamental biological process, phenomenon as well as for producing products useful for mankind. Give one example for each type.</div>

answer:
<div class="solution">The animals in which foreign genes are expressed are transgenic animals. The fundamental biological process/ phenomenon, like using model organisms like mouse and determining how genes are regulated (gene regulation), how they affect the normal functions of the body and its development, etc. are studied using such animals.<br/>Transgenic animals are also used for producing products useful for mankind, e.g., transgenic cow (rosie) produced human protein enriched milk $(2.4 \mathrm{~g} / \mathrm{L})$. The milk contained the human alpha-lactalbumin protein and was nutritionally a more balanced product for human babies that cow-milk.</div>


tags: ['Biology', 'Anatomy of Flowering Plants']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
trunks-of-some-of-the-aged-tree-species-appear-to-be-composed-of-several-fused-trunks-is-it-a-physiological-or-anatomical-abnormality-explain-in-detai

slug:
trunks-of-some-of-the-aged-tree-species-appear-to-be-composed-of-several-fused-trunks-is-it-a-physiological-or-anatomical-abnormality-explain-in-detai-60981

question:
<div class="question">Trunks of some of the aged tree species appear to be composed of several fused trunks. Is it a physiological or anatomical abnormality? Explain in detail.</div>

answer:
<div class="solution">The appearance of several fused trunks is anatomical abnormality. It is an abnormal type of secondary growth where a regular vascular cambium or cork cambium is not formed in its normal position. Anomalous secondary growth produces cortical and medullary vascular bundles in case of old tree trunks.<br/>Thus, the additional or accessory vascular bundles given appurtenance of several fused trunks.</div>


tags: ['Physics', 'Waves and Sound', 'VITEEE', 'VITEEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2: 1$</span> </div>
source_db: MarksBatch2_P3.db
title:
tube-a-has-both-ends-open-while-tube-b-has-one-end-closed-otherwise-they-are-identical-the-ratio-of-fundamental-frequency-of-tube-a-and-b-is-1

slug:
tube-a-has-both-ends-open-while-tube-b-has-one-end-closed-otherwise-they-are-identical-the-ratio-of-fundamental-frequency-of-tube-a-and-b-is-1-43618

question:
<div class="question">Tube $A$ has both ends open while tube $B$ has one end closed, otherwise they are identical. The ratio of fundamental frequency of tube $A$ and $B$ is</div>

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

answer:
<div class="solution">Fundamental freq. for,
<br/>Tube $\mathrm{A} \Rightarrow \mathrm{f}_{\mathrm{A}}=\frac{\mathrm{v}}{2 \mathrm{~L}}$
<br/>Tube $\mathrm{B} \Rightarrow \mathrm{f}_{\mathrm{B}}=\frac{\mathrm{v}}{4 \mathrm{~L}}$
<br/>Now,
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{\mathrm{f}_{\mathrm{A}}}{\mathrm{f}_{\mathrm{B}}}=\frac{\mathrm{v}}{2 \mathrm{~L}} \times \frac{4 \mathrm{~L}}{\mathrm{v}}=\frac{2}{1} \\
<br/>\mathrm{f}_{\mathrm{A}}: \mathrm{f}_{\mathrm{B}}=2: 1
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2: 1$</span> </div>
source_db: MarksBatch2_P3.db
title:
tube-a-has-both-ends-open-while-tube-b-has-one-end-closed-otherwise-they-are-identical-the-ratio-of-fundamental-frequency-of-tube-a-and-b-is-2

slug:
tube-a-has-both-ends-open-while-tube-b-has-one-end-closed-otherwise-they-are-identical-the-ratio-of-fundamental-frequency-of-tube-a-and-b-is-2-38341

question:
<div class="question">Tube $A$ has both ends open while tube $B$ has one end closed, otherwise they are identical. The ratio of fundamental frequency of tube $A$ and $B$ is</div>

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

answer:
<div class="solution">Fundamental freq. for,
<br/>Tube $\mathrm{A} \Rightarrow \mathrm{f}_{\mathrm{A}}=\frac{\mathrm{v}}{2 \mathrm{~L}}$
<br/>Tube $\mathrm{B} \Rightarrow \mathrm{f}_{\mathrm{B}}=\frac{\mathrm{v}}{4 \mathrm{~L}}$
<br/>Now,
<br/>$$
<br/>\begin{array}{l}
<br/>\frac{\mathrm{f}_{\mathrm{A}}}{\mathrm{f}_{\mathrm{B}}}=\frac{\mathrm{v}}{2 \mathrm{~L}} \times \frac{4 \mathrm{~L}}{\mathrm{v}}=\frac{2}{1} \\
<br/>\mathrm{f}_{\mathrm{A}}: \mathrm{f}_{\mathrm{B}}=2: 1
<br/>\end{array}
<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2002']
source_db: MarksBatch2_P3.db
title:
tube-a-has-both-ends-open-while-tube-b-has-one-end-closed-otherwise-they-are-identical-the-ratio-of-fundamental-frequency-of-tube-a-and-b-is

slug:
tube-a-has-both-ends-open-while-tube-b-has-one-end-closed-otherwise-they-are-identical-the-ratio-of-fundamental-frequency-of-tube-a-and-b-is-61525

question:
<div class="question">Tube A has both ends open while tube B has one end closed, otherwise they are identical. The ratio of fundamental frequency of tube $A$ and $B$ is</div>

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

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

answer:
<div class="solution">$\frac{\lambda_{\mathrm{A}}}{\lambda_{\mathrm{B}}}=\frac{1}{2} \Rightarrow \frac{\mathrm{n}_{\mathrm{A}}}{\mathrm{n}_{\mathrm{B}}}=\frac{2}{1}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/UbF_Q73kI0x8nlWIq8oX1rlrrpCiam7tgYd5j-PqROE.original.fullsize.png"/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/_pakG1ZjC7gWfapWtmxjvJSP8XEJSLiBGUW7E32UxJI.original.fullsize.png"/><br/></div>


title: two-angles-of-a-triangle-are-tan-1-2-1-and-tan-1-3-1-what-is-the-third-angle
slug: two-angles-of-a-triangle-are-tan-1-2-1-and-tan-1-3-1-what-is-the-third-angle-70002
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">Two angles of a triangle are $\tan ^{-1} \frac{1}{2}$ and $\tan ^{-1} \frac{1}{3}$. What is the third angle?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$30^{\circ}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$45^{\circ}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$90^{\circ}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$135^{\circ}$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$135^{\circ}$</span> </div>

answer:
<div class="solution">In any $\Delta \mathrm{ABC}$ $\angle \mathrm{A}+\angle \mathrm{B}+\angle \mathrm{C}=\pi$<br/>$\Rightarrow$ let $A=\tan ^{-1} \frac{1}{2}$ and $B=\tan ^{-1} \frac{1}{3}$<br/>$\Rightarrow \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{3}+\angle \mathrm{C}=\pi$<br/>$\Rightarrow \tan ^{-1}\left(\frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{6}}\right)+\angle \mathrm{C}=\pi$<br/>$\left[\because \tan ^{-1} \mathrm{x}+\tan ^{-1} \mathrm{y}=\tan ^{-1}\left(\frac{\mathrm{x}+\mathrm{y}}{1-\mathrm{xy}}\right)\right]$<br/>$\Rightarrow \tan ^{-1}\left(\frac{5 / 6}{5 / 6}\right)+\angle \mathrm{C}=\pi \Rightarrow \frac{\pi}{4}+\angle \mathrm{C}=\pi$<br/>$\Rightarrow \angle \mathrm{C}=\pi-\frac{\pi}{4}=\frac{3 \pi}{4}=135^{\circ}$</div>


title: two-bacteria-found-to-be-very-useful-in-genetic-engineering-experiments-are
slug: two-bacteria-found-to-be-very-useful-in-genetic-engineering-experiments-are-62858
tags: ['Biology', 'Biotechnology - Principles and Processes', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">Two bacteria found to be very useful in genetic engineering experiments are:</div>

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

answer:
<div class="solution"><em>E.coli</em> contains many important standard cloning vectors widely used in gene cloning experiments like pBR322 contains origin of replication (ori). Other cloning vectors like PACYC177, pBR324, PRK 64.6 contain ampicillin resistance gene they are also found in <em>E.coli.</em> Among higher plants, Ti plasmid of <em>Agrobacterium tumefaciens</em> and Ri plasmid of <em>A. rhizogenes</em> is the best-known vector. TDNA from Ti or Ri plasmid of <em>Agrobacterium</em> is considered to be a very potential vector for cloning experiments with higher plants.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-balls-are-selected-from-a-box-containing-2-blue-and-7-red-balls-what-is-the-probability-that-at-least-one-ball-is-blue

slug:
two-balls-are-selected-from-a-box-containing-2-blue-and-7-red-balls-what-is-the-probability-that-at-least-one-ball-is-blue-76320

question:
<div class="question">Two balls are selected from a box containing 2 blue and 7 red balls. What is the probability that at least one ball is blue?</div>

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

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

answer:
<div class="solution">No. of blue balls $=2$ No. of red balls $=7$<br/>Total no. of balls $=9$<br/>Required probability $=P($ one ball is blue $)+P$ (both ball is blue)<br/>$$<br/>=\frac{2}{9} \times \frac{7}{8}+\frac{2}{9} \times \frac{1}{8}=\frac{14}{72}+\frac{2}{72}=\frac{16}{72}=\frac{2}{9}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-cards-are-chosen-at-random-from-a-deck-of-52-playig-cards-what-is-the-probability-that-both-of-them-have-the-same-value

slug:
two-cards-are-chosen-at-random-from-a-deck-of-52-playig-cards-what-is-the-probability-that-both-of-them-have-the-same-value-18001

question:
<div class="question">Two cards are chosen at random from a deck of 52 playig cards. What is the probability that both of them have the same value?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{17}$</span> </div>

answer:
<div class="solution">Number of ways of selecting 2 cards from deck of cards $={ }^{52} \mathrm{C}_{2}$<br/>favourable cases $={ }^{13} \mathrm{C}_{2}$<br/>Probability $=\frac{13}{52} C_{2}=\frac{\frac{13 \times 12}{2}}{\frac{52 \times 51}{2}}=\frac{13 \times 12}{52 \times 51}=\frac{1}{17}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-cards-are-drawn-successively-without-replacement-from-a-wellshuffled-pack-of-52-cards-the-probability-of-drawing-two-aces-is

slug:
two-cards-are-drawn-successively-without-replacement-from-a-wellshuffled-pack-of-52-cards-the-probability-of-drawing-two-aces-is-57198

question:
<div class="question">Two cards are drawn successively without replacement from a wellshuffled pack of 52 cards. The probability of drawing two aces is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{26}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{221}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{4}{223}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{13}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{221}$</span> </div>

answer:
<div class="solution">For the nexttwo (02) items that follow<br/>$\begin{aligned} &amp; \mathrm{P}(\mathrm{A})=\frac{{ }^{4} \mathrm{C}_{1}}{{ }^{52} \mathrm{C}_{1}}=\frac{4}{52}=\frac{1}{13} \\ &amp; \mathrm{P}(\mathrm{B} / \mathrm{A})=\frac{{ }^{3} \mathrm{C}_{1}}{{ }^{51} \mathrm{C}_{1}}=\frac{3}{51}=\frac{1}{17} \\ \therefore \quad &amp; \text { Required probability }=\frac{1}{13} \times \frac{1}{17}=\frac{1}{221} \\ \therefore \quad &amp; \text { Option (b) is correct. } \end{aligned}$</div>


tags: ['Physics', 'Current Electricity', 'AP EAMCET', 'AP EAMCET 2004']
source_db: MarksBatch2_P3.db
title:
two-cells-a-and-b-are-connected-in-the-secondary-circuit-of-a-potentiometer-one-at-a-time-and-the-balancing-length-are-respectively-400-cm-and-440-cm--1

slug:
two-cells-a-and-b-are-connected-in-the-secondary-circuit-of-a-potentiometer-one-at-a-time-and-the-balancing-length-are-respectively-400-cm-and-440-cm-1-75453

question:
<div class="question">Two cells $A$ and. $B$ are connected in the secondary circuit of a potentiometer one at a time and the balancing length are respectively $400 \mathrm{~cm}$ and $440 \mathrm{~cm}$. The emf of the cell $A$ is 1.08 volt. The emf of the second cell $B$ in volts is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.08$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.188$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$11.88$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$12.8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.188$</span> </div>

answer:
<div class="solution">$\begin{array}{lc}\frac{E_1}{E_2}=\frac{l_1}{l_2} &amp; \\ \Rightarrow &amp; \frac{1.08}{E_2}=\frac{400}{440} \\ \Rightarrow &amp; E_2=\frac{440 \times 1.08}{400}=1.188 \mathrm{~V}\end{array}$</div>


tags: ['Physics', 'Current Electricity', 'TS EAMCET', 'TS EAMCET 2004']
source_db: MarksBatch2_P3.db
title:
two-cells-a-and-b-are-connected-in-the-secondary-circuit-of-a-potentiometer-one-at-a-time-and-the-balancing-length-are-respectively-400-cm-and-440-cm-

slug:
two-cells-a-and-b-are-connected-in-the-secondary-circuit-of-a-potentiometer-one-at-a-time-and-the-balancing-length-are-respectively-400-cm-and-440-cm-20785

question:
<div class="question">Two cells $A$ and. $B$ are connected in the secondary circuit of a potentiometer one at a time and the balancing length are respectively $400 \mathrm{~cm}$ and $440 \mathrm{~cm}$. The emf of the cell $A$ is 1.08 volt. The emf of the second cell $B$ in volts is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1.08$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$1.188$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$11.88$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$12.8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.188$</span> </div>

answer:
<div class="solution">$\begin{array}{lc}\frac{E_1}{E_2}=\frac{l_1}{l_2} &amp; \\ \Rightarrow &amp; \frac{1.08}{E_2}=\frac{400}{440} \\ \Rightarrow &amp; E_2=\frac{440 \times 1.08}{400}=1.188 \mathrm{~V}\end{array}$</div>


tags: ['Physics', 'Current Electricity', 'AP EAMCET', 'AP EAMCET 2022 (07 Jul Shift 2)']
source_db: MarksBatch2_P3.db
title:
two-cells-a-and-b-are-connected-in-the-secondary-circuit-of-a-potentiometer-one-at-a-time-and-the-balancing-lengths-are-respectively-360-cm-and-420-cm

slug:
two-cells-a-and-b-are-connected-in-the-secondary-circuit-of-a-potentiometer-one-at-a-time-and-the-balancing-lengths-are-respectively-360-cm-and-420-cm-72045

question:
<div class="question">Two cells $A$ and $B$ are connected in the secondary circuit of a potentiometer one at a time and the balancing lengths are respectively $360 \mathrm{~cm}$ and $420 \mathrm{~cm}$. If emf of $A$ is $2.4 \mathrm{~V}$, the emf of the second cell $B$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2.8 \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$3.2 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3.0 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.6 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.8 \mathrm{~V}$</span> </div>

answer:
<div class="solution">If $E_1$ and $E_2$ are emf's of cells with $l_1$ and $l_2$ balance lengths in a potentiometer experiment, then, $\frac{E_1}{E_2}=\frac{l_1}{l_2}$<br/>or $\quad E_2=E_1 \cdot \frac{l_2}{l_1}$<br/>Here, $E_1=24 \mathrm{~V}, l_1=360 \mathrm{~cm}$ and $l_2=420 \mathrm{~cm}$ Substituting values in Eq. (i), we get<br/>$$<br/>E_2=24 \times \frac{420}{360}=28 \mathrm{~V}<br/>$$<br/><br/>So, emf of cell $B$ is $2.8 \mathrm{~V}$.</div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
two-cells-of-emf-e-1-and-e-2-e-2-e-1-are-connected-in-series-in-the-secondary-circuit-of-a-potentiometer-experiment-for-determination-of-emf-the-balan

slug:
two-cells-of-emf-e-1-and-e-2-e-2-e-1-are-connected-in-series-in-the-secondary-circuit-of-a-potentiometer-experiment-for-determination-of-emf-the-balan-44905

question:
<div class="question">Two cells of E.M.F. <math><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal"> </mi><mfenced separators="|"><mrow><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal"> </mi><mo>&gt;</mo><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfenced></math> are connected in series in the secondary circuit of a potentiometer experiment for determination of E.M.F. The balancing length is found to be <math><mn>825</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">m</mi></math>. Now when the terminals of cell <math><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>1</mn></mrow></msub></math> are reversed, then the balancing length is found to be <math><mn>225</mn><mi mathvariant="normal"> </mi><mi mathvariant="normal">c</mi><mi mathvariant="normal">m</mi></math>. The ratio of <math><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>2</mn></mrow></msub></math> is</div>

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

answer:
<div class="solution"><math><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal"> </mi><mo>+</mo><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal"> </mi><mo>=</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">x</mi><mfenced separators="|"><mrow><mn>825</mn></mrow></mfenced></math><br/><br/><math><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>2</mn></mrow></msub><mi mathvariant="normal"> </mi><mo>â</mo><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">E</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal"> </mi><mo>=</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal">x</mi><mfenced separators="|"><mrow><mn>225</mn></mrow></mfenced></math><br/><br/>Dividing above equations, we get<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msub><mi mathvariant="normal">E</mi><mn>2</mn></msub><mo>+</mo><msub><mi mathvariant="normal">E</mi><mn>1</mn></msub></mrow><mrow><msub><mi mathvariant="normal">E</mi><mn>2</mn></msub><mo>-</mo><msub><mi mathvariant="normal">E</mi><mn>1</mn></msub></mrow></mfrac><mo>=</mo><mfrac><mn>11</mn><mn>3</mn></mfrac></math><br/><br/>Appling componendo and dividendo, we get<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi mathvariant="normal">E</mi><mn>1</mn></msub><msub><mi mathvariant="normal">E</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><mn>4</mn><mn>7</mn></mfrac></math></div>


tags: ['Physics', 'Electrostatics', 'NEET', 'NEET 2005']
source_db: MarksBatch2_P3.db
title:
two-charge-q-1-and-q-2-are-placed-30-cm-apart-as-shown-in-the-figure-a-third-charge-q-3-is-moved-along-the-are-of-a-circle-of-radius-40-cm-from-c-to-d

slug:
two-charge-q-1-and-q-2-are-placed-30-cm-apart-as-shown-in-the-figure-a-third-charge-q-3-is-moved-along-the-are-of-a-circle-of-radius-40-cm-from-c-to-d-58772

question:
<div class="question">Two charge $q_1$ and $q_2$ are placed $30 \mathrm{~cm}$ apart, as shown in the figure. A third charge $q_3$ is moved along the are of a circle of radius $40 \mathrm{~cm}$ from $\mathrm{C}$ to $\mathrm{D}$. The change in the potential energy of the system is \(\frac{q_3 k}{4 \pi \in_0}\), where $k$ is:<br/><img src="https://cdn-question-pool.getmarks.app/pyq/neet/sXTwOhTvNzPIVwBPTsyiCApMZIg5kKhj3Zz7zTvlmkY.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$8 q_1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$6 q_1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$8 q_2$</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">$6 q_2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$8 q_2$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; V_i=\frac{1}{4 \pi \epsilon_0}\left[\frac{q_1 q_3}{(0.4)}+\frac{q_1 q_2}{(0.3)}+\frac{q_2 q_3}{(0.5)}\right] \\<br/>&amp; v_f=\frac{1}{4 \pi \epsilon_0}\left[\frac{q_1 q_3}{(0.4)}+\frac{q_1 q_2}{(0.3)}+\frac{q_2 q_3}{(0.1)}\right]<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/neet/b-AshgV8dWUrC-Ijg1D2Sqx_z4S_pv6TJbwVknDaZ4o.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>&amp; \Rightarrow \Delta V=V f-V_1 \\<br/>&amp; =\frac{1}{4 \pi \epsilon_0} q_2 q_3\left(\frac{1}{0.1}-\frac{1}{0.5}\right) \\<br/>&amp; =\frac{q_2 q_3}{\pi \epsilon_0}\left(10^{-2}\right)=\frac{q_3}{4 \pi \epsilon_0}\left(8 q_2\right) \\<br/>&amp; \Rightarrow k=8 q_2<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'AIIMS', 'AIIMS 2008']
source_db: MarksBatch2_P3.db
title:
two-closed-organ-pipes-of-length-100-mathrmcm-and-101-mathrmcm-produces-16-beats-in-20-mathrmsec-when-each-pipe-is-sounded-in-its-fundamental-mode-cal

slug:
two-closed-organ-pipes-of-length-100-mathrmcm-and-101-mathrmcm-produces-16-beats-in-20-mathrmsec-when-each-pipe-is-sounded-in-its-fundamental-mode-cal-30117

question:
<div class="question">Two closed organ pipes of length \(100 \mathrm{~cm}\) and \(101 \mathrm{~cm}\) produces 16 beats in \(20 \mathrm{sec}\). When each pipe is sounded in its fundamental mode calculate the velocity of sound.</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(303 \mathrm{~ms}^{-1}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(332 \mathrm{~ms}^{-1}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(323.2 \mathrm{~ms}^{-1}\)</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">\(300 \mathrm{~ms}^{-1}\).</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(323.2 \mathrm{~ms}^{-1}\)</span> </div>

answer:
<div class="solution">Frequency of fundamental mode of vibrations of $1^{\text {st }}$ closed organ pipe,<br/><img src="https://cdn-question-pool.getmarks.app/pyq/aiims/xT5eWEEKlK4X0eJtTWJB8jpNP_PUO1ydSeSA47fFs3Q.original.fullsize.png"/><br/><br/>From question, $v_1-v_2=\frac{16}{20}$<br/>$\begin{aligned} &amp; \Rightarrow \frac{v}{4 \times 100}-\frac{v}{4 \times 101}=\frac{16}{20} \\ &amp; \Rightarrow \frac{v}{4} \cdot\left[\frac{1}{100}-\frac{1}{101}\right]=\frac{16}{20} \\ &amp; \Rightarrow \frac{v}{4}\left[\frac{101-100}{101 \times 100}\right]=\frac{16}{20} \Rightarrow \frac{v}{4} \times \frac{1}{101 \times 100}=\frac{16}{20} \\ &amp; \Rightarrow v=\frac{16 \times 101 \times 4 \times 100}{20}=32320 \mathrm{~cm} / \mathrm{s} . \\ &amp; =323.2 \mathrm{~m} / \mathrm{s} .\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1:4$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-closed-pipes-have-the-same-fundamental-frequency-one-is-filled-with-oxygen-and-the-other-with-hydrogen-at-the-same-temperature-ratio-of-their-leng

slug:
two-closed-pipes-have-the-same-fundamental-frequency-one-is-filled-with-oxygen-and-the-other-with-hydrogen-at-the-same-temperature-ratio-of-their-leng-76890

question:
<div class="question">Two closed pipes have the same fundamental frequency. One is filled with oxygen and the other with hydrogen at the same temperature. Ratio of their lengths respectively is</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">$4:1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1:2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2:1$</span> </li> </ul>

answer:
<div class="solution">The fundamental frequency,<br/>$\begin{aligned}<br/>&amp; f=\frac{v}{4 l}=\frac{1}{4 e} \sqrt{\frac{\gamma R T}{M_0}} \\<br/>&amp; \ell \propto \frac{1}{\sqrt{M}} \Rightarrow \frac{\ell_1}{\ell_2}=\sqrt{\frac{M_1}{M_2}}=\sqrt{\frac{2}{32}}=\frac{1}{4}<br/>\end{aligned}$<br/>So, $\ell_1: \ell_2=1: 4$</div>


tags: ['Physics', 'Electromagnetic Induction', 'MHT CET', 'MHT CET 2020 (12 Oct Shift 1)']
source_db: MarksBatch2_P3.db
title:
two-coaxial-coils-a-and-b-of-radii-r-and-rz-are-placed-in-the-same-plane-r-2-r-1-if-a-current-is-passed-through-coil-b-the-coefficient-of-mutual-induc

slug:
two-coaxial-coils-a-and-b-of-radii-r-and-rz-are-placed-in-the-same-plane-r-2-r-1-if-a-current-is-passed-through-coil-b-the-coefficient-of-mutual-induc-63276

question:
<div class="question">Two coaxial coils A and B of radii 'R ' and 'Rz' are placed in the same plane. (R $_{2}&gt;$<br/>$\mathrm{R}_{1}$ ). If a current is passed through coil $\mathrm{B}$, the coefficient of mutual inductance<br/>between the coils is proportional to</div>

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

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

answer:
<div class="solution">Magnetic field at the centre of primary coil<br/>$$<br/>B=\mu_{0} i_{1} / 2 R_{1} .<br/>$$<br/>Considering it to be uniform, magnetic flux passing through secondary coil is<br/>$$<br/>\phi_{2}=B A=\frac{\mu_{0} i_{1}}{2 R_{1}}\left(\pi R_{2}^{2}\right)<br/>$$<br/>Now, $M=\frac{\phi_{2}}{i_{1}}=\frac{\mu_{0} \pi R_{2}^{2}}{2 R_{1}}$<br/>$$<br/>\therefore \quad M \propto \frac{R_{2}^{2}}{R_{1}} .<br/>$$</div>


title: two-common-characters-found-in-centipede-cockroach-and-crab-are
slug: two-common-characters-found-in-centipede-cockroach-and-crab-are-71359
question: <div class="question">Two common characters found in centipede, cockroach and crab are:</div>
tags: ['Biology', 'Animal Kingdom', 'NEET', 'NEET 2006']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">book lungs and antennae</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">compound eyes and anal cerci</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">jointed legs and chitinous exoskeleton</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">green gland and tracheae</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">jointed legs and chitinous exoskeleton</span> </div>

answer:
<div class="solution">Crab, centipede and cockroach belong to Phylum Arthropoda. These have jointed appendages and a chitinous exoskeleton.</div>


tags: ['Chemistry', 'Aldehydes and Ketones', 'JEE Main', 'JEE Main 2020 (02 Sep Shift 2)']
source_db: MarksBatch2_P3.db
title:
two-compounds-a-and-b-with-same-molecular-formula-c-3-h-6-o-undergo-grignard-reaction-with-methylmagnesium-bromide-to-give-products-c-and-d-products-c

slug:
two-compounds-a-and-b-with-same-molecular-formula-c-3-h-6-o-undergo-grignard-reaction-with-methylmagnesium-bromide-to-give-products-c-and-d-products-c-33517

question:
<div class="question"><p>Two compounds <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> with same molecular formula <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi mathvariant="normal">C</mi><mn>3</mn></msub><msub><mi mathvariant="normal">H</mi><mn>6</mn></msub><mi mathvariant="normal">O</mi></mrow></mfenced></math> undergo Grignard reaction with methylmagnesium bromide to give products <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">D</mi></math>. Products <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">D</mi></math> show following chemical tests.</p><table border="1" cellpadding="1" cellspacing="1" style="width: 500px;"> <tbody> <tr> <td style="text-align: center;"><strong>Test</strong></td> <td style="text-align: center;"><strong>C</strong></td> <td style="text-align: center;"><strong>D</strong></td> </tr> <tr> <td>Ceric ammonium nitrate Test</td> <td>Positive</td> <td>Positive</td> </tr> <tr> <td>Lucas Test</td> <td>Turbidity obtained after five minutes</td> <td>Turbidity obtained Immediately</td> </tr> <tr> <td>Iodoform Test</td> <td>Positive</td> <td>Negative</td> </tr> </tbody></table><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">C</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">D</mi></math> respectively are</p></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/727034c2-f87f-4240-aa35-c9ff30abc6c3-image.png" style="width: 400px; height: 104px;"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/6f8cc3c4-6435-4f84-be88-0b6fc6b2d9a8-image.png" style="width: 402px; height: 106px;"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/4159a267-278d-4f70-9400-a785c11403db-image.png" style="width: 440px; height: 97px;"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/a1c06898-3148-429a-90f4-0bf3acec4a53-image.png" style="width: 520px; height: 60px;"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/727034c2-f87f-4240-aa35-c9ff30abc6c3-image.png" style="width: 400px; height: 104px;"/></span> </div>

answer:
<div class="solution"><p>B will be ketone as it gives tertiary alcohol after reaction with grignard while A will be aldehyde which produces secondary alcohol after reaction with grignard reagent. </p><p><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/050585e9-d9e0-4396-a754-1f4cc85df7f7-image.png" style="width: 525px; height: 390px;"/></p></div>


tags: ['Physics', 'Electromagnetic Induction', 'AIIMS', 'AIIMS 2014']
source_db: MarksBatch2_P3.db
title:
two-conducting-circular-loops-of-radii-r-1-and-r-2-are-placed-in-the-same-plane-with-their-centres-coinciding-if-r-1-r-2-the-mutual-inductance-m-betwe-2

slug:
two-conducting-circular-loops-of-radii-r-1-and-r-2-are-placed-in-the-same-plane-with-their-centres-coinciding-if-r-1-r-2-the-mutual-inductance-m-betwe-2-35900

question:
<div class="question">Two conducting circular loops of radii $R_1$ and $R_2$ are placed in the same plane with their centres coinciding. If $R_1&gt;R_2$, the mutual inductance $M$ between them will be directly proportional to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{R_1}{R_2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{R_2}{R_1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{R_1^2}{R_2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{R_2^2}{R_1}$</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">$\frac{R_2^2}{R_1}$</span> </div>

answer:
<div class="solution">Magnetic field at the centre of primary coil $B=\mu_0 i_1 / 2 R_1$. Considering it to be uniform, magnetic flux passing through secondary coil is<br/>$\phi_2=B A=\frac{\mu_0 i_1}{2 R_1}\left(\pi R_2^2\right)$<br/>Now, $M=\frac{\phi_2}{i_1}=\frac{\mu_0 \pi R_2^2}{2 R_1} \quad \therefore \quad M \propto \frac{R_2^2}{R_1}$.</div>


tags: ['Physics', 'Electromagnetic Induction', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-conducting-circular-loops-of-radii-r-1-and-r-2-are-placed-in-the-same-plane-with-their-centres-coinciding-if-r-1-r-2-the-mutual-inductance-m-betwe

slug:
two-conducting-circular-loops-of-radii-r-1-and-r-2-are-placed-in-the-same-plane-with-their-centres-coinciding-if-r-1-r-2-the-mutual-inductance-m-betwe-70218

question:
<div class="question">Two conducting circular loops of radii <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>R</mi><mn>2</mn></msub></math> are placed in the same plane with their centres coinciding. If <math><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&gt;</mo><msub><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo></math> the mutual inductance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math> between them will be directly proportional to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfrac><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></math></span> </div>

answer:
<div class="solution">Let the magnetic field is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> at the centre of the primary coil. Considering it to be uniform, the magnetic flux passing through the secondary coil is<br/><br/><math><mi>ϕ</mi><mo>=</mo><mi>B</mi><mi>A</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mn>2</mn><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mfenced separators="|"><mrow><mi>π</mi><msubsup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></mfenced></math><br/><br/>Now, <math><mi>M</mi><mo>=</mo><mfrac><mrow><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>μ</mi></mrow><mrow><mn>0</mn></mrow></msub><mi>π</mi><msubsup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><mn>2</mn><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></math><br/><br/><math><mo>∴</mo><mi> </mi><mi> </mi><mi>M</mi><mo>∝</mo><mfrac><mrow><msubsup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msubsup></mrow><mrow><msub><mrow><mi>R</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></math></div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2021 (20 Sep Shift 1)']
source_db: MarksBatch2_P3.db
title:
two-consecutive-harmonics-of-an-air-column-in-pipe-closed-at-one-end-are-of-frequencies-150-hz-and-250-hz-the-fundament-frequency-of-an-air-column-is

slug:
two-consecutive-harmonics-of-an-air-column-in-pipe-closed-at-one-end-are-of-frequencies-150-hz-and-250-hz-the-fundament-frequency-of-an-air-column-is-16395

question:
<div class="question">Two consecutive harmonics of an air column in pipe closed at one end are of frequencies $150 \mathrm{~Hz}$ and $250 \mathrm{~Hz}$. The fundament frequency of an air column is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$25 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$75 \mathrm{~Hz}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$100 \mathrm{~Hz}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$50 \mathrm{~Hz}$</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> </ul>

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

answer:
<div class="solution">In a pipe closed at one end only odd harmonics of the fundamental are produced, i.e., $50 \mathrm{~Hz}, 150 \mathrm{~Hz}, 250 \mathrm{~Hz}$. Hence fundamental frequency is $50 \mathrm{~Hz}$</div>


tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-crosssections-of-stem-and-root-appear-simple-when-viewed-by-naked-eye-but-under-microscope-they-can-be-differentiated-by

slug:
two-crosssections-of-stem-and-root-appear-simple-when-viewed-by-naked-eye-but-under-microscope-they-can-be-differentiated-by-50550

question:
<div class="question">Two cross-sections of stem and root appear simple, when viewed by naked eye. But under microscope, they can be differentiated by</div>

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

answer:
<div class="solution">Characteristic feature of stem is endarch condition of xylem tissue. In endarch, protoxylem is present towards the centre of stem, while metaxylem towards the pericycle. Characteristic feature of root is exarch condition of xylem tissue. In exarch condition, protoxylem is present towards the pericycle and metaxylem towards the center of the root.</div>


tags: ['Physics', 'Thermodynamics', 'NEET', 'NEET 2020 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-cylinders-a-and-b-of-equal-capacity-are-connected-to-each-other-via-a-stop-cock-a-contains-an-ideal-gas-at-standard-temperature-and-pressure-b-is-

slug:
two-cylinders-a-and-b-of-equal-capacity-are-connected-to-each-other-via-a-stop-cock-a-contains-an-ideal-gas-at-standard-temperature-and-pressure-b-is-26921

question:
<div class="question">Two cylinders A and B of equal capacity are connected to each other via a stop cock. A contains an ideal gas at standard temperature and pressure. B is completely evacuated. The entire system is thermally insulated. The stop cock is suddenly opened. The process is:</div>

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

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

answer:
<div class="solution">Free expansion i.e. expansion against vacuum is adiabatic in nature for all type of gases. It should be noted that temperature final temperature is equal to initial temperature for adiabatic process.</div>


tags: ['Physics', 'Thermodynamics']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
two-cylinders-a-and-b-of-equal-capacity-are-connected-to-each-other-via-stopcock-a-contains-gas-at-standard-temperature-and-pressure-b-is-completely-e

slug:
two-cylinders-a-and-b-of-equal-capacity-are-connected-to-each-other-via-stopcock-a-contains-gas-at-standard-temperature-and-pressure-b-is-completely-e-10038

question:
<div class="question">Two cylinders $A$ and $B$ of equal capacity are connected to each other via stopcock. A contains gas at standard temperature and pressure, $B$ is completely evacuated. The entire system is thermally insulated. The stopcock is suddenly opened. Answer the following:<br/>(a) What is the final pressure of the gas in $A$ and $B$ ?<br/>(b) What is the change in internal energy of the gas?<br/>(c) What is the change in the temperature of the gas?<br/>(d) Do the intermediate states of the system (before settling to the final equilibrium state) lie on its $P-V-T$ surface?</div>

answer:
<div class="solution">(a) If the stopcock opens suddenly, volume of gas would become double at 1 atmospheric pressure, therefore, pressure would become half (i.e. $1 / 2$ atmosphere).<br/>(b) As no work is done on the system by the gas, there will be no change in internal energy.<br/>(c) Gas does not do any work in expanding, thereby, no change in temperature of gas would occur.<br/>(d) No, the process of the expansion is rapid and cannot be controlled. The intermediate state would be a nonequilibrium state, which does not follow gas equation.<br/>In due course, gas will come back to equilibrium state.</div>


tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
two-cylindrical-conductors-with-equal-crosssections-and-different-resistivities-1-and-2-are-put-end-to-end-find-the-charge-at-the-boundary-of-the-cond

slug:
two-cylindrical-conductors-with-equal-crosssections-and-different-resistivities-1-and-2-are-put-end-to-end-find-the-charge-at-the-boundary-of-the-cond-18469

question:
<div class="question">Two cylindrical conductors with equal cross-sections and different resistivities <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Ï</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Ï</mi><mn>2</mn></msub></math> are put end to end. Find the charge at the boundary of the conductors if a current <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math> flows from conductor <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> to conductor <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math>.<sub>â</sub></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"><msub><mtext mathvariant="italic">ε</mtext><mn>0</mn></msub><mfenced><mrow><msub><mtext mathvariant="italic">ρ</mtext><mn>2</mn></msub><mo>-</mo><msub><mtext mathvariant="italic">ρ</mtext><mn>1</mn></msub></mrow></mfenced><mtext mathvariant="italic">I</mtext></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"><msub><mtext mathvariant="italic">ε</mtext><mn>0</mn></msub><mfenced><mrow><mn>2</mn><msub><mtext mathvariant="italic">ρ</mtext><mn>2</mn></msub><mo>+</mo><msub><mtext mathvariant="italic">ρ</mtext><mn>1</mn></msub></mrow></mfenced><mtext mathvariant="italic">I</mtext></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>2</mtext><msub><mtext mathvariant="italic">ε</mtext><mn>0</mn></msub><mfenced><mrow><msub><mtext mathvariant="italic">ρ</mtext><mn>1</mn></msub><mo>-</mo><msub><mtext mathvariant="italic">ρ</mtext><mn>2</mn></msub></mrow></mfenced><mtext mathvariant="italic">I</mtext></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext mathvariant="italic">ε</mtext><mn>0</mn></msub><mfenced><mrow><msub><mtext mathvariant="italic">ρ</mtext><mn>1</mn></msub><mo>+</mo><msub><mtext mathvariant="italic">ρ</mtext><mn>2</mn></msub></mrow></mfenced><mtext mathvariant="italic">I</mtext></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext mathvariant="italic">ε</mtext><mn>0</mn></msub><mfenced><mrow><msub><mtext mathvariant="italic">ρ</mtext><mn>2</mn></msub><mo>-</mo><msub><mtext mathvariant="italic">ρ</mtext><mn>1</mn></msub></mrow></mfenced><mtext mathvariant="italic">I</mtext></math></span> </div>

answer:
<div class="solution">According to ohm's law,<math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace><mover><mtext>E</mtext><mo>â</mo></mover><mo>=</mo><mtext>Ï</mtext><mover><mtext>J</mtext><mo>â</mo></mover></math><br/><br/>The problem is solved by the following steps:<br/><br/><strong>Step I:</strong> Determine the electric field in both conductors.<br/><br/>The current density in the first conductor is<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace><msub><mi>J</mi><mn>1</mn></msub><mo>=</mo><mfrac><mtext mathvariant="italic">I</mtext><mrow><mtext>Ï</mtext><msup><mtext mathvariant="italic">R</mtext><mn>2</mn></msup></mrow></mfrac><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace></math>(Along the length of the conductor)<br/><br/><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/EM0014652_AnsExp_1.jpg" style="width: 243px; height: 58px;"/><br/><br/>Similarly,<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msub><mi>J</mi><mn>2</mn></msub><mo>=</mo><mfrac><mi>I</mi><mrow><mtext>Ï</mtext><msup><mtext mathvariant="italic">R</mtext><mn>2</mn></msup></mrow></mfrac><mo> </mo></math> (along the length of the conductor)<br/><br/>The electric field inside the first conductor is<br/>      <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext mathvariant="italic">E</mtext><mn>2</mn></msub><mo>=</mo><mfrac><mrow><msub><mtext mathvariant="italic">Ï</mtext><mn>2</mn></msub><mtext mathvariant="italic">I</mtext></mrow><mrow><mtext>Ï</mtext><msup><mtext mathvariant="italic">R</mtext><mn>2  </mn></msup></mrow></mfrac><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace><mspace width="0.5em"></mspace></math>(along the length of the conductor)<br/><br/><strong>Step II:</strong> Consider small cylindrical region near the boundary and apply Gauss's law:<br/><br/>We consider a small cylindrical region at the boundary<br/><br/><img alt="" src="https://cdn-question-pool.getmarks.app/nta_abhyas/jee_main/images/EM0014652_AnsExp_2.jpg" style="width: 193px; height: 81px;"/><br/><br/>The area of the cylindrical region is<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo> </mo><mo>=</mo><mo> </mo><mi mathvariant="normal">Ï</mi><msup><mi>R</mi><mn>2</mn></msup></math>â   <br/>The net flux through this region is<br/><br/><math> <mrow> <mtext>Ï</mtext> <mo>=</mo> <msub> <mtext>Ï</mtext> <mn>1</mn> </msub> <mo>+</mo> <msub> <mtext>Ï</mtext> <mn>2</mn> </msub> <mo>=</mo> <msub> <mrow> <mover accent="true"> <mtext>E</mtext> <mo>â</mo> </mover> </mrow> <mn>1</mn> </msub> <mo>.</mo> <mover accent="true"> <mtext>S</mtext> <mo>â</mo> </mover> <mo>+</mo> <msub> <mrow> <mover accent="true"> <mtext>E</mtext> <mo>â</mo> </mover> </mrow> <mn>2</mn> </msub> <mover accent="true"> <mtext>S</mtext> <mo>â</mo> </mover> </mrow> </math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><msub><mi>E</mi><mn>1</mn></msub><mi>S</mi><mo> </mo><mi>cos</mi><mo> </mo><mn>180</mn><mo>Â°</mo><mo> </mo><mo>+</mo><mo> </mo><msub><mi>E</mi><mn>2</mn></msub><mi>S</mi><mo> </mo><mi>cos</mi><mn>0</mn><mo>Â°</mo><mo>=</mo><mo> </mo><mo>-</mo><mo> </mo><msub><mi>E</mi><mn>1</mn></msub><mi>S</mi><mo> </mo><mo>+</mo><mo> </mo><msub><mi>E</mi><mn>2</mn></msub><mi>S</mi><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><msub><mi>E</mi><mn>2</mn></msub><mo> </mo><mo>-</mo><mo> </mo><msub><mi>E</mi><mn>1</mn></msub><mo>)</mo><mo> </mo><mi>S</mi></math><br/><br/>According to Gauss's Law,<br/><br/><math> <mrow> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mtext>Ï</mtext> <mo>=</mo> <mfrac> <mrow> <mtext>q</mtext> </mrow> <mrow> <msub> <mtext>Îµ</mtext> <mn>0</mn> </msub> </mrow> </mfrac> </mrow> </math><br/><br/><math> <mrow> <mtext>or</mtext> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mfenced close=")" open="(" separators=","> <mrow> <msub> <mtext>E</mtext> <mn>2</mn> </msub> <mo>-</mo> <msub> <mtext>E</mtext> <mn>1</mn> </msub> </mrow> </mfenced> <mo>=</mo> <mfrac> <mrow> <mtext>q</mtext> </mrow> <mrow> <mtext>S</mtext> <msub> <mtext>Îµ</mtext> <mn>0</mn> </msub> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mtext>Ï</mtext> </mrow> <mrow> <msub> <mtext>Îµ</mtext> <mn>0</mn> </msub> </mrow> </mfrac> </mrow> </math><br/><br/>â´ The surface charge density at the boundary is<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Ï</mi><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><msub><mi>E</mi><mn>2</mn></msub><mo> </mo><mo>-</mo><mo> </mo><msub><mi>E</mi><mn>1</mn></msub><mo>)</mo><mo> </mo><msub><mi>Îµ</mi><mn>0</mn></msub></math><br/><br/>Putting the value of E<sub>1</sub><sub> </sub>and E<sub>2</sub>, we get<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>â´</mo><mo> </mo><mi>Ï</mi><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><msub><mi>Ï</mi><mn>2</mn></msub><mi mathvariant="normal">j</mi><mo> </mo><mo>-</mo><mo> </mo><msub><mi>Ï</mi><mn>1</mn></msub><mi mathvariant="normal">j</mi><mo>)</mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>o</mi><mi>r</mi><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mi>Ï</mi><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><msub><mi>Ï</mi><mn>2</mn></msub><mo> </mo><mo>-</mo><mo> </mo><msub><mi>Ï</mi><mn>1</mn></msub><mo>)</mo><mi mathvariant="normal">j</mi></math><br/><br/><math> <mrow> <mo>â´</mo> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mspace width="0.5em"></mspace> <mtext>Ï</mtext> <mo>=</mo> <msub> <mtext>Îµ</mtext> <mn>0</mn> </msub> <mfenced close=")" open="(" separators=","> <mrow> <msub> <mtext>Ï</mtext> <mn>2</mn> </msub> <mo>-</mo> <msub> <mtext>Ï</mtext> <mn>1</mn> </msub> </mrow> </mfenced> <mfrac> <mrow> <mtext>I</mtext> </mrow> <mrow> <mtext>Ï</mtext> <msup> <mtext>R</mtext> <mn>2</mn> </msup> </mrow> </mfrac> </mrow> </math><br/><br/>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi>Ï</mi><mi mathvariant="normal">Ï</mi><msup><mi>R</mi><mn>2</mn></msup><mo>=</mo><mo> </mo><msub><mi>Îµ</mi><mn>0</mn></msub><mo>(</mo><msub><mi>Ï</mi><mn>2</mn></msub><mo> </mo><mo>-</mo><mo> </mo><msub><mi>Ï</mi><mn>1</mn></msub><mo>)</mo><mi>I</mi></math><br/><br/>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo> </mo><mo>=</mo><mo> </mo><msub><mi>Îµ</mi><mn>0</mn></msub><mo>(</mo><msub><mi>Ï</mi><mn>2</mn></msub><mo> </mo><mo>-</mo><mo> </mo><msub><mi>Ï</mi><mn>1</mn></msub><mo>)</mo><mi>I</mi></math><br/><br/>Charge at the boundary <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo> </mo><mo>=</mo><mo> </mo><msub><mi>Îµ</mi><mn>0</mn></msub><mo>(</mo><msub><mi>Ï</mi><mn>2</mn></msub><mo> </mo><mo>-</mo><mo> </mo><msub><mi>Ï</mi><mn>1</mn></msub><mo>)</mo><mi>I</mi></math></div>


title: two-dice-are-thrown-what-is-the-probability-that-the-sumof-the-faces-equals-or-exceeds-10
slug: two-dice-are-thrown-what-is-the-probability-that-the-sumof-the-faces-equals-or-exceeds-10-55083
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">Two dice are thrown. What is the probability that the sumof the faces equals or exceeds $10 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 / 12$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1 / 4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 3$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1 / 6$</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> </ul>

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

answer:
<div class="solution">Let $\mathrm{E}$ be the sum of the faces equals or exceeds<br/>Then, $\mathrm{E}=\{(5,5),(4,6),(6,4),(5,6),(6,5),(6,6)\}$<br/>$\therefore \quad n(E)=6$<br/>Hence, $P(E)=\frac{n(E)}{n(S)}=\frac{6}{36}=\frac{1}{6}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2012 (Phase 2)']
answer: <div class="solution">$\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{6}{36}=\frac{1}{6}$</div>
source_db: MarksBatch2_P3.db
title:
two-dice-each-numbered-from-1-to-6-are-thrown-together-let-a-and-b-be-two-events-given-by-a-even-number-on-the-first-die-b-number-on-the-second-die-is-1

slug:
two-dice-each-numbered-from-1-to-6-are-thrown-together-let-a-and-b-be-two-events-given-by-a-even-number-on-the-first-die-b-number-on-the-second-die-is-1-84546

question:
<div class="question">Two dice each numbered from 1 to 6 are thrown together. Let $A$ and $\mathrm{B}$ be two events given by<br/>A: even number on the first die.<br/>B: number on the second die is greater than 4<br/>What is $\mathrm{P}(\mathrm{A} \cap \mathrm{B})$ equal to ?</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">$\frac{1}{4}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{2}{3}$</span> </li><li class="correct"> <span class="option-label">D</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> </ul>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
two-dice-each-numbered-from-1-to-6-are-thrown-together-let-a-and-b-be-two-events-given-by-a-even-number-on-the-first-die-b-number-on-the-second-die-is

slug:
two-dice-each-numbered-from-1-to-6-are-thrown-together-let-a-and-b-be-two-events-given-by-a-even-number-on-the-first-die-b-number-on-the-second-die-is-41218

question:
<div class="question">Two dice each numbered from 1 to 6 are thrown together. Let $A$ and $\mathrm{B}$ be two events given by<br/>A: even number on the first die.<br/>B: number on the second die is greater than 4<br/>What is $\mathrm{P}(\mathrm{A} \cup \mathrm{B})$ equal to $?$</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">$\frac{1}{4}$</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">$\frac{1}{6}$</span> </li> </ul>

answer:
<div class="solution">In A there are 18 events possible.<br/>$\therefore \quad \mathrm{P}(\mathrm{A})=\frac{18}{36}=\frac{1}{2}$<br/>In B there are 12 events are possible<br/>$\therefore \quad \mathrm{P}(\mathrm{B})=\frac{12}{36}=\frac{1}{3}$<br/>In $A \cap B$ there are 6 events are possible. Now,<br/>$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}=\frac{3+2-1}{6}=\frac{4}{6}=\frac{2}{3}$</div>


title: two-different-species-cannot-live-for-long-duration-in-the-same-niche-or-habitat-this-law-is
slug: two-different-species-cannot-live-for-long-duration-in-the-same-niche-or-habitat-this-law-is-29676
tags: ['Biology', 'Organisms and Populations', 'NEET', 'NEET 2002']
source_db: MarksBatch2_P3.db
question:
<div class="question">Two different species cannot live for long duration in the same niche or habitat. This law is:</div>

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

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

answer:
<div class="solution">The Principle of Competitive Exclusion states that two species or populations cannot inhabit the same niche: one will consistently outcompete the other.<br/>Allen rule states that in a warmblooded animal species having distinct geographic populations, limbs, ears and other appendages of animals living in cold climate tend to be shorter than in animals of same species living in warm climates.<br/>Weismann proposed the theory of the continuity of germ-plasm, a theory of heredity.</div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2018 (04 May Shift 2)']
source_db: MarksBatch2_P3.db
title:
two-distributions-a-and-b-have-the-same-mean-if-their-coefficients-of-variation-are-6-and-2-respectively-and-a-b-are-their-standard-deviations-then

slug:
two-distributions-a-and-b-have-the-same-mean-if-their-coefficients-of-variation-are-6-and-2-respectively-and-a-b-are-their-standard-deviations-then-29850

question:
<div class="question">Two distributions $A$ and $B$ have the same mean. If their coefficients of variation are 6 and 2 respectively and $\sigma_{A^{\prime}} \sigma_B$ are their standard deviations, then</div>

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

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

answer:
<div class="solution">Let $\bar{x}_A=\bar{x}_B=x$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{\sigma_A}{\bar{x}_A} \times 100=6, \frac{\sigma_B}{\bar{x}_B} \times 100=2 \\<br/>&amp; \frac{\sigma_A}{x} \times 100=6, \frac{\sigma_B}{x} \times 100=2 \\<br/>&amp; \frac{\sigma_A \times 100}{6}=x, \frac{\sigma_B \times 100}{2}=x \\<br/>\Rightarrow \quad &amp; \frac{\sigma_A \times 100}{6}=\frac{\sigma_B \times 100}{2} \Rightarrow \sigma_A=3 \sigma_B<br/>\end{aligned}<br/>$$</div>


title: two-events-a-and-b-are-such-that-p-not-b-08-p-a-b-05-and-p-a-b-04-then-p-a-is-equal-to
slug: two-events-a-and-b-are-such-that-p-not-b-08-p-a-b-05-and-p-a-b-04-then-p-a-is-equal-to-32963
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.38$</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">Two events $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{P}(\mathrm{not} \mathrm{B})=0.8, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=$<br/>$0.5$ and $\mathrm{P}(\mathrm{A} \mid \mathrm{B})=0.4$. Then $\mathrm{P}(\mathrm{A})$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.28$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.32$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.38$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \because \mathrm{P}(\overline{\mathrm{B}})=0.8 \Rightarrow \mathrm{P}(\mathrm{B})=0.2 \\ &amp; \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.5 \&amp; \mathrm{P}(\mathrm{A} \mid \mathrm{B})=0.4 \\ &amp; \because \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\mathrm{P}(\mathrm{B}) \times \mathrm{P}(\mathrm{A} \mid \mathrm{B})=0.2 \times 0.4=0.08 \\ &amp; \&amp; \mathrm{P}(\mathrm{A})=\mathrm{P}(\mathrm{A} \cup \mathrm{B})-\mathrm{P}(\mathrm{B})+\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \\ &amp; \mathrm{P}(\mathrm{A})=0.5-0.2+0.08=0.38 \end{aligned}$</div>


title: two-fair-dice-are-rolled-what-is-the-probability-of-getting-a-sum-of-7
slug: two-fair-dice-are-rolled-what-is-the-probability-of-getting-a-sum-of-7-34211
tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">Two fair dice are rolled. What is the probability of getting a sum of $7 ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{36}$</span> </li><li class="correct"> <span class="option-label">B</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">C</span> <span class="option-data">$\frac{7}{12}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{5}{12}$</span> </li> </ul>

answer:
<div class="solution">When two dice are rolled, the events where we get sum of 7 is $\mathrm{E}=\{(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\}$<br/>$\therefore \mathrm{n}(\mathrm{E})=6$<br/>Total number of events, $\mathrm{n}(\mathrm{S})=36$.<br/>$\therefore$ Probability $=\frac{\mathrm{n}(\mathrm{E})}{\mathrm{n}(\mathrm{S})}=\frac{6}{36}=\frac{1}{6}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2008 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-forces-are-equal-to-2-o-a-and-3-bo-their-resultant-being-og-where-g-is-the-point-on-a-b-such-that-a-g-bg-3-2-what-is-the-value-of

slug:
two-forces-are-equal-to-2-o-a-and-3-bo-their-resultant-being-og-where-g-is-the-point-on-a-b-such-that-a-g-bg-3-2-what-is-the-value-of-56730

question:
<div class="question">Two forces are equal to $2 \overrightarrow{O A}$ and $3 \overrightarrow{B O}$, their resultant<br/>being $\lambda \overrightarrow{O G}$, where $G$ is the point on $A B$ such that<br/>$\frac{B G}{A G}=-\frac{2}{3} .$ What is the value of $\lambda ?$</div>

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

answer:
<div class="solution">$\overrightarrow{O G}=\frac{2 \overrightarrow{O A}-3 \overrightarrow{O B}}{2-3}$<br/>$\overrightarrow{O G}=\frac{2 \overrightarrow{O A}-3 \overrightarrow{O B}}{-1}$<br/>$-\overrightarrow{O G}=2 \overrightarrow{O A}-3 \overrightarrow{O B}$ ... (1)<br/>$\lambda \overrightarrow{O G}=2 \overrightarrow{O A}+3 \overrightarrow{O B} \quad \cdots$ (2)<br/>Adding (1) and (2)<br/>$(\lambda-1) \overrightarrow{O G}=4 \overrightarrow{O A}$<br/>$\Rightarrow \overrightarrow{O A}=\left(\frac{\lambda-1}{4}\right) \overrightarrow{O G}$ ... (3)<br/>Subtracting (2) from (1)<br/>$(-1-\lambda) \overrightarrow{O G}=-6 \overrightarrow{O B}$<br/>$\overrightarrow{O B}=\frac{(1+\lambda)}{6} \overrightarrow{O G}$ ... (4)<br/>From equ (2), (3) and (4)<br/>$\lambda \overrightarrow{O G}=2\left(\frac{\lambda-1}{4}\right) \overrightarrow{O G}+3\left(\frac{\lambda+1}{6}\right) \overrightarrow{O G}$<br/>$\Rightarrow \lambda=\frac{\lambda-1}{2}=\frac{\lambda+1}{3}$<br/>$\therefore \lambda=\frac{\lambda-1}{2}$ or $\lambda=\frac{\lambda+1}{3}$<br/>$\Rightarrow 2 \lambda-\lambda=-1$ or $3 \lambda-\lambda=1$<br/>$\Rightarrow \lambda=-1$ or $\lambda=\frac{1}{2}$</div>


tags: ['Chemistry', 'Electrochemistry', 'NEET', 'NEET 2022 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
two-half-cell-reactions-are-given-below-co-3-e-co-2-e-co-2-co-3-181-v-2-al-3-6-e-2-al-s-e-o-al-al-3-166-v-the-standard-emf-of-a-cell-with-feasible-red

slug:
two-half-cell-reactions-are-given-below-co-3-e-co-2-e-co-2-co-3-181-v-2-al-3-6-e-2-al-s-e-o-al-al-3-166-v-the-standard-emf-of-a-cell-with-feasible-red-26260

question:
<div class="question">Two half cell reactions are given below :<br/>$\mathrm{CO}^{3+}+\mathrm{e}^{-} \rightarrow \mathrm{Co}^{2+}, \mathrm{E}^{\circ} \mathrm{Co}^{2+} / \mathrm{Co}^{3+}=-1.81 \mathrm{~V}$<br/>$2 \mathrm{Al}^{3+}+6 \mathrm{e}^{-} \rightarrow 2 \mathrm{Al}(\mathrm{s}), \mathrm{E}^{\mathrm{o}} \mathrm{Al} / \mathrm{Al}^{3+}=+1.66 \mathrm{~V}$<br/>The standard EMF of a cell with feasible redox reaction will be :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$+7.09 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$+0.15 \mathrm{~V}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$+3.47 \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-3.47 \mathrm{~V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$+3.47 \mathrm{~V}$</span> </div>

answer:
<div class="solution">The cell reaction is written as :<br/>$\begin{aligned}<br/>&amp; \mathrm{E}^{\circ} \mathrm{Co}^{2+} / \mathrm{Co}^{3+}=-1.81 \mathrm{~V}(\mathrm{SOP}) \\<br/>&amp; \therefore \quad \mathrm{E}^{\circ} \mathrm{Co}^{3+} / \mathrm{Co}^{2+}=+1.81 \mathrm{~V} \text { (SRP) } \\<br/>&amp; \mathrm{E}_{\mathrm{A} / / \mathrm{Al}^{3+}}^{\circ}=+1.66 \mathrm{~V}(\mathrm{SOP}) \\<br/>&amp; \therefore \mathrm{E}_{\mathrm{Al}^{3+} / \mathrm{Al}}^{\circ}=-1.66 \mathrm{~V}(\mathrm{SRP}) \\<br/>&amp;<br/>\end{aligned}$<br/>Here $\mathrm{E}^{\circ}$ is given in standard Reduction Potential<br/>$\therefore \mathrm{Co}^{3+} / \mathrm{Co}$ will undergo reduction acting as cathode and $\mathrm{Al}^{3+} / \mathrm{Al}$ will undergo oxidation acting as anode.<br/>$\begin{aligned}<br/>\therefore \quad \mathrm{E}_{\text {cell }}^{\circ} &amp; =\mathrm{E}_{\text {cathode (Red.) }}^{\circ}-\mathrm{E}_{\text {Anode (oxide) }}^{\circ} \\<br/>\mathrm{E}_{\text {cell }}^{\circ} &amp; =1.81-(-1.66) \\<br/>\mathrm{E}_{\text {cell }}^{\circ} &amp; =+3.47 \mathrm{~V}<br/>\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
two-identical-piano-strings-of-length-0750-m-are-each-tuned-exactly-to-500-h-z-the-tension-in-one-of-the-strings-is-then-increased-by-21-if-they-are-n

slug:
two-identical-piano-strings-of-length-0750-m-are-each-tuned-exactly-to-500-h-z-the-tension-in-one-of-the-strings-is-then-increased-by-21-if-they-are-n-54908

question:
<div class="question">Two identical piano strings of length <math><mn>0.750</mn></math> <math><mi mathvariant="normal">m</mi></math> are each, tuned exactly to <math><mn>500</mn></math> <math><mi mathvariant="normal">H</mi><mi mathvariant="normal">z</mi></math> . The tension in one of the strings is then increased by <math><mn>21</mn><mi mathvariant="normal">%</mi></math> . If they are now struck, what is the beat frequency between the fundamental mode of the two strings?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math><mn>45</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math><mn>50</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mn>60</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mn>40</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mn>50</mn></math></span> </div>

answer:
<div class="solution"><math><msub><mrow><mi mathvariant="italic">f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi mathvariant="script">l</mi></mrow></mfrac><msqrt><mfrac><mrow><mi mathvariant="italic">T</mi></mrow><mrow><mi mathvariant="italic">Î¼</mi></mrow></mfrac></msqrt><mo>=</mo><mn>500</mn></math> <br/><math><msub><mrow><mi mathvariant="italic">f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi mathvariant="script">l</mi></mrow></mfrac><msqrt><mfrac><mrow><mn>1.21</mn><mi mathvariant="italic">T</mi></mrow><mrow><mi mathvariant="italic">Î¼</mi></mrow></mfrac></msqrt><mo>=</mo><mn>1.1</mn><mfenced separators="|"><mrow><mn>500</mn></mrow></mfenced></math> <br/><math><msub><mrow><mi mathvariant="italic">f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>-</mo><msub><mrow><mi mathvariant="italic">f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>50</mn></math></div>


tags: ['Physics', 'Waves and Sound', 'VITEEE', 'VITEEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.02$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio-1

slug:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio-1-41632

question:
<div class="question">Two identical piano wires have a fundamental frequency of 600 cycle per second when kept under the same tension. What fractional increase in the tension of one wires will lead to the occurrence of 6 beats per second when both wires vibrate simultaneously?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.01$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.02$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.03$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.04$</span> </li> </ul>

answer:
<div class="solution">When both the wires vibrate simultaneously, beats per second,<br/>$$<br/>\begin{array}{c}<br/>n_{1} \pm n_{2}=6 \\<br/>\text { or } \frac{1}{2 l} \sqrt{\frac{T}{m}} \pm \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}=6 \\<br/>\Rightarrow \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}-\frac{1}{2 l} \sqrt{\frac{T}{m}}=6 \\<br/>\Rightarrow \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}-600=6 \Rightarrow \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}=606...(i)<br/>\end{array}<br/>$$<br/>Given that fundamental frequency<br/>$$<br/>\frac{1}{2 l} \sqrt{\frac{T}{m}}=600...(ii)<br/>$$<br/>Dividing Eq. (i) by(ii),<br/>$$<br/>\begin{array}{l}<br/>\frac{\frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}}{\frac{1}{2 l} \sqrt{\frac{T}{m}}}=\frac{606}{600} \\<br/>\Rightarrow \sqrt{\frac{T^{\prime}}{T}}=(1.01) \Rightarrow \frac{T^{\prime}}{T}=(1.02) \% \\<br/>\Rightarrow T^{\prime}=T(1.02)<br/>\end{array}<br/>$$<br/>Increase in tension<br/>$$<br/>\begin{array}{l}<br/>\Delta T^{\prime}=T \times 1.02-T=(0.02 T) \\<br/>\therefore \Delta T=0.02<br/>\end{array}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'TS EAMCET', 'TS EAMCET 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.01</span> </div>
source_db: MarksBatch2_P3.db
title:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio-2

slug:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio-2-57771

question:
<div class="question">Two identical piano wires have a fundamental frequency of 600 cycle per second when kept under the same tension. What fractional increase in the tension of one wires will lead to the occurrence of 6 beats per second when both wires vibrate simultaneously?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.01</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.02</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.03</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.04</span> </li> </ul>

answer:
<div class="solution">Beats per second when both the wires vibrate simultaneously.<br/>$n_1 \pm n_2=6$<br/>$\begin{aligned} \text { or } &amp; \frac{1}{2 l} \sqrt{\frac{T}{m}} \pm \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}} &amp; =6 \\ \text { or } &amp; \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}-\frac{1}{2 l} \sqrt{\frac{T}{m}} &amp; =6 \\ \text { or } &amp; \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}-600 &amp; =6\end{aligned}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/rimsbR8H_jWGomzwQNyuakUpRY9jvmSGj7jBmnw4cyQ.original.fullsize.png"/><br/><br/>Given that fundamental frequency<img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/NjOjhjegpYf_UBLoO5s6FfzJUOV2kTC62bqC4gtP5gA.original.fullsize.png"/><br/><br/>Dividing Eq. (i) by Eq. (ii), we get<br/>$\frac{\frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}}{\frac{1}{2 l} \sqrt{\frac{T}{m}}}=\frac{606}{600}$<br/>$\begin{array}{ll}\text { or } &amp; \sqrt{\frac{T^{\prime}}{T}}=(1.01) \\ \text { or } &amp; \frac{T^{\prime}}{T}=(1.02) \% \\ \text { or } &amp; T^{\prime}=T(1.02)\end{array}$<br/>Increase in tension<br/>$\begin{aligned}<br/>\Delta T^{\prime} &amp; =T \times 1.02-T \\<br/>&amp; =(0.02 T) \\<br/>\text{Hence,} \quad \Delta T^{\prime}=0.02<br/>\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.01</span> </div>
source_db: MarksBatch2_P3.db
title:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio-3

slug:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio-3-29816

question:
<div class="question">Two identical piano wires have a fundamental frequency of 600 cycle per second when kept under the same tension. What fractional increase in the tension of one wires will lead to the occurrence of 6 beats per second when both wires vibrate simultaneously?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.01</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.02</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.03</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.04</span> </li> </ul>

answer:
<div class="solution">Beats per second when both the wires vibrate simultaneously.<br/>$n_1 \pm n_2=6$<br/>$\begin{aligned} \text { or } &amp; \frac{1}{2 l} \sqrt{\frac{T}{m}} \pm \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}} &amp; =6 \\ \text { or } &amp; \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}-\frac{1}{2 l} \sqrt{\frac{T}{m}} &amp; =6 \\ \text { or } &amp; \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}-600 &amp; =6\end{aligned}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/rimsbR8H_jWGomzwQNyuakUpRY9jvmSGj7jBmnw4cyQ.original.fullsize.png"/><br/><br/>Given that fundamental frequency<img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/NjOjhjegpYf_UBLoO5s6FfzJUOV2kTC62bqC4gtP5gA.original.fullsize.png"/><br/><br/>Dividing Eq. (i) by Eq. (ii), we get<br/>$\frac{\frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}}{\frac{1}{2 l} \sqrt{\frac{T}{m}}}=\frac{606}{600}$<br/>$\begin{array}{ll}\text { or } &amp; \sqrt{\frac{T^{\prime}}{T}}=(1.01) \\ \text { or } &amp; \frac{T^{\prime}}{T}=(1.02) \% \\ \text { or } &amp; T^{\prime}=T(1.02)\end{array}$<br/>Increase in tension<br/>$\begin{aligned}<br/>\Delta T^{\prime} &amp; =T \times 1.02-T \\<br/>&amp; =(0.02 T) \\<br/>\text{Hence,} \quad \Delta T^{\prime}=0.02<br/>\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.02$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio-4

slug:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio-4-48080

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.01$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.02$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.03$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.04$</span> </li> </ul>

answer:
<div class="solution">When both the wires vibrate simultaneously, beats per second,<br/>$$<br/>\begin{array}{c}<br/>n_{1} \pm n_{2}=6 \\<br/>\text { or } \frac{1}{2 l} \sqrt{\frac{T}{m}} \pm \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}=6 \\<br/>\Rightarrow \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}-\frac{1}{2 l} \sqrt{\frac{T}{m}}=6 \\<br/>\Rightarrow \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}-600=6 \Rightarrow \frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}=606...(i)<br/>\end{array}<br/>$$<br/>Given that fundamental frequency<br/>$$<br/>\frac{1}{2 l} \sqrt{\frac{T}{m}}=600...(ii)<br/>$$<br/>Dividing Eq. (i) by(ii),<br/>$$<br/>\begin{array}{l}<br/>\frac{\frac{1}{2 l} \sqrt{\frac{T^{\prime}}{m}}}{\frac{1}{2 l} \sqrt{\frac{T}{m}}}=\frac{606}{600} \\<br/>\Rightarrow \sqrt{\frac{T^{\prime}}{T}}=(1.01) \Rightarrow \frac{T^{\prime}}{T}=(1.02) \% \\<br/>\Rightarrow T^{\prime}=T(1.02)<br/>\end{array}<br/>$$<br/>Increase in tension<br/>$$<br/>\begin{array}{l}<br/>\Delta T^{\prime}=T \times 1.02-T=(0.02 T) \\<br/>\therefore \Delta T=0.02<br/>\end{array}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio

slug:
two-identical-piano-wires-have-a-fundamental-frequency-of-600-cycle-per-second-when-kept-under-the-same-tension-what-fractional-increase-in-the-tensio-38934

question:
<div class="question">Two identical piano wires have a fundamental frequency of<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>600</mn></math> cycle per second when kept under the same tension. What fractional increase in the tension of wires will lead to the occurrence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> beats per second when both wires vibrate simultaneously?</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>01</mn></math></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>02</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>03</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>04</mn></math></span> </li> </ul>

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

answer:
<div class="solution">Beats per second when both the wires vibrate simultaneously<br/> <br/>n<sub>1</sub> ± n<sub>2</sub> = 6<br/> <br/>or    <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>2</mn><mi>l</mi><mo>⁡</mo></mrow></mfrac><msqrt><mfrac><mtext mathvariant="italic">T</mtext><mtext mathvariant="italic">m</mtext></mfrac></msqrt><mo>±</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>l</mi><mo>⁡</mo></mrow></mfrac><msqrt><mfrac><msup><mtext mathvariant="italic">T</mtext><mo mathvariant="italic">'</mo></msup><mtext mathvariant="italic">m</mtext></mfrac></msqrt><mo>=</mo><mn>6</mn></math><br/> <br/>or   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>2</mn><mi>l</mi><mo>⁡</mo></mrow></mfrac><msqrt><mfrac><msup><mtext mathvariant="italic">T</mtext><mo mathvariant="italic">'</mo></msup><mtext mathvariant="italic">m</mtext></mfrac></msqrt><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>l</mi><mo>⁡</mo></mrow></mfrac><msqrt><mfrac><mtext mathvariant="italic">T</mtext><mtext mathvariant="italic">m</mtext></mfrac></msqrt><mo>=</mo><mn>6</mn></math><br/> <br/>or   <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mn>2</mn><mi>l</mi><mo>⁡</mo></mrow></mfrac><msqrt><mfrac><msup><mtext mathvariant="italic">T</mtext><mo mathvariant="italic">'</mo></msup><mtext mathvariant="italic">m</mtext></mfrac></msqrt><mo>-</mo><mn>6</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>6</mn></math> ...(i)<br/><br/>Given that fundamental frequency<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mfrac linethickness="1"><mn>1</mn><mrow><mn>2</mn><mi>l</mi></mrow></mfrac></mtd><mtd><msqrt><mfrac><mtext mathvariant="italic">T</mtext><mtext mathvariant="italic">m</mtext></mfrac></msqrt><mo>=</mo><mn>600</mn></mtd><mtd></mtd></mtr></mtable></math><br/> <br/>Dividing Eq.(i) by Eq. (ii), we get<br/> <br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mi>l</mi><mo>⁡</mo></mrow></mfrac><msqrt><mfrac><msup><mtext mathvariant="italic">T</mtext><mo mathvariant="italic">'</mo></msup><mtext mathvariant="italic">m</mtext></mfrac></msqrt></mrow><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mi>l</mi><mo>⁡</mo></mrow></mfrac><msqrt><mfrac><mtext mathvariant="italic">T</mtext><mtext mathvariant="italic">m</mtext></mfrac></msqrt></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>6</mn><mn>0</mn><mn>6</mn></mrow><mrow><mn>6</mn><mn>0</mn><mn>0</mn></mrow></mfrac></math><br/> <br/>Or <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><msup><mtext mathvariant="italic">T</mtext><mo mathvariant="italic">'</mo></msup><mtext mathvariant="italic">T</mtext></mfrac></msqrt><mo>=</mo><mtext>1.01</mtext></math><br/> <br/>Or <math> <mrow> <mfrac> <mrow> <msup> <mrow> <mi></mi><mi></mi><mi></mi><mi>T</mi><mo>⁡</mo> </mrow> <mo>′</mo> </msup> </mrow> <mrow> <mi></mi><mi></mi><mi></mi><mi>T</mi><mo>⁡</mo> </mrow> </mfrac> <mo>=</mo> <mn>1</mn> <mtext>.</mtext> <mn>0</mn> <mn>2</mn> <mtext>%</mtext> </mrow> </math><br/> <br/>Or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>'</mo><mo> </mo><mo>=</mo><mi>T</mi><mo> </mo><mo>(</mo><mn>1</mn><mo>.</mo><mn>02</mn><mo>)</mo></math><br/> <br/>Increase in tension<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Δ</mi><mi>T</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><mi>T</mi><mo> </mo><mo>×</mo><mo> </mo><mn>1</mn><mo>.</mo><mn>02</mn><mo>)</mo><mo> </mo><mo>–</mo><mo> </mo><mi>T</mi></math><br/><br/>       <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>02</mn><mi>T</mi></math><br/><br/>Hence,<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi>Δ</mi><mi>T</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>02</mn></math></div>


tags: ['Physics', 'Waves and Sound', 'NEET', 'NEET 2011 (Mains)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.02</span> </div>
source_db: MarksBatch2_P3.db
title:
two-identical-piano-wires-kept-under-the-same-tension-t-have-a-fundamental-frequency-of-600-hz-the-fractional-increase-in-the-tension-of-one-of-the-wi

slug:
two-identical-piano-wires-kept-under-the-same-tension-t-have-a-fundamental-frequency-of-600-hz-the-fractional-increase-in-the-tension-of-one-of-the-wi-80969

question:
<div class="question">Two identical piano wires kept under the same tension $T$ have a fundamental frequency of $600 \mathrm{~Hz}$. The fractional increase in the tension of one of the wires which will lead to occurrence of 6 beat/s when both the wires oscillate together would be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.02</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.03</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.04</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0.01</span> </li> </ul>

answer:
<div class="solution">According to law of tension the frequency of the string varies directly as the square root of its tension<br/>$$<br/>\text { or } \quad \begin{aligned}<br/>n &amp; \propto \sqrt{T} \\<br/>\frac{\Delta n}{n} &amp; =\frac{1}{2} \cdot \frac{\Delta T}{T} \\<br/>\frac{\Delta T}{T} &amp; =2 \times \frac{\Delta n}{n} \\<br/>\frac{\Delta T}{T} &amp; =2 \times \frac{6}{600} \\<br/>&amp; =0.02<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2020 (16 Oct Shift 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1:9</span> </div>
source_db: MarksBatch2_P3.db
title:
two-identical-strings-of-length-and-2-vibrate-with-fundamental-frequencies-n-hertz-and-15-n-hertz-respectively-the-ratio-of-tensions-for-smaller-lengt

slug:
two-identical-strings-of-length-and-2-vibrate-with-fundamental-frequencies-n-hertz-and-15-n-hertz-respectively-the-ratio-of-tensions-for-smaller-lengt-89050

question:
<div class="question">Two identical strings of length ' $\ell$ ' and '2 $\ell$ ' vibrate with fundamental frequencies 'N' hertz and '1:5 N' hertz, respectively. The ratio of tensions for smaller length to larger length is</div>

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

answer:
<div class="solution">$\mathrm{N}=\frac{1}{2 \ell} \sqrt{\frac{\mathrm{T}_{1}}{\mathrm{~m}}}$<br/>$1.5 \mathrm{~N}=\frac{1}{4 \ell} \sqrt{\frac{\mathrm{T}_{2}}{\mathrm{~m}}}$<br/>$\therefore \sqrt{\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}}=\frac{\mathrm{N} \times 2 \ell}{1.5 \mathrm{~N} \times 4 \ell}=\frac{1}{3}$<br/>$\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\frac{1}{9}$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2020 (02 Sep Shift 1)']
source_db: MarksBatch2_P3.db
title:
two-identical-strings-x-and-z-made-of-same-material-have-tension-t-x-and-t-z-in-then-if-their-fundamental-frequencies-are-450-hz-and-300-hz-respective

slug:
two-identical-strings-x-and-z-made-of-same-material-have-tension-t-x-and-t-z-in-then-if-their-fundamental-frequencies-are-450-hz-and-300-hz-respective-28432

question:
<div class="question">Two identical strings <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">X</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">Z</mi></math> made of same material have tension <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">X</mi></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">Z</mi></msub></math> in then if their fundamental frequencies are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>450</mn><mo> </mo><mi>Hz</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mo> </mo><mi>Hz</mi></math>, respectively, then the ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">X</mi></msub><mo>/</mo><msub><mi mathvariant="normal">T</mi><mi mathvariant="normal">Z</mi></msub></math> 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>2</mn><mo>.</mo><mn>25</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>44</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>25</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>25</mn></math></span> </div>

answer:
<div class="solution"><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><msub><mi>f</mi><mi>X</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ℓ</mi></mrow></mfrac><msqrt><mfrac><msub><mi>T</mi><mi>X</mi></msub><mi>μ</mi></mfrac></msqrt></mtd></mtr></mtable></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><msub><mi>f</mi><mi>Z</mi></msub><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>ℓ</mi></mrow></mfrac><msqrt><mfrac><msub><mi>T</mi><mi>Z</mi></msub><mi>μ</mi></mfrac></msqrt></mtd></mtr></mtable></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable columnalign="left"><mtr><mtd><mfrac><msub><mi>f</mi><mi>X</mi></msub><msub><mi>f</mi><mi>Z</mi></msub></mfrac><mo>=</mo><mfrac><mn>450</mn><mn>300</mn></mfrac><mo>=</mo><msqrt><mfrac><msub><mi>T</mi><mi>X</mi></msub><msub><mi>T</mi><mi>Z</mi></msub></mfrac></msqrt></mtd></mtr><mtr><mtd></mtd></mtr></mtable></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msub><mi>T</mi><mi>X</mi></msub><mo>/</mo><msub><mi>T</mi><mi>Z</mi></msub><mo>=</mo><mn>9</mn><mo>/</mo><mn>4</mn><mo>=</mo><mn>2</mn><mo>.</mo><mn>25</mn></math></p></div>


tags: ['Physics', 'Waves and Sound', 'TS EAMCET', 'TS EAMCET 2021 (06 Aug Shift 2)']
source_db: MarksBatch2_P3.db
title:
two-identical-wires-have-a-fundamental-frequency-f-0-when-kept-under-the-same-tension-t-if-the-tension-of-one-wire-is-increased-by-t-then-the-n-beats-

slug:
two-identical-wires-have-a-fundamental-frequency-f-0-when-kept-under-the-same-tension-t-if-the-tension-of-one-wire-is-increased-by-t-then-the-n-beats-36166

question:
<div class="question">Two identical wires have a fundamental frequency $f_0$ when kept under the same tension $T$. If the tension of one wire is increased by $\Delta T$, then the $N$ beats occur when both wires oscillate simultaneously.</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\Delta T}{T}=\left(\frac{f_0+N}{f_0}\right)^2-1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\Delta T}{T}=\left(\frac{f_0}{f_0-N}\right)^2-1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\Delta T}{T}=\left(\frac{f_0-N}{f_0}\right)^2+1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\Delta T}{T}=\left(\frac{f_0+N}{f_0}\right)^2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\Delta T}{T}=\left(\frac{f_0+N}{f_0}\right)^2-1$</span> </div>

answer:
<div class="solution">Fundamental frequency of the wire,<br/>$$<br/>f_0=\frac{v}{2 L}, \text { where } v=\sqrt{\frac{T}{\mu}}<br/>$$<br/>$\begin{array}{ll}\Rightarrow &amp; f_0=\frac{1}{2 L} \sqrt{\frac{T}{\mu}} \\ \Rightarrow &amp; f_0 \propto \sqrt{T}\end{array}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ts_eamcet/p__XzQtcBpVryGKxE9kx-UB4DPA9sFrMRE24fZY11w0.original.fullsize.png"/><br/><br/>Dividing Eq. (ii) by Eq. (i), we get<br/>$$<br/>\begin{aligned}<br/>\frac{\left(f_0+N\right)^2}{f_0^2} &amp; =\frac{T+\Delta T}{T} \\<br/>\Rightarrow \quad \frac{\Delta T}{T} &amp; =\frac{\left(f_0+N\right)^2}{f_0^2}-1=\left(\frac{f_0+N}{f_0}\right)^2-1<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-identical-wires-have-the-same-fundamental-frequency-of-400-hz-when-kept-under-the-same-tension-if-the-tension-in-one-wire-is-increased-by-2-the-nu

slug:
two-identical-wires-have-the-same-fundamental-frequency-of-400-hz-when-kept-under-the-same-tension-if-the-tension-in-one-wire-is-increased-by-2-the-nu-25231

question:
<div class="question">Two identical wires have the same fundamental frequency of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>400</mn><mo> </mo><mi>Hz</mi></math> when kept under the same tension. If the tension in one wire is increased by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>%</mo></math>, the number of beats produced will be</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>4</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math></span> </div>

answer:
<div class="solution"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∝</mo><msqrt><mtext>T</mtext></msqrt><mo>⇒</mo><mfrac><mrow><mtext>Δ</mtext><mi>n</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><mtext>Δ</mtext><mtext mathvariant="italic">T</mtext></mrow><mtext mathvariant="italic">T</mtext></mfrac></math><br/><br/>Beat frequency  <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>Δ</mtext><mi>n</mi><mo>=</mo><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><mtext>Δ</mtext><mtext mathvariant="italic">T</mtext></mrow><mtext mathvariant="italic">T</mtext></mfrac></mrow></mfenced><mi>n</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>2</mn><mrow><mn>1</mn><mn>0</mn><mn>0</mn></mrow></mfrac><mo>×</mo><mn>4</mn><mn>0</mn><mn>0</mn><mo>=</mo><mn>4</mn></math></div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-independent-events-a-and-b-are-such-that-p-a-b-3-2-and-p-a-b-6-1-if-p-b-p-a-then-what-is-p-b-equal-to

slug:
two-independent-events-a-and-b-are-such-that-p-a-b-3-2-and-p-a-b-6-1-if-p-b-p-a-then-what-is-p-b-equal-to-28734

question:
<div class="question">Two independent events $\mathrm{A}$ and $\mathrm{B}$ are such that $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=$<br/>$\frac{2}{3}$ and $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{6}$. If $\mathrm{P}(\mathrm{B})   &lt; \mathrm{P}(\mathrm{A})$, then what is $\mathrm{P}(\mathrm{B})$<br/>equal to ?</div>

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

answer:
<div class="solution">$P(A \cup B)=\frac{2}{3}$<br/>$P(A \cap B)=\frac{1}{6}$<br/>Since, $A$ and $B$ are independent events.<br/>$P(A \cap B)=P(A) \cdot P(B)=\frac{1}{6}$ ... (1)<br/>$P(A \cup B)=\frac{2}{3} \Rightarrow P(A)+P(B)-P(A \cap B)=\frac{2}{3}$<br/>$\Rightarrow P(A)+P(B)-\frac{1}{6}=\frac{2}{3}$<br/>$\Rightarrow P(A)+P(B)=\frac{2}{3}+\frac{1}{6}=\frac{5}{6}$ .... (2)<br/>from $(1),(2), P(B)=\frac{1}{3}$ or $\frac{1}{2}$<br/>$\because P(B)   &lt; P(A), P(B)=\frac{1}{3}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-independent-events-a-and-b-have-p-a-3-1-and-p-b-4-3-what-is-the-probability-that-exactly-one-of-the-two-events-a-or-b-occurs

slug:
two-independent-events-a-and-b-have-p-a-3-1-and-p-b-4-3-what-is-the-probability-that-exactly-one-of-the-two-events-a-or-b-occurs-92297

question:
<div class="question">Two independent events $\mathrm{A}$ and $\mathrm{B}$ have $\mathrm{P}(\mathrm{A})=\frac{1}{3}$<br/>and $\mathrm{P}(\mathrm{B})=\frac{3}{4}$. What is the probability that exactly one of the two events A or B occurs?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{7}{12}$</span> </div>

answer:
<div class="solution">$A$ and $B$ are independent.<br/>$P(A)=\frac{1}{3}$<br/>$P(B)=\frac{3}{4}$<br/>We want to find probability that exactly one of the two events $A$ or $B$ occurs i.e. when $A$ occurs $B$ does not and vice-versa. Lets take desired prob. is $P$. $\therefore P=P(A)(1-P(B))+P(B)(1-P(A))$<br/>$$<br/>\begin{array}{l}<br/>=\frac{1}{3}\left(1-\frac{3}{4}\right)+\frac{3}{4}\left(1-\frac{1}{3}\right) \\<br/>=\frac{1}{3} \times \frac{1}{4}+\frac{3 \times 2}{12} \\<br/>P=\frac{7}{12}<br/>\end{array}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
two-integers-x-and-y-are-chosen-with-replacement-from-the-set-0-1-2-10-the-probability-that-x-y-5-is

slug:
two-integers-x-and-y-are-chosen-with-replacement-from-the-set-0-1-2-10-the-probability-that-x-y-5-is-68469

question:
<div class="question">Two integers $\mathrm{x}$ and $\mathrm{y}$ are chosen with replacement from the<br/>set $(0,1,2, \ldots \ldots, 10)$. The probability that $|\mathrm{x}-\mathrm{y}|&gt;5$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{6}{11}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{35}{121}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{30}{121}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{25}{121}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{30}{121}$</span> </div>

answer:
<div class="solution">Probability $=\frac{2 \times \frac{1}{2} \times 5 \times 6}{11 \times 11}=\frac{30}{121}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/c-Pl5PDmDzYUfHjLbGSXq-8DW3yDqW0FhNuwYx4COC4.original.fullsize.png"/></div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
two-letters-are-drawn-at-random-from-the-word-home-what-is-the-probability-that-both-the-letters-are-vowels

slug:
two-letters-are-drawn-at-random-from-the-word-home-what-is-the-probability-that-both-the-letters-are-vowels-33579

question:
<div class="question">Two letters are drawn at random from the word 'HOME'. What is the probability that both the letters are vowels?</div>

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

answer:
<div class="solution">Total number of letters $=4$ Total number of vowels $=2(\mathrm{O}$ and $\mathrm{E})$<br/>Required Probability $=\frac{2}{4}=\frac{1}{2}$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-men-hit-at-a-target-with-probabilities-2-1-and-3-1-respectively-what-is-the-probability-that-exactly-one-of-them-hits-the-target

slug:
two-men-hit-at-a-target-with-probabilities-2-1-and-3-1-respectively-what-is-the-probability-that-exactly-one-of-them-hits-the-target-42127

question:
<div class="question">Two men hit at a target with probabilities $\frac{1}{2}$ and $\frac{1}{3}$ respectively. What is the probability that exactly one of them hits the target?</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">$\frac{1}{6}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{2}{3}$</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \mathrm{P}=\mathrm{P}\left(\mathrm{E}_{1}\right) \mathrm{P}\left(\overline{\mathrm{E}}_{2}\right)+\mathrm{P}\left(\overline{\mathrm{E}}_{1}\right) \mathrm{P}\left(\mathrm{E}_{2}\right) \\ &amp;=\frac{1}{2}\left(1-\frac{1}{3}\right)+\left(1-\frac{1}{2}\right) \frac{1}{3} \\ &amp;=\frac{1}{2} \times \frac{2}{3}+\frac{1}{2} \times \frac{1}{3} \\ &amp;=\frac{1}{3}+\frac{1}{6}=\frac{1}{2} \\ \therefore &amp; \text { Option (a) is correct. } \end{aligned}$</div>


tags: ['Physics', 'Thermodynamics', 'JEE Advanced', 'JEE Advanced 2012 (Paper 2)']
source_db: MarksBatch2_P3.db
title:
two-moles-of-ideal-helium-gas-are-in-a-rubber-balloon-at-3-0-c-the-balloon-is-fully-expandable-and-can-be-assumed-to-require-no-energy-in-its-expansio

slug:
two-moles-of-ideal-helium-gas-are-in-a-rubber-balloon-at-3-0-c-the-balloon-is-fully-expandable-and-can-be-assumed-to-require-no-energy-in-its-expansio-17073

question:
<div class="question">Two moles of ideal helium gas are in a rubber balloon at $30^{\circ} \mathrm{C}$. The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to $35^{\circ} \mathrm{C}$. The amount of heat required in raising the temperature is nearly (take $R=8.31 \mathrm{~J} / \mathrm{mol} . \mathrm{K}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$62 \mathrm{~J}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$104 \mathrm{~J}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$124 \mathrm{~J}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$208 \mathrm{~J}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$208 \mathrm{~J}$</span> </div>

answer:
<div class="solution">The heat is supplied at constant pressure. i.e., the process is isobaric
<br/> <br/>$\begin{array}{l}
<br/> <br/>\therefore Q=n C_{p} \Delta t \\
<br/> <br/>=2\left[\frac{5}{2} R\right] \times \Delta t=2 \times \frac{5}{2} \times 8.31 \times 5=208 \mathrm{~J} \\
<br/> <br/>\left(\because C_{p}=\frac{5}{2} R \text { for mono-atomic gas }\right)
<br/> <br/>\end{array}$</div>


tags: ['Physics', 'Thermodynamics', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-moles-of-ideal-helium-gas-is-in-a-rubber-balloon-at-30-c-the-balloon-is-fully-expandable-and-can-be-assumed-to-require-no-energy-in-its-expansion-

slug:
two-moles-of-ideal-helium-gas-is-in-a-rubber-balloon-at-30-c-the-balloon-is-fully-expandable-and-can-be-assumed-to-require-no-energy-in-its-expansion-82323

question:
<div class="question">Two moles of ideal helium gas is in a rubber balloon at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo> </mo><mo>Â°</mo><mi mathvariant="normal">C</mi></math>.The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>35</mn><mo> </mo><mo>Â°</mo><mi mathvariant="normal">C</mi></math>. The amount of heat required in raising the temperature is nearly ( take <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">R</mi><mo> </mo><mo>=</mo><mo> </mo><mn>8</mn><mo>.</mo><mn>31</mn><mo> </mo><mi mathvariant="normal">J</mi><mo> </mo><msup><mi>mol</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mi mathvariant="normal">K</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>62</mn><mo> </mo><mi mathvariant="normal">J</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>140</mn><mo> </mo><mi mathvariant="normal">J</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>124</mn><mo> </mo><mi mathvariant="normal">J</mi></math></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>208</mn><mo> </mo><mi mathvariant="normal">J</mi></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>208</mn><mo> </mo><mi mathvariant="normal">J</mi></math></span> </div>

answer:
<div class="solution"><math> <mrow> <mo>Î</mo> <mtext>Q</mtext> <mo>=</mo> <mi mathvariant="normal">n</mi> <msub> <mtext>C</mtext> <mtext>p</mtext> </msub> <mo>Î</mo> <mtext>T</mtext> <mspace width="0.5em"></mspace> <mfenced close=")" open="(" separators=","> <mrow> <mtext>Isobaric process</mtext> </mrow> </mfenced> </mrow> </math><br/><br/><math> <mrow> <mo>=</mo> <mn>2</mn> <mo>Ã</mo> <mfrac> <mrow> <mn>5</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mtext>R</mtext> <mo>Ã</mo> <mfenced close=")" open="(" separators=","> <mrow> <mn>3</mn> <mn>5</mn> <mo>-</mo> <mn>3</mn> <mn>0</mn> </mrow> </mfenced> </mrow> </math><br/><br/><math> <mrow> <mo>=</mo> <mn>2</mn> <mn>0</mn> <mn>8</mn> <mspace width="0.5em"></mspace> <mtext>J</mtext> </mrow> </math></div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">42</span> </div>
source_db: MarksBatch2_P3.db
title:
two-numbers-are-successively-drawn-from-the-set-1-2-3-4-5-6-7-without-replacement-and-the-outcomes-recorded-in-that-order-what-is-the-number-of-elemen

slug:
two-numbers-are-successively-drawn-from-the-set-1-2-3-4-5-6-7-without-replacement-and-the-outcomes-recorded-in-that-order-what-is-the-number-of-elemen-61563

question:
<div class="question">Two numbers are successively drawn from the set $\{1,2,3,4$, 5. 6. 7\} without replacement and the outcomes recorded in<br/>that order. What is the number of elementary events in the randomexperiment?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">49</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">42</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">14</span> </li> </ul>

answer:
<div class="solution">Given set is $\{1,2,3,4,5,6,7\}$ Total numbers in set $=7$ Since, two numbers are drawn without replacement $\therefore$ Total number of elementary events $={ }^{7} C_{1} \times{ }^{6} C_{1}$ $=7 \times 6$<br/>$=42$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2013 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">56</span> </div>
source_db: MarksBatch2_P3.db
title:
two-numbers-are-successively-drawn-from-the-set-u-1-2-3-4-5-6-7-8-the-second-being-drawn-without-replacing-the-first-the-number-of-elementary-events-i

slug:
two-numbers-are-successively-drawn-from-the-set-u-1-2-3-4-5-6-7-8-the-second-being-drawn-without-replacing-the-first-the-number-of-elementary-events-i-46271

question:
<div class="question">Two numbers are successively drawn from the set $\mathrm{U}=\{1,2,3,4,5,6,7,8\}$, the second being drawn without replacing the first. The number of elementary events in the sample is:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">64</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">56</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">32</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">14</span> </li> </ul>

answer:
<div class="solution">Required no. of elementary events $={ }^{8} \mathrm{C}_{2} \times 2 !=56$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2009 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
title:
two-numbers-x-and-y-are-simultaneously-drawn-from-the-set-1-2-3-4-5-6-7-8-9-10-what-is-the-conditional-probability-of-exactly-one-of-the-two-numbers-x

slug:
two-numbers-x-and-y-are-simultaneously-drawn-from-the-set-1-2-3-4-5-6-7-8-9-10-what-is-the-conditional-probability-of-exactly-one-of-the-two-numbers-x-79621

question:
<div class="question">Two numbers $X$ and $Y$ are simultaneously drawn from the set $\{1,2,3,4,5,6,7,8,9,10\}$. What is the conditional probability of exactly one of the two numbers $X$ and $Y$ bems even, given $(X+Y)=15$ ?</div>

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

answer:
<div class="solution">Given, $X+Y=15$ The total number of ordered pairs which satisfies $X+Y=15$ is<br/>$\therefore(5,10),(6,9),(7,8),(8,7),(9,6),(10,5)$<br/>$\therefore n(S)=6$ where $S$ denotes the sample space. In each above pairs exactly one of the two numbers is even number. Therefore $E=\{(5,10),(6,9),(7,8),(8,7),(9,6),(10,5)\}$<br/>where $E$ is an event.<br/>$\therefore$<br/>$n(E)=6$<br/>$\therefore$ Required probability $=\frac{n(E)}{n(S)}=\frac{6}{6}=1$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2020 (14 Oct Shift 1)']
source_db: MarksBatch2_P3.db
title:
two-open-organ-pipes-of-fundamental-frequencies-n-1-and-n-2-are-joined-in-series-the-fundamental-frequency-of-the-new-pipe-is

slug:
two-open-organ-pipes-of-fundamental-frequencies-n-1-and-n-2-are-joined-in-series-the-fundamental-frequency-of-the-new-pipe-is-23697

question:
<div class="question">Two open organ pipes of fundamental frequencies $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$ are joined in series. The fundamental frequency of the new pipe is</div>

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

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

answer:
<div class="solution">$\mathfrak{n}=\frac{\mathrm{V}}{2 \ell_{1}} \quad \therefore \quad \ell_{1}=\frac{\mathrm{V}}{2 \mathrm{n}_{1}}$<br/>$\mathfrak{n}_{2}=\frac{\mathrm{V}}{2 \ell_{2}} \quad \therefore \quad \ell_{2}=\frac{\mathrm{V}}{2 \mathrm{n}_{2}}$<br/>$\mathfrak{n}=\frac{\mathrm{V}}{2 \ell} \quad \therefore \quad \ell=\frac{\mathrm{V}}{2 \mathrm{n}}$<br/>$\ell=\ell_{1}+\ell_{2}$<br/>$\frac{\mathrm{V}}{2 \mathrm{n}}=\frac{\mathrm{V}}{2 \mathrm{n}_{1}}+\frac{\mathrm{V}}{2 \mathrm{n}_{2}}$<br/>$\therefore \frac{1}{\mathrm{n}}=\frac{1}{\mathrm{n}_{1}}+\frac{1}{\mathrm{n}_{2}}$<br/>$\therefore \mathrm{n}=\frac{\mathrm{n}_{1} \mathrm{n}_{2}}{\mathrm{n}_{1}+\mathrm{n}_{2}}$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2022 (06 Jul Shift 2)']
source_db: MarksBatch2_P3.db
title:
two-open-organ-pipes-of-lengths-50-cm-and-51-cm-are-totally-immersed-in-a-medium-they-are-found-to-give-40-beats-in-10-s-when-each-is-sounding-at-its-

slug:
two-open-organ-pipes-of-lengths-50-cm-and-51-cm-are-totally-immersed-in-a-medium-they-are-found-to-give-40-beats-in-10-s-when-each-is-sounding-at-its-46931

question:
<div class="question">Two open organ pipes of lengths $50 \mathrm{~cm}$ and $51 \mathrm{~cm}$ are totally immersed in a medium. They are found to give 40 beats in $10 \mathrm{~s}$ when each is sounding at its fundamental note. The speed of sound in this medium is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$275 \mathrm{~ms}^{-1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$310 \mathrm{~ms}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$258 \mathrm{~ms}^{-1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$204 \mathrm{~ms}^{-1}$</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">$204 \mathrm{~ms}^{-1}$</span> </div>

answer:
<div class="solution">We know that, fundamental frequency<br/>$$<br/>\begin{aligned}<br/>f_0 &amp; =\frac{v}{2 l} \\<br/>\text { Beat } &amp; =f_1-f_2 \\<br/>\Rightarrow \frac{40}{10} &amp; =\frac{v}{2 l_1}-\frac{v}{2 l_2} \Rightarrow 4=\frac{v}{2}\left(\frac{1}{l_1}-\frac{1}{l_2}\right) \\<br/>\Rightarrow \quad 4 &amp; =\frac{v}{2}\left(\frac{1}{50}-\frac{1}{51}\right) \times \frac{1}{10^{-2}} \\<br/>\Rightarrow 8 \times 10^{-2} &amp; =v\left(\frac{51-50}{50 \times 51}\right) \\<br/>\Rightarrow 8 \times 10^{-2} &amp; =\frac{v}{2550} \\<br/>\Rightarrow \quad v &amp; =2550 \times 8 \times 10^{-2} \\<br/>&amp; =204 \mathrm{~ms}^{-1}<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2019 (02 May Shift 1)']
source_db: MarksBatch2_P3.db
title:
two-open-pipes-of-different-lengths-and-of-same-diameter-in-which-the-air-column-vibrates-with-fundamental-frequencies-n-1-and-n-2-respectively-when-b

slug:
two-open-pipes-of-different-lengths-and-of-same-diameter-in-which-the-air-column-vibrates-with-fundamental-frequencies-n-1-and-n-2-respectively-when-b-91837

question:
<div class="question">Two open pipes of different lengths and of same diameter in which the air column vibrates with fundamental frequencies â <math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></math> â, and â <math><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></math> â respectively. When both pipes are joined to form a single pipe, its fundamental frequency will be</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></math></span> </div>

answer:
<div class="solution"><math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>;</mo><mo>â</mo><mi mathvariant="normal"></mi><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><mo>â</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mo>â</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>2</mn><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></math> <br/><math><mo>â´</mo><mo>â</mo><mi>n</mi><mo>â</mo><mo>=</mo><mo>â</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>2</mn><mfenced separators="|"><mrow><mi>r</mi><mo>+</mo><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfenced></mrow></mfrac><mo>â</mo><mo>=</mo><mo>â</mo><mo>â</mo><mo>â</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mn>2</mn><mi>r</mi><mo>+</mo><mn>2</mn><msub><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></math> <br/><math><mi>n</mi><mo>â</mo><mo>=</mo><mo>â</mo><mfrac><mrow><mi>v</mi></mrow><mrow><mfrac><mrow><mi>v</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>v</mi></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></mrow></mfrac><mo>â</mo><mo>â</mo><mo>â</mo><mo>=</mo><mo>â</mo><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></math></div>


title: two-plants-can-be-conclusively-said-to-belong-to-the-same-species-if-they-1
slug: two-plants-can-be-conclusively-said-to-belong-to-the-same-species-if-they-1-30950
tags: ['Biology', 'The Living World', 'NEET', 'NEET 2007']
source_db: MarksBatch2_P3.db
question:
<div class="question">Two plants can be conclusively said to belong to the same species if they:</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">have more than 90 per cent similar genes</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">look similar and possess identical secondary metabolites</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">have same number of chromosomes</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">can reproduce freely with each other and form seeds.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">can reproduce freely with each other and form seeds.</span> </div>

answer:
<div class="solution">Two organisms are said to belong to same species if they can interbreed among themselves to produce fertile offspring. In the case of plants, seeds represent the embryonic structure of the plant which later develops into offspring. So, the correct option is 'Can reproduce freely with each other and form seeds'.</div>


title: two-plants-can-be-conclusively-said-to-belong-to-the-same-species-if-they
slug: two-plants-can-be-conclusively-said-to-belong-to-the-same-species-if-they-46321
tags: ['Biology', 'The Living World', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">Two plants can be conclusively said to belong to the same species if they</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Have more than 90 percent similar genes</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Look similar and possess identical secondary metabolites</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Have same number of chromosomes</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Can reproduce freely with each other and form seeds.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Can reproduce freely with each other and form seeds.</span> </div>

answer:
<div class="solution">When the members of a species are inter-fertile and produce fertile offsprings.</div>


tags: ['Mathematics', 'Heights and Distances', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-poles-are-10-m-and-20-m-high-the-line-joining-their-tips-makes-an-angle-of-1-5-with-the-horizontal-what-is-the-distance-between-the-poles

slug:
two-poles-are-10-m-and-20-m-high-the-line-joining-their-tips-makes-an-angle-of-1-5-with-the-horizontal-what-is-the-distance-between-the-poles-37498

question:
<div class="question">Two poles are $10 \mathrm{~m}$ and $20 \mathrm{~m}$ high. The line joining their tips makes an angle of $15^{\circ}$ with the horizontal. What is the distance between the poles? $\quad$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10(\sqrt{3}-1) \mathrm{m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$5(4+2 \sqrt{3}) \mathrm{m}$</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">$20(\sqrt{3}+1) \mathrm{m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10(\sqrt{3}+1) \mathrm{m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$5(4+2 \sqrt{3}) \mathrm{m}$</span> </div>

answer:
<div class="solution">Let $\mathrm{AB}$ and $\mathrm{CD}$ be two poles of height $10 \mathrm{~m}$ and $20 \mathrm{~m}$ respectively. $\operatorname{In} \Delta \mathrm{AEC}$<br/>$\frac{\mathrm{CE}}{\mathrm{AE}}=\tan 15^{\circ}$<br/>$\frac{10}{\mathrm{AE}}=\tan \left(45^{\circ}-30^{\circ}\right)$<br/>$=\frac{\tan 45^{\circ}-\tan 30^{\circ}}{1+\tan 45^{\circ} \tan 30^{\circ}}$<br/>$=\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}$<br/>$\frac{10}{\mathrm{AE}}=\frac{\sqrt{3}-1}{\sqrt{3}+1}$<br/>$\mathrm{AE}=\frac{10(\sqrt{3}+1)}{\sqrt{3}-1} \times \frac{\sqrt{3}+1}{\sqrt{3}+1}=5(4+2 \sqrt{3})$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/XAC9haTduiyB1x9QJs2WSa5h9CGBwlZG4snklvr1nE0.original.fullsize.png"/></div>


tags: ['Mathematics', 'Heights and Distances', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
two-poles-are-10-m-and-20-m-high-the-line-joining-their-tops-makes-an-angle-of-1-5-with-the-horizontal-the-distance-between-the-poles-is-approximately

slug:
two-poles-are-10-m-and-20-m-high-the-line-joining-their-tops-makes-an-angle-of-1-5-with-the-horizontal-the-distance-between-the-poles-is-approximately-72280

question:
<div class="question">Two poles are $10 \mathrm{~m}$ and $20 \mathrm{~m}$ high. The line joining their tops makes an angle of $15^{\circ}$ with the horizontal. The distance between the poles is approximately equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$36.3 \mathrm{~m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$37.3$ in</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$38.3 \mathrm{~m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$39.3$ in</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$37.3$ in</span> </div>

answer:
<div class="solution">$\tan 15^{\circ}=\mathrm{DE} / \mathrm{AE}$<br/>$\mathrm{AE}=10 \cot 15^{\circ}$<br/>$\cot \left(15^{\circ}\right)=\cot \left(45-30^{\circ}\right)$<br/>$=\frac{\cot 45^{\circ} \cot 30^{\circ}+1}{\cot 30^{\circ}-\cot 45^{\circ}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/YHZepjMP5ItXG6fi8FuhUtCl7sJdt771DK2KV6qJxrE.original.fullsize.png"/><br/><br/>$\cot 15^{\circ}=\frac{1 \cdot \sqrt{3}+1}{\sqrt{3}-1}=2+\sqrt{3}$<br/>Putting $\cot 15^{\circ}$ in eq (1) $\begin{aligned} \mathrm{AE} &amp;=10 \cot 15^{\circ} \\ &amp;=10(2+\sqrt{3}) \\ &amp;=10(3.73)=37.3 \mathrm{~m} \end{aligned}$</div>


tags: ['Mathematics', 'Heights and Distances', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
two-poles-are-10-mand-20-mhigh-the-line-joining-their-tops-makes-an-angle-of-1-5-with-the-horizontal-what-is-the-approximate-distance-between-the-pole

slug:
two-poles-are-10-mand-20-mhigh-the-line-joining-their-tops-makes-an-angle-of-1-5-with-the-horizontal-what-is-the-approximate-distance-between-the-pole-22397

question:
<div class="question">Two poles are 10 mand 20 mhigh. The line joining their tops makes an angle of $15^{\circ}$ with the horizontal. What is the approximate distance between the poles?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$35.3 \mathrm{~m}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$37.3 \mathrm{~m}$</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">$41 \mathrm{~m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$44 \mathrm{~m}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$37.3 \mathrm{~m}$</span> </div>

answer:
<div class="solution">Let $\mathrm{AC}$ and $\mathrm{ED}$ be two poles of height $20 \mathrm{~m}$ and $10 \mathrm{~m}$ respectively.<br/>Let $\angle \mathrm{AEB}=15^{\circ}$<br/>To find $: \mathrm{DC}$.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/doFUjLAA5ekNZ35WG3mzjBxNZeNW6KXWzluTlXoKviE.original.fullsize.png"/><br/><br/>Now, $\mathrm{AB}=\mathrm{AC}-\mathrm{BC}=\mathrm{AC}-\mathrm{ED}=20-10=10 \mathrm{~m}$<br/>Now in $\triangle \mathrm{ABE}, \quad(\because \mathrm{BC}=\mathrm{ED})$<br/>$\tan 15^{\circ}=\frac{\mathrm{AB}}{\mathrm{BE}}$<br/>$\Rightarrow \tan \left(45^{\circ}-30^{\circ}\right)=\frac{10}{\mathrm{BE}}$<br/>$\Rightarrow \frac{\tan 45^{\circ}-\tan 30^{\circ}}{1+\tan 45^{\circ} \tan 30^{\circ}}=\frac{10}{\mathrm{BE}}$<br/>$\left(\because \tan (\mathrm{A}-\mathrm{B})=\frac{\tan \mathrm{A}-\tan \mathrm{B}}{1+\tan \mathrm{A} \tan \mathrm{B}}\right)$<br/>$\Rightarrow \frac{\sqrt{3}-1}{\sqrt{3}+1}=\frac{10}{\mathrm{BE}}$<br/>$\Rightarrow \quad \mathrm{BE}=10\left(\frac{\sqrt{3}+1}{\sqrt{3}-1}\right)=\frac{10 \times(\sqrt{3}+1)^{2}}{2}$<br/>(Byrationalizing)<br/>$5(3+1+2 \sqrt{3})=5(4+2 \times 1.73)=5(4+3.46)$<br/>$\Rightarrow \mathrm{CD}=\mathrm{BE}=5 \times 7.46=37.3 \mathrm{~m}$</div>


tags: ['Chemistry', 'Thermodynamics (C)', 'JEE Main', 'JEE Main 2024 (30 Jan Shift 2)']
options: None
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">492</span> </div>
source_db: MarksBatch2_P3.db
title:
two-reactions-are-given-below-2-fe-s-3-2-o-2-g-fe-2-o-3-s-h-o-822-kj-mol-c-s-1-2-o-2-g-co-g-h-o-110-kj-mol-then-enthalpy-change-for-following-reaction

slug:
two-reactions-are-given-below-2-fe-s-3-2-o-2-g-fe-2-o-3-s-h-o-822-kj-mol-c-s-1-2-o-2-g-co-g-h-o-110-kj-mol-then-enthalpy-change-for-following-reaction-35597

question:
<div class="question"><p>Two reactions are given below:<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi>Fe</mi><mrow><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msub><mi mathvariant="normal">O</mi><mrow><mn>2</mn><mo>(</mo><mo>Â </mo><mi mathvariant="normal">g</mi><mo>)</mo></mrow></msub><mo>â</mo><msub><mi>Fe</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mrow><mn>3</mn><mo>(</mo><mo>Â </mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>,</mo><mi>Î</mi><msup><mi mathvariant="normal">H</mi><mi mathvariant="normal">o</mi></msup><mo>=</mo><mo>-</mo><mn>822</mn><mo>Â </mo><mi>kJ</mi><mo>/</mo><mi>mol</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mrow><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi mathvariant="normal">O</mi><mrow><mn>2</mn><mo>(</mo><mo>Â </mo><mi mathvariant="normal">g</mi><mo>)</mo></mrow></msub><mo>â</mo><msub><mi>CO</mi><mrow><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo></mrow></msub><mo>,</mo><mi>Î</mi><msup><mi mathvariant="normal">H</mi><mi mathvariant="normal">o</mi></msup><mo>=</mo><mo>-</mo><mn>110</mn><mo>Â </mo><mi>kJ</mi><mo>/</mo><mi>mol</mi></math></p><p>Then enthalpy change for following reaction<br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi mathvariant="normal">C</mi><mrow><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>+</mo><msub><mi>Fe</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mrow><mn>3</mn><mo>(</mo><mo>Â </mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>â</mo><mn>2</mn><msub><mi>Fe</mi><mrow><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>+</mo><mn>3</mn><msub><mi>CO</mi><mrow><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo></mrow></msub></math></p></div>

answer:
<div class="solution"><blockquote><p>Enthalpy change is the standard enthalpy of formation, in any general chemical reaction, the reactants undergo chemical changes and combine to give products. It can be represented by the following equation:</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Reactant</mi><mo>â</mo><mi>Product</mi></math></p><p>From the given equations,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msub><mi>Fe</mi><mrow><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msub><mi mathvariant="normal">O</mi><mrow><mn>2</mn><mo>(</mo><mo>Â </mo><mi mathvariant="normal">g</mi><mo>)</mo></mrow></msub><mo>â</mo><msub><mi>Fe</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mrow><mn>3</mn><mo>(</mo><mo>Â </mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>,</mo><mi>Î</mi><msup><mi mathvariant="normal">H</mi><mi mathvariant="normal">o</mi></msup><mo>=</mo><mo>-</mo><mn>822</mn><mo>Â </mo><mi>kJ</mi><mo>/</mo><mi>mol</mi></math>Â .........(i)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">C</mi><mrow><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi mathvariant="normal">O</mi><mrow><mn>2</mn><mo>(</mo><mo>Â </mo><mi mathvariant="normal">g</mi><mo>)</mo></mrow></msub><mo>â</mo><msub><mi>CO</mi><mrow><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo></mrow></msub><mo>,</mo><mi>ÎH</mi><mo>Â°</mo><mo>=</mo><mo>-</mo><mn>110</mn><mo>Â </mo><mi>kJ</mi><mo>/</mo><mi>mol</mi></math>....(ii)</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi mathvariant="normal">C</mi><mrow><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>+</mo><msub><mi>Fe</mi><mn>2</mn></msub><msub><mi mathvariant="normal">O</mi><mrow><mn>3</mn><mo>(</mo><mo>Â </mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>â</mo><mn>2</mn><msub><mi>Fe</mi><mrow><mo>(</mo><mi mathvariant="normal">s</mi><mo>)</mo></mrow></msub><mo>+</mo><mn>3</mn><msub><mi>CO</mi><mrow><mo>(</mo><mi mathvariant="normal">g</mi><mo>)</mo></mrow></msub><mo>,</mo><msub><mi>ÎH</mi><mn>3</mn></msub><mo>=</mo><mo>?</mo></math></p><p>The value ofÂ <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ÎH</mi><mn>3</mn></msub></math>Â can be calculated as,</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>ÎH</mi><mn>3</mn></msub><mo>=</mo><mn>3</mn><mo>Ã</mo><msub><mi>ÎH</mi><mn>2</mn></msub><mo>-</mo><msub><mi>ÎH</mi><mn>1</mn></msub></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>(</mo><mo>-</mo><mn>110</mn><mo>)</mo><mo>+</mo><mn>822</mn></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>492</mn><mo>Â </mo><mi>kJ</mi><mo>/</mo><mi>mole</mi></math></p></blockquote></div>


tags: ['Mathematics', 'Statistics']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
title:
two-sets-each-of-20-observations-have-the-same-standard-deviation-5-the-first-set-has-a-mean-17-and-the-second-a-mean-22-determine-the-standatd-deviat

slug:
two-sets-each-of-20-observations-have-the-same-standard-deviation-5-the-first-set-has-a-mean-17-and-the-second-a-mean-22-determine-the-standatd-deviat-80680

question:
<div class="question">Two sets each of 20 observations, have the same standard deviation 5 . The first set has a mean 17 and the second a mean 22. Determine the standatd deviation of the set obtained by combining the given two sets.</div>

answer:
<div class="solution">We have, $n_1=20, \sigma_1=5, \bar{x}_1=17$<br/>$$<br/>\begin{aligned}<br/>n_2 &amp;=20, \sigma_2=5, \bar{x}_2=22 \\<br/>\bar{x}_{12} &amp;=\frac{n_1 \bar{x}_1+n_2 \bar{x}_2}{n_1+n_2} \\<br/>&amp;=\frac{(20)(17)+(20)(22)}{20+20}=\frac{340+440}{40}=19.5 \\<br/>d_1 &amp;=\bar{x}_1-\bar{x}_{12}=17-19.5=-2.5<br/>\end{aligned}<br/>$$<br/><br/><br/>$$<br/>\begin{aligned}<br/>d_2 &amp;=\bar{x}_2-\bar{x}_{12}=22-19.5=2.5 \\<br/>\sigma_{12} &amp;=\sqrt{\frac{n_1\left(\sigma_1^2+d_1^2\right)+n_2\left(\sigma_2^2+d_2^2\right)}{n_1+n_2}} \\<br/>&amp;=\sqrt{\frac{20(25+6.25)+20(25+6.25)}{20+20}} \\<br/>&amp;=\sqrt{\frac{1250}{40}}=\sqrt{31.25}=5.59<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2015 (Phase 1)']
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P3.db
title:
two-similar-boxes-b-i-i-1-2-contain-i-1-red-and-5-i-1-black-balls-one-box-is-chosen-at-random-and-two-balls-are-drawn-randomly-what-is-the-probability

slug:
two-similar-boxes-b-i-i-1-2-contain-i-1-red-and-5-i-1-black-balls-one-box-is-chosen-at-random-and-two-balls-are-drawn-randomly-what-is-the-probability-96453

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{5}$</span> </div>


title: two-straight-line-paths-are-represented-by-the-equation-2-x-y-2-and-4-x-2-y-6-then-the-paths-will
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
slug:
two-straight-line-paths-are-represented-by-the-equation-2-x-y-2-and-4-x-2-y-6-then-the-paths-will-28663

question:
<div class="question">Two straight line paths are represented by the equation $2 \mathrm{x}-\mathrm{y}=2$ and $-4 \mathrm{x}+2 \mathrm{y}=6$. Then the paths will</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">not cross each other</span> </div>

answer:
<div class="solution">Given lines are $2 \mathrm{x}-\mathrm{y}=2$ and $2 \mathrm{x}-\mathrm{y}=-3$ since,<br/>Both lines are parallel so they will never meet.</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
two-straight-lines-passing-through-the-point-a-3-2-cut-the-line-2-y-x-3-and-x-axis-perpendicularly-at-p-and-q-respectively-the-equation-of-the-line-pq

slug:
two-straight-lines-passing-through-the-point-a-3-2-cut-the-line-2-y-x-3-and-x-axis-perpendicularly-at-p-and-q-respectively-the-equation-of-the-line-pq-57524

question:
<div class="question">Two straight lines passing through the point $\mathrm{A}(3,2)$ cut the line $2 \mathrm{y}=\mathrm{x}+3$ and $\mathrm{x}$ -axis perpendicularly at $\mathrm{P}$ and $\mathrm{Q}$ respectively. The equation of the line PQ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$7 x+y-21=0$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x+7 y+21=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 x+y-8=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x+2 y+8=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$7 x+y-21=0$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/mCpHvB-8hbUf0fmez_XADlfcojkSBrCtWyRUoOfnNOQ.original.fullsize.png"/><br/><br/>$\because$ Coordinates of $Q$ are $(3,0) \&amp;$ it passes through $P Q$.<br/>$\therefore$ Putting the values of $(x=3) \&amp;(y=0)$ in options we get:<br/>Equation of line $\mathrm{PQ}=7 \mathrm{x}+\mathrm{y}-21=0$</div>


title: two-straight-lines-x-3-y-2-0-and-2-x-6-y-6-0
slug: two-straight-lines-x-3-y-2-0-and-2-x-6-y-6-0-76624
question: <div class="question">Two straight lines $x-3 y-2=0$ and $2 x-6 y-6=0$</div>
tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">never intersect</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">intersect at a single point</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">intersect at infinite number of points</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">intersect at more than one point (but finite number of points)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">never intersect</span> </div>

answer:
<div class="solution">Given equation of straight lines are $x-3 y-2=0$ and $2 x$ $-6 y-6=0$<br/>Here, $a_{1}=1, a_{2}=2, b_{1}=-3, b_{2}=-6$,<br/>$c_{1}=-2, c_{2}=-6$<br/>Now, $\frac{a_{1}}{a_{2}}=\frac{1}{2}, \frac{b_{1}}{b_{2}}=\frac{1}{2}, \frac{c_{1}}{c_{2}}=\frac{1}{3}$<br/>$\Rightarrow \quad \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$<br/>$\therefore \quad$ Both straight lines never intersect.</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2006']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">144</span> </div>
source_db: MarksBatch2_P3.db
title:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-d-a-d-b-is-081-the--1

slug:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-d-a-d-b-is-081-the-1-39658

question:
<div class="question">Two strings $A$ and $B$ of lengths, $L_A=80 \mathrm{~cm}$ and $L_B=x \mathrm{~cm}$ respectively are used separately in a sonometer. The ratio of their densities $\left(d_A / d_B\right)$ is 0.81 . The diameter of $B$ is one-half that of $A$. If the strings have the same tension and fundamental frequency the value of $x$ is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">33</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">102</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">144</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">130</span> </li> </ul>

answer:
<div class="solution">Given, $\frac{T_A}{T_B}=1, \quad \frac{L_A}{L_B}=\frac{80}{x}$<br/>$\frac{D_A}{D_B}=\frac{2}{1}, \quad \frac{d_A}{d_B}=\frac{0.81}{1}$<br/>Let $\mu_1$ and $\mu_2$ be the linear densities.<br/>$\therefore \quad \frac{\mu_A}{\mu_B}=\left(\frac{D_A}{D_B}\right)^2 \times \frac{d_A}{d_B}=\left(\frac{2}{1}\right)^2 \times 0.81$<br/>$=4 \times 0.81=3.24$<br/>$\therefore \quad \frac{f_1}{f_2}=\frac{L_B}{L_A} \times \sqrt{\frac{T_A}{T_B} \times \frac{\mu_B}{\mu_{\mathrm{A}}}}$<br/>$1=\frac{x}{80} \times \sqrt{1 \times \frac{1}{3.24}}$<br/>$\therefore \quad x=80 \times \sqrt{3.24}$<br/>or $\quad x=144$</div>


tags: ['Physics', 'Waves and Sound', 'TS EAMCET', 'TS EAMCET 2006']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">144</span> </div>
source_db: MarksBatch2_P3.db
title:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-d-a-d-b-is-081-the-

slug:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-d-a-d-b-is-081-the-33995

question:
<div class="question">Two strings $A$ and $B$ of lengths, $L_A=80 \mathrm{~cm}$ and $L_B=x \mathrm{~cm}$ respectively are used separately in a sonometer. The ratio of their densities $\left(d_A / d_B\right)$ is 0.81 . The diameter of $B$ is one-half that of $A$. If the strings have the same tension and fundamental frequency the value of $x$ is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">33</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">102</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">144</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">130</span> </li> </ul>

answer:
<div class="solution">Given, $\frac{T_A}{T_B}=1, \quad \frac{L_A}{L_B}=\frac{80}{x}$<br/>$\frac{D_A}{D_B}=\frac{2}{1}, \quad \frac{d_A}{d_B}=\frac{0.81}{1}$<br/>Let $\mu_1$ and $\mu_2$ be the linear densities.<br/>$\therefore \quad \frac{\mu_A}{\mu_B}=\left(\frac{D_A}{D_B}\right)^2 \times \frac{d_A}{d_B}=\left(\frac{2}{1}\right)^2 \times 0.81$<br/>$=4 \times 0.81=3.24$<br/>$\therefore \quad \frac{f_1}{f_2}=\frac{L_B}{L_A} \times \sqrt{\frac{T_A}{T_B} \times \frac{\mu_B}{\mu_{\mathrm{A}}}}$<br/>$1=\frac{x}{80} \times \sqrt{1 \times \frac{1}{3.24}}$<br/>$\therefore \quad x=80 \times \sqrt{3.24}$<br/>or $\quad x=144$</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-is-0-81-the-diamete-1

slug:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-is-0-81-the-diamete-1-86505

question:
<div class="question">Two strings <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> of lengths, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>A</mi></msub><mo>=</mo><mn>80</mn><mi mathvariant="normal"> </mi><mi>cm</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>B</mi></msub><mo>=</mo><mi>x</mi><mi mathvariant="normal"> </mi><mi>cm</mi></math> respectively are used separately in a sonometer. The ratio of their densities is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>81</mn></math>. the diameter of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> is one-half that of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>. if the strings have the same tension and fundamental frequency the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>33</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>102</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>144</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>130</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>144</mn></math></span> </div>

answer:
<div class="solution">Given,<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>T</mi><mi mathvariant="normal">A</mi></msub><msub><mi>T</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mfrac><msub><mi>L</mi><mi mathvariant="normal">A</mi></msub><msub><mi>L</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mfrac><mn>80</mn><mi>x</mi></mfrac></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>D</mi><mi mathvariant="normal">A</mi></msub><msub><mi>D</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>1</mn></mfrac><mo>,</mo><mfrac><msub><mi>d</mi><mi mathvariant="normal">A</mi></msub><msub><mi>d</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mfrac><mn>0.81</mn><mn>1</mn></mfrac></math><br/><br/>Let <math><msub><mrow><mi mathvariant="normal">μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">μ</mi></mrow><mrow><mn>2</mn></mrow></msub></math> be the linear densities.<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfrac><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">A</mi></msub><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><msup><mfenced separators="|"><mfrac><msub><mi>D</mi><mi mathvariant="normal">A</mi></msub><msub><mi>D</mi><mi mathvariant="normal">B</mi></msub></mfrac></mfenced><mn>2</mn></msup><mo>×</mo><mfrac><msub><mi>d</mi><mi mathvariant="normal">A</mi></msub><msub><mi>d</mi><mi mathvariant="normal">B</mi></msub></mfrac></math><br/><br/><math><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mn>0.81</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>81</mn><mo>=</mo><mn>3</mn><mo>.</mo><mn>24</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mfrac><msub><mi>v</mi><mn>1</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>L</mi><mi mathvariant="normal">B</mi></msub><msub><mi>L</mi><mi mathvariant="normal">A</mi></msub></mfrac><mo>×</mo><msqrt><mfrac><msub><mi>T</mi><mi mathvariant="normal">A</mi></msub><msub><mi>T</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>×</mo><mfrac><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">B</mi></msub><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">A</mi></msub></mfrac></msqrt></math><br/><br/><math><mn>1</mn><mo>=</mo><mfrac><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>80</mn></mrow></mfrac><mo>×</mo><msqrt><mn>1</mn><mo>×</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3.21</mn></mrow></mfrac></msqrt><mi mathvariant="normal"> </mi></math><br/><br/>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>144</mn></math></div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-is-0-81-the-diamete-2

slug:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-is-0-81-the-diamete-2-54962

question:
<div class="question">Two strings <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> of lengths, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>A</mi></msub><mo>=</mo><mn>80</mn><mi mathvariant="normal"> </mi><mi>cm</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>B</mi></msub><mo>=</mo><mi>x</mi><mi mathvariant="normal"> </mi><mi>cm</mi></math> respectively are used separately in a sonometer. The ratio of their densities is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>81</mn></math>. the diameter of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> is one-half that of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>. if the strings have the same tension and fundamental frequency the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>33</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>102</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>144</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>130</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>144</mn></math></span> </div>

answer:
<div class="solution">Given,<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>T</mi><mi mathvariant="normal">A</mi></msub><msub><mi>T</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mfrac><msub><mi>L</mi><mi mathvariant="normal">A</mi></msub><msub><mi>L</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mfrac><mn>80</mn><mi>x</mi></mfrac></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>D</mi><mi mathvariant="normal">A</mi></msub><msub><mi>D</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>1</mn></mfrac><mo>,</mo><mfrac><msub><mi>d</mi><mi mathvariant="normal">A</mi></msub><msub><mi>d</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mfrac><mn>0.81</mn><mn>1</mn></mfrac></math><br/><br/>Let <math><msub><mrow><mi mathvariant="normal">μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">μ</mi></mrow><mrow><mn>2</mn></mrow></msub></math> be the linear densities.<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfrac><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">A</mi></msub><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><msup><mfenced separators="|"><mfrac><msub><mi>D</mi><mi mathvariant="normal">A</mi></msub><msub><mi>D</mi><mi mathvariant="normal">B</mi></msub></mfrac></mfenced><mn>2</mn></msup><mo>×</mo><mfrac><msub><mi>d</mi><mi mathvariant="normal">A</mi></msub><msub><mi>d</mi><mi mathvariant="normal">B</mi></msub></mfrac></math><br/><br/><math><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mn>0.81</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>81</mn><mo>=</mo><mn>3</mn><mo>.</mo><mn>24</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mfrac><msub><mi>v</mi><mn>1</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>L</mi><mi mathvariant="normal">B</mi></msub><msub><mi>L</mi><mi mathvariant="normal">A</mi></msub></mfrac><mo>×</mo><msqrt><mfrac><msub><mi>T</mi><mi mathvariant="normal">A</mi></msub><msub><mi>T</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>×</mo><mfrac><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">B</mi></msub><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">A</mi></msub></mfrac></msqrt></math><br/><br/><math><mn>1</mn><mo>=</mo><mfrac><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>80</mn></mrow></mfrac><mo>×</mo><msqrt><mn>1</mn><mo>×</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3.21</mn></mrow></mfrac></msqrt><mi mathvariant="normal"> </mi></math><br/><br/>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>144</mn></math></div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-is-0-81-the-diamete

slug:
two-strings-a-and-b-of-lengths-l-a-80-cm-and-l-b-x-cm-respectively-are-used-separately-in-a-sonometer-the-ratio-of-their-densities-is-0-81-the-diamete-71713

question:
<div class="question">Two strings <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> of lengths, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>A</mi></msub><mo>=</mo><mn>80</mn><mi mathvariant="normal"> </mi><mi>cm</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mi>B</mi></msub><mo>=</mo><mi>x</mi><mi mathvariant="normal"> </mi><mi>cm</mi></math> respectively are used separately in a sonometer. The ratio of their densities is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>81</mn></math>. the diameter of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> is one-half that of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math>. if the strings have the same tension and fundamental frequency the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>33</mn></math></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>102</mn></math></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>144</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>130</mn></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>144</mn></math></span> </div>

answer:
<div class="solution">Given,<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>T</mi><mi mathvariant="normal">A</mi></msub><msub><mi>T</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mfrac><msub><mi>L</mi><mi mathvariant="normal">A</mi></msub><msub><mi>L</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mfrac><mn>80</mn><mi>x</mi></mfrac></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>D</mi><mi mathvariant="normal">A</mi></msub><msub><mi>D</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mfrac><mn>2</mn><mn>1</mn></mfrac><mo>,</mo><mfrac><msub><mi>d</mi><mi mathvariant="normal">A</mi></msub><msub><mi>d</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><mfrac><mn>0.81</mn><mn>1</mn></mfrac></math><br/><br/>Let <math><msub><mrow><mi mathvariant="normal">μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">a</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi><mi mathvariant="normal"> </mi><msub><mrow><mi mathvariant="normal">μ</mi></mrow><mrow><mn>2</mn></mrow></msub></math> be the linear densities.<br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mfrac><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">A</mi></msub><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>=</mo><msup><mfenced separators="|"><mfrac><msub><mi>D</mi><mi mathvariant="normal">A</mi></msub><msub><mi>D</mi><mi mathvariant="normal">B</mi></msub></mfrac></mfenced><mn>2</mn></msup><mo>×</mo><mfrac><msub><mi>d</mi><mi mathvariant="normal">A</mi></msub><msub><mi>d</mi><mi mathvariant="normal">B</mi></msub></mfrac></math><br/><br/><math><mo>=</mo><msup><mrow><mfenced separators="|"><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></mfrac></mrow></mfenced></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><mn>0.81</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>×</mo><mn>0</mn><mo>.</mo><mn>81</mn><mo>=</mo><mn>3</mn><mo>.</mo><mn>24</mn></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴</mo><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mi mathvariant="normal"> </mi><mfrac><msub><mi>v</mi><mn>1</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>L</mi><mi mathvariant="normal">B</mi></msub><msub><mi>L</mi><mi mathvariant="normal">A</mi></msub></mfrac><mo>×</mo><msqrt><mfrac><msub><mi>T</mi><mi mathvariant="normal">A</mi></msub><msub><mi>T</mi><mi mathvariant="normal">B</mi></msub></mfrac><mo>×</mo><mfrac><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">B</mi></msub><msub><mi mathvariant="normal">μ</mi><mi mathvariant="normal">A</mi></msub></mfrac></msqrt></math><br/><br/><math><mn>1</mn><mo>=</mo><mfrac><mrow><mi mathvariant="normal">x</mi></mrow><mrow><mn>80</mn></mrow></mfrac><mo>×</mo><msqrt><mn>1</mn><mo>×</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3.21</mn></mrow></mfrac></msqrt><mi mathvariant="normal"> </mi></math><br/><br/>or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>144</mn></math></div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2020 (12 Oct Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 1$</span> </div>
answer: <div class="solution">No Solution</div>
source_db: MarksBatch2_P3.db
title:
two-strings-of-same-material-having-lengths-as-l-2l-and-radii-2r-r-respectively-are-vibrating-in-the-fundamental-mode-tension-applied-to-both-the-stri

slug:
two-strings-of-same-material-having-lengths-as-l-2l-and-radii-2r-r-respectively-are-vibrating-in-the-fundamental-mode-tension-applied-to-both-the-stri-87470

question:
<div class="question">Two strings of same material having lengths as 'L', '2L', and radii '2r', 'r' respectively, are vibrating in the fundamental mode. Tension applied to both the<br/>strings is same. The ratio of their respective fundamental frequencies is</div>

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


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$0.11$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-students-x-and-y-appeared-in-an-examination-the-probability-that-x-will-qualify-the-examination-is-005-and-y-will-qualify-the-examination-is-010-t

slug:
two-students-x-and-y-appeared-in-an-examination-the-probability-that-x-will-qualify-the-examination-is-005-and-y-will-qualify-the-examination-is-010-t-56249

question:
<div class="question">Two students $X$ and $Y$ appeared in an examination. The probability that $\mathrm{X}$ will qualify the examination is $0.05$ and $\mathrm{Y}$ will qualify the examination is $0.10$. The probability that both will qualify the examination is $0.02$. What is the probability that only one of them will qualify the examination?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\quad 0.15$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.14$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.12$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$0.11$</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> </ul>

answer:
<div class="solution">Let $\mathrm{A}$ and $\mathrm{B}$ be the events $=\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})+\mathrm{P}(\mathrm{B} \cap \overline{\mathrm{A}})$<br/>$=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-2 \mathrm{P}(\mathrm{A} \cap \mathrm{B})$<br/>$=0.05+0.1-2(0.02)$<br/>$=0.15-0.04=0.11$<br/>Hence the pobability that only one of them will qualify the examination is $0.11$.</div>


tags: ['Mathematics', 'Probability', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-symmetric-dice-flipped-with-each-dice-having-two-sides-painted-red-two-painted-black-one-painted-yellow-and-the-other-painted-white-what-is-the-pr

slug:
two-symmetric-dice-flipped-with-each-dice-having-two-sides-painted-red-two-painted-black-one-painted-yellow-and-the-other-painted-white-what-is-the-pr-16930

question:
<div class="question">Two symmetric dice flipped with each dice having two sides painted red, two painted black, one painted yellow and the other painted white. What is the probability that both land on the same colour?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{18}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{2}{9}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{5}{18}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{5}{18}$</span> </div>

answer:
<div class="solution">Sides of dice $=\mathrm{R}, \mathrm{R}, \mathrm{B}, \mathrm{B}, \mathrm{Y}, \mathrm{W}$<br/>Total events with dice $=6 \times 6=36$ Favourable events $={ }^{2} \mathrm{C}_{1} \cdot{ }^{2} \mathrm{C}_{1}+{ }^{2} \mathrm{C}_{1} \cdot{ }^{2} \mathrm{C}_{1}+{ }^{1} \mathrm{C}_{1} \cdot{ }^{1} \mathrm{C}_{1}+$<br/>${ }^{1} \mathrm{C}_{1}{ }^{1} \mathrm{C}_{1}$<br/>$=4+4+2+2=10$<br/>$\therefore$ Probability $=\frac{10}{36}=\frac{5}{18}$</div>


tags: ['Mathematics', 'Statistics', 'TS EAMCET', 'TS EAMCET 2015']
source_db: MarksBatch2_P3.db
title:
two-teams-a-and-b-have-the-same-mean-and-their-coefficients-of-variation-are-42-respectively-if-a-b-are-the-standard-deviations-of-teams-a-b-respectiv

slug:
two-teams-a-and-b-have-the-same-mean-and-their-coefficients-of-variation-are-42-respectively-if-a-b-are-the-standard-deviations-of-teams-a-b-respectiv-65262

question:
<div class="question">Two teams $A$ and $B$ have the same mean and their coefficients of variation are 4,2 , respectively. If $\sigma_A, \sigma_B$ are the standard deviations of teams $A, B$ respectively, then the relation between them is</div>

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

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

answer:
<div class="solution">We have,<br/>$$<br/>\bar{X}_A=\bar{X}_B \text { and } \mathrm{CV}_A=4, \mathrm{CV}_B=2<br/>$$<br/>Let $\sigma_A$ and $\sigma_B$ be the standard deviations of teams $A$ and $B$ respectively.<br/>Then,<br/>$$<br/>\mathrm{CV}=\frac{\sigma}{\bar{X}} \times 100 \text { or } \sigma=\frac{\mathrm{CV} \times \bar{X}}{100}<br/>$$<br/>$$<br/>\therefore \quad \sigma_A=\frac{4 \bar{X}_A}{100} \text { and } \sigma_B=\frac{2 \bar{X}_B}{100}<br/>$$<br/>$$<br/>\Rightarrow \quad \bar{X}_A=25 \sigma_A \text { and } \bar{X}_B=50 \sigma_B<br/>$$<br/>But $\quad \bar{X}_A=\bar{X}_B$<br/>$$<br/>\begin{array}{rlrl} <br/>&amp; \therefore &amp; 25 \sigma_A &amp; =50 \sigma_B \\<br/>\Rightarrow &amp; \sigma_A &amp; =2 \sigma_B<br/>\end{array}<br/>$$</div>


tags: ['Physics', 'Ray Optics', 'NEET', 'NEET 2022 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-transparent-media-a-and-b-are-separated-by-a-plane-boundary-the-speed-of-light-in-those-media-are-1-5-10-8-m-s-1-and-2-0-10-8-m-s-1-respectively-t

slug:
two-transparent-media-a-and-b-are-separated-by-a-plane-boundary-the-speed-of-light-in-those-media-are-1-5-10-8-m-s-1-and-2-0-10-8-m-s-1-respectively-t-56130

question:
<div class="question">Two transparent media <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">B</mi></math> are separated by a plane boundary. The speed of light in those media are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>0</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><msup><mi mathvariant="normal">s</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, respectively. The critical angle for a ray of light for these two media 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"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>750</mn></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">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>500</mn></mrow></mfenced></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>750</mn></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"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>500</mn></mrow></mfenced></math></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>750</mn></mrow></mfenced></math></span> </div>

answer:
<div class="solution"><p>Refractive index <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>μ</mi></mfenced></math> in terms of speed of light <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>c</mi></mfenced></math> in vacuum and wave speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>v</mi></mfenced></math> in medium is given by, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mi>c</mi><mi>v</mi></mfrac></math>.</p><p>So, for medium <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>A</mi></msub><mo>=</mo><mfrac><mi>c</mi><msub><mi>v</mi><mi>A</mi></msub></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup></mrow><mrow><mn>1</mn><mo>.</mo><mn>5</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn></math> and for medium <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mi>B</mi></msub><mo>=</mo><mfrac><mi>c</mi><msub><mi>v</mi><mi>B</mi></msub></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup></mrow><mrow><mn>2</mn><mo>×</mo><msup><mn>10</mn><mn>8</mn></msup></mrow></mfrac><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math></p><p>For critical angle, angle of refraction <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>90</mn><mo>°</mo></math>.</p><p>Using Snell's law, </p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>μ</mi><mn>1</mn></msub><mi>sin</mi><mi>i</mi><mo>=</mo><msub><mi>μ</mi><mn>2</mn></msub><mi>sin</mi><mi>r</mi></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mn>2</mn><mi>sin</mi><mi>i</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mi>sin</mi><mn>90</mn><mo>°</mo><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>sin</mi><mi>i</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>750</mn><mspace linebreak="newline"></mspace><mo>⇒</mo><mi>i</mi><mo>=</mo><msup><mi>sin</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>750</mn></mrow></mfenced></math></p></div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2003']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 3$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-uniform-strings-a-and-b-made-of-steel-are-made-to-vibrate-under-the-same-tension-if-the-first-overtone-of-a-is-equal-to-the-second-overtone-of-b-a-1

slug:
two-uniform-strings-a-and-b-made-of-steel-are-made-to-vibrate-under-the-same-tension-if-the-first-overtone-of-a-is-equal-to-the-second-overtone-of-b-a-1-86658

question:
<div class="question">Two uniform strings $A$ and $B$ made of steel are made to vibrate under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B$, the ratio of the lengths of the strings is :</div>

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

answer:
<div class="solution">Fundamental frequency of string,<br/>$n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>where, $m=$ mass per unit length of the string<br/>$=\pi r^2 \rho$<br/>$\therefore \quad n=\frac{1}{2 l} \sqrt{\frac{T}{\pi r^2 \rho}}$<br/>For string $A$ first overtone,<br/>$n_A=\frac{1}{l_1} \sqrt{\frac{T_1}{m_1}}=\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 \rho}}$<br/>For string $B$ second overtone,<br/>$n_B=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 \rho_2}}$<br/>According to question,<br/>$n_A=n_B$<br/>$\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 l_1}}=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 l_2}}$<br/>but $\quad T_1=T_2$<br/>$\therefore \quad \frac{2 l_2}{3 l_1}=\sqrt{\frac{\pi r_1^2 \rho}{\pi r_2^2 \rho}}$<br/>$\frac{4 l_2^2}{9 l_1^2}=\frac{r_1^2}{r_2^2}$<br/>$\left(\frac{l_1}{l_2}\right)^2=\frac{4 r_2^2}{9 r_1^2}$<br/>$\frac{2 l_2}{3 l_1}=\frac{r_1}{r_2}$<br/>$\frac{l_2}{l_1}=\frac{3 r_1}{2 r_2}$<br/>$\frac{l_1}{l_2}=\frac{2 r_2}{3 r_1}$<br/>but $\quad r_1=2 r_2$<br/>$\therefore \quad \frac{l_1}{l_2}=\frac{2 r_2}{3\left(2 r_2\right)}$<br/>$l_1: l_2=1: 3$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 3$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-uniform-strings-a-and-b-made-of-steel-are-made-to-vibrate-under-the-same-tension-if-the-first-overtone-of-a-is-equal-to-the-second-overtone-of-b-a-2

slug:
two-uniform-strings-a-and-b-made-of-steel-are-made-to-vibrate-under-the-same-tension-if-the-first-overtone-of-a-is-equal-to-the-second-overtone-of-b-a-2-53103

question:
<div class="question">Two uniform strings $A$ and $B$ made of steel are made to vibrate under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B$, the ratio of the lengths of the strings is :</div>

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

answer:
<div class="solution">Fundamental frequency of string,<br/>$n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>where, $m=$ mass per unit length of the string<br/>$=\pi r^2 \rho$<br/>$\therefore \quad n=\frac{1}{2 l} \sqrt{\frac{T}{\pi r^2 \rho}}$<br/>For string $A$ first overtone,<br/>$n_A=\frac{1}{l_1} \sqrt{\frac{T_1}{m_1}}=\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 \rho}}$<br/>For string $B$ second overtone,<br/>$n_B=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 \rho_2}}$<br/>According to question,<br/>$n_A=n_B$<br/>$\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 l_1}}=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 l_2}}$<br/>but $\quad T_1=T_2$<br/>$\therefore \quad \frac{2 l_2}{3 l_1}=\sqrt{\frac{\pi r_1^2 \rho}{\pi r_2^2 \rho}}$<br/>$\frac{4 l_2^2}{9 l_1^2}=\frac{r_1^2}{r_2^2}$<br/>$\left(\frac{l_1}{l_2}\right)^2=\frac{4 r_2^2}{9 r_1^2}$<br/>$\frac{2 l_2}{3 l_1}=\frac{r_1}{r_2}$<br/>$\frac{l_2}{l_1}=\frac{3 r_1}{2 r_2}$<br/>$\frac{l_1}{l_2}=\frac{2 r_2}{3 r_1}$<br/>but $\quad r_1=2 r_2$<br/>$\therefore \quad \frac{l_1}{l_2}=\frac{2 r_2}{3\left(2 r_2\right)}$<br/>$l_1: l_2=1: 3$</div>


tags: ['Physics', 'Waves and Sound', 'TS EAMCET', 'TS EAMCET 2003']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 3$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-uniform-strings-a-and-b-made-of-steel-are-made-to-vibrate-under-the-same-tension-if-the-first-overtone-of-a-is-equal-to-the-second-overtone-of-b-a

slug:
two-uniform-strings-a-and-b-made-of-steel-are-made-to-vibrate-under-the-same-tension-if-the-first-overtone-of-a-is-equal-to-the-second-overtone-of-b-a-41168

question:
<div class="question">Two uniform strings $A$ and $B$ made of steel are made to vibrate under the same tension. If the first overtone of $A$ is equal to the second overtone of $B$ and if the radius of $A$ is twice that of $B$, the ratio of the lengths of the strings is :</div>

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

answer:
<div class="solution">Fundamental frequency of string,<br/>$n=\frac{1}{2 l} \sqrt{\frac{T}{m}}$<br/>where, $m=$ mass per unit length of the string<br/>$=\pi r^2 \rho$<br/>$\therefore \quad n=\frac{1}{2 l} \sqrt{\frac{T}{\pi r^2 \rho}}$<br/>For string $A$ first overtone,<br/>$n_A=\frac{1}{l_1} \sqrt{\frac{T_1}{m_1}}=\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 \rho}}$<br/>For string $B$ second overtone,<br/>$n_B=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 \rho_2}}$<br/>According to question,<br/>$n_A=n_B$<br/>$\frac{1}{l_1} \sqrt{\frac{T_1}{\pi r_1^2 l_1}}=\frac{3}{2 l_2} \sqrt{\frac{T_2}{\pi r_2^2 l_2}}$<br/>but $\quad T_1=T_2$<br/>$\therefore \quad \frac{2 l_2}{3 l_1}=\sqrt{\frac{\pi r_1^2 \rho}{\pi r_2^2 \rho}}$<br/>$\frac{4 l_2^2}{9 l_1^2}=\frac{r_1^2}{r_2^2}$<br/>$\left(\frac{l_1}{l_2}\right)^2=\frac{4 r_2^2}{9 r_1^2}$<br/>$\frac{2 l_2}{3 l_1}=\frac{r_1}{r_2}$<br/>$\frac{l_2}{l_1}=\frac{3 r_1}{2 r_2}$<br/>$\frac{l_1}{l_2}=\frac{2 r_2}{3 r_1}$<br/>but $\quad r_1=2 r_2$<br/>$\therefore \quad \frac{l_1}{l_2}=\frac{2 r_2}{3\left(2 r_2\right)}$<br/>$l_1: l_2=1: 3$</div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2023 (10 May Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 3$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-uniform-wires-of-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-first-wire-is-equal-to-the-2-nd-overtone-of-2-nd-wire

slug:
two-uniform-wires-of-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-first-wire-is-equal-to-the-2-nd-overtone-of-2-nd-wire-78846

question:
<div class="question">Two uniform wires of same material are vibrating under the same tension. If the first overtone of first wire is equal to the $2^{\text {nd }}$ overtone of $2^{\text {nd }}$ wire and radius of the first wire is twice the radius of the $2^{\text {nd }}$ wire then the ratio of length of first wire to $2^{\text {nd }}$ wire is</div>

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

answer:
<div class="solution">Fundamental frequency of the first wire is<br/>$\mathrm{n}=\frac{1}{2 l_1} \sqrt{\frac{\mathrm{T}}{\mathrm{m}}}=\frac{1}{2 l_1} \sqrt{\frac{\mathrm{T}}{\pi \mathrm{r}_1^2 \rho}}=\frac{1}{2 l_1 \mathrm{r}_1} \sqrt{\frac{\mathrm{T}}{\pi \rho}}$<br/>The first overtone $\mathrm{n}_1=2 \mathrm{n}=\frac{1}{l_1 \mathrm{r}_1} \sqrt{\frac{\mathrm{T}}{\pi \rho}}$<br/>Similarly, the second overtone of the second wire will be,<br/>$\mathrm{n}_2=\frac{3}{2 l_2 \mathrm{r}_2} \sqrt{\frac{\mathrm{T}}{\pi \rho}}$<br/>Given that $\mathrm{n}_1=\mathrm{n}_2$<br/>$\begin{aligned}<br/>&amp; \therefore \quad \frac{1}{l_1 \mathrm{r}_1} \sqrt{\frac{\mathrm{T}}{\pi \rho}}=\frac{3}{2 l_2 \mathrm{r}_2} \sqrt{\frac{\mathrm{T}}{\pi \rho}} \\<br/>&amp; \therefore \quad 3 l_1 \mathrm{r}_1=2 l_2 \mathrm{r}_2 \\<br/>&amp; \\<br/>&amp; \quad \frac{l_1}{l_2}=\frac{2 \mathrm{r}_2}{3 \mathrm{r}_1}<br/>\end{aligned}$<br/>$=\frac{2 r_2}{3\left(2 r_2\right)} \quad \ldots .\left(\because r_1=2 r_2\right)$<br/>$=\frac{1}{3}$</div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
two-uniform-wires-of-the-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-the-first-wire-is-equal-to-the-second-overtone-of-1

slug:
two-uniform-wires-of-the-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-the-first-wire-is-equal-to-the-second-overtone-of-1-33178

question:
<div class="question">Two uniform wires of the same material are vibrating under the same tension. If the first overtone of the first wire is equal to the second overtone of the second wire and radius of the first wire is twice the radius of the second wire then the ratio of the lengths of the first wire to second wire is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</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><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></math></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><mn>3</mn></mrow></mfrac></math></span> </div>

answer:
<div class="solution">Fundamental frequency of the first wire<br/><br/><math><mi>f</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><msubsup><mrow><mi>Ï</mi><mi>r</mi></mrow><mrow><mn>1</mn><mi> </mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/>The first overtone <math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mi>f</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/>The second overtone of the second wire<br/><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><br/><math><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mi> </mi><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/><math><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><mn>3</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><br/><math><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>.</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></math><br/><br/><math><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>.</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></math> <math><mo>[</mo><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo></math><br/><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-uniform-wires-of-the-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-the-first-wire-is-equal-to-the-second-overtone-of-2

slug:
two-uniform-wires-of-the-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-the-first-wire-is-equal-to-the-second-overtone-of-2-78752

question:
<div class="question">Two uniform wires of the same material are vibrating under the same tension. If the first overtone of the first wire is equal to the second overtone of the second wire and radius of the first wire is twice the radius of the second wire then the ratio of the lengths of the first wire to second wire is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</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><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></math></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><mn>3</mn></mrow></mfrac></math></span> </div>

answer:
<div class="solution">Fundamental frequency of the first wire<br/><br/><math><mi>f</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><msubsup><mrow><mi>Ï</mi><mi>r</mi></mrow><mrow><mn>1</mn><mi> </mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/>The first overtone <math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mi>f</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/>The second overtone of the second wire<br/><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><br/><math><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mi> </mi><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/><math><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><mn>3</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><br/><math><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>.</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></math><br/><br/><math><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>.</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></math> <math><mo>[</mo><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo></math><br/><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></div>


tags: ['Physics', 'Waves and Sound', 'MHT CET', 'MHT CET 2017']
source_db: MarksBatch2_P3.db
title:
two-uniform-wires-of-the-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-the-first-wire-is-equal-to-the-second-overtone-of-3

slug:
two-uniform-wires-of-the-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-the-first-wire-is-equal-to-the-second-overtone-of-3-22969

question:
<div class="question">Two uniform wires of the same material are vibrating under the same tension. If the first overtone of the first wire is equal to the second overtone of the second wire and radius of the first wire is twice the radius of the second wire then the ratio of the lengths of the first wire to second wire is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</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><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>5</mn></mrow></mfrac></math></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>6</mn></mrow></mfrac></math></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><mn>3</mn></mrow></mfrac></math></span> </div>

answer:
<div class="solution">Fundamental frequency of the first wire<br/><math><mi>f</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><msubsup><mrow><mi>Ï</mi><mi>r</mi></mrow><mrow><mn>1</mn><mi> </mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/>The first overtone <math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mi>f</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/>The second overtone of the second wire<br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><math><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mi> </mi><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><math><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><mn>3</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><math><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>.</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></math><br/><math><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>.</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></math> <math><mo>[</mo><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo></math><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
two-uniform-wires-of-the-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-the-first-wire-is-equal-to-the-second-overtone-of

slug:
two-uniform-wires-of-the-same-material-are-vibrating-under-the-same-tension-if-the-first-overtone-of-the-first-wire-is-equal-to-the-second-overtone-of-63106

answer:
<div class="solution">Fundamental frequency of the first wire<br/><br/><math><mi>f</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>m</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><msubsup><mrow><mi>Ï</mi><mi>r</mi></mrow><mrow><mn>1</mn><mi> </mi></mrow><mrow><mn>2</mn></mrow></msubsup><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/>The first overtone <math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><mi>f</mi><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/>The second overtone of the second wire<br/><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><br/><math><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt><mo>=</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mi> </mi><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>Ï</mi><mi>Ï</mi></mrow></mfrac><mi> </mi></msqrt></math><br/><br/><math><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><mn>3</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><br/><math><mfrac><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>.</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></math><br/><br/><math><mo>=</mo><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>.</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></math> <math><mo>[</mo><mo>â´</mo><mi> </mi><mi> </mi><mi> </mi><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>2</mn><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>]</mo></math><br/><br/><math><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></math></div>


tags: ['Mathematics', 'Statistics', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
two-variates-x-and-y-are-uncorrelated-and-have-standard-deviations-x-and-y-respectively-what-is-the-correlation-coefficient-between-x-y-and-x-y

slug:
two-variates-x-and-y-are-uncorrelated-and-have-standard-deviations-x-and-y-respectively-what-is-the-correlation-coefficient-between-x-y-and-x-y-35243

question:
<div class="question">Two variates, $\mathrm{x}$ and $\mathrm{y}$, are uncorrelated and have standard<br/>deviations $\sigma_{x}$ and $\sigma_{y}$ respectively. What is the correlation coefficient between $x+y$ and $x-y ?$</div>

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

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

answer:
<div class="solution">Let $u=(x+y) ; v=(x-y)$<br/>$\therefore \bar{u}=(\bar{x}+\bar{y}) ; \bar{v}=(\bar{x}-\bar{y})$<br/>$\operatorname{cov}(u, v)=E\{(u-\bar{u})(v-\bar{v})\}$<br/>$=E\{(x-\bar{x})+(y-\bar{y})\} \cdot\{(x-\bar{x})-(y-\bar{y})\}$<br/>$=E\left\{(x-\bar{x})^{2}-(y-\bar{y})^{2}\right\}=\sigma_{x}^{2}-\sigma_{y}^{2}$<br/>$\operatorname{var}(u)=E(u-\bar{u})^{2}=E\{(x-\bar{x})+(y-\bar{y})\}^{2}=\sigma_{x}^{2}+\sigma_{y}^{2}$<br/>Therefore $x$ and $y$ are uncorrelated. $E(x-\bar{x})(y-\bar{y})=0$<br/>Similarly, $\operatorname{var}(v)=\sigma_{x}^{2}+\sigma_{y}^{2}$<br/>Thus, $r(u, v)=\frac{\operatorname{cov}(u, v)}{\sqrt{\operatorname{var}(u) \cdot \operatorname{var}(v)}}=\frac{\sigma_{x}^{2}-\sigma_{y}^{2}}{\sigma_{x}^{2}+\sigma_{y}^{2}}$</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-vector-2-i-m-j-3-n-k-and-5-i-3-m-j-n-k-are-such-that-their-magnitudes-are-respectively-14-and-35-where-m-n-are-integers-which-one-of-the-following

slug:
two-vector-2-i-m-j-3-n-k-and-5-i-3-m-j-n-k-are-such-that-their-magnitudes-are-respectively-14-and-35-where-m-n-are-integers-which-one-of-the-following-30835

question:
<div class="question">Two vector $2 \hat{\mathbf{i}}+m \hat{\mathbf{j}}-3 n \hat{\mathbf{k}}$ and $5 \hat{\mathbf{i}}+3 m \hat{\mathbf{j}}+n \hat{\mathbf{k}}$ are such<br/>that their magnitudes are respectively $\sqrt{14}$ and $\sqrt{35}$, where<br/>$\mathrm{m}, \mathrm{n}$ are integers. Which one of the following is correct?</div>

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

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

answer:
<div class="solution">Vectors are $2 \mathrm{i}+\mathrm{mj}-3 \mathrm{nk}$ and $5 \mathrm{i}+3 \mathrm{mj}+\mathrm{nk}$<br/>$|2 \mathrm{i}+\mathrm{mj}-3 \mathrm{nk}|=\sqrt{14}$<br/>and $|5 \mathrm{i}+3 \mathrm{mj}+\mathrm{nk}|=\sqrt{35}$<br/>$\sqrt{2^{2}+m^{2}+(-3 n)^{2}}=\sqrt{14}$<br/>or, $4+\mathrm{m}^{2}+9 \mathrm{n}^{2}=14$<br/>or, $\mathrm{m}^{2}+9 \mathrm{n}^{2}=10$<br/>From (ii)<br/>$\sqrt{5^{2}+(3 m)^{2}+n^{2}}=\sqrt{35}$<br/>or, $25+9 \mathrm{~m}^{2}+\mathrm{n}^{2}=35$<br/>or, $9 \mathrm{~m}^{2}+\mathrm{n}^{2}=10$<br/>From (iii) and (iv) $m^{2}+9 n^{2}=9 m^{2}+n^{2}$<br/>or, $8 \mathrm{n}^{2}=8 \mathrm{~m}^{2}$<br/>or, $\mathrm{n}^{2}=\mathrm{m}^{2}$<br/>$\mathrm{n}=\pm \mathrm{m}$<br/>n takes 2 values and $\mathrm{m}$ takes 2 values.</div>


tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2007 (Phase 1)']
source_db: MarksBatch2_P3.db
title:
two-vectors-a-and-b-are-nonzero-and-noncollinear-what-is-the-value-of-x-for-which-the-vectors-p-x-2-a-b-and-q-x-1-a-b-are-collinear

slug:
two-vectors-a-and-b-are-nonzero-and-noncollinear-what-is-the-value-of-x-for-which-the-vectors-p-x-2-a-b-and-q-x-1-a-b-are-collinear-88821

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

answer:
<div class="solution">Since, $\overrightarrow{\mathrm{p}}$ and $\overrightarrow{\mathrm{q}}$ are collinear, then<br/>$\overrightarrow{\mathrm{p}}=\mathrm{k} \overrightarrow{\mathrm{q}}$<br/>[where $\mathrm{k}$ is a scalar $]$<br/>$\Rightarrow(\mathrm{x}-2) \overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}=\mathrm{k}(\mathrm{x}+1) \overrightarrow{\mathrm{a}}-\mathrm{k} \overrightarrow{\mathrm{b}}$<br/>On equating the coefficients $x-2=k(x+1)$ and $-k=1$<br/>putting value of $\mathrm{k}$ we get, $\mathrm{x}-2=-(\mathrm{x}+1)$<br/>$\Rightarrow 2 \mathrm{x}=1$<br/>$\Rightarrow \mathrm{x}=\frac{1}{2}$</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-vibrating-strings-of-the-same-material-but-length-l-and-2-l-have-radii-2-r-and-r-respectively-they-are-stretched-under-the-same-tension-both-the-s

slug:
two-vibrating-strings-of-the-same-material-but-length-l-and-2-l-have-radii-2-r-and-r-respectively-they-are-stretched-under-the-same-tension-both-the-s-63191

question:
<div class="question">Two vibrating strings of the same material but length <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">L</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">L</mi></math> have radii <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">r</mi></math> respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">L</mi></math> with frequency <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ï</mi><mn>1</mn></msub></math> and the other with frequency <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="normal">Ï</mi><mn>2</mn></msub></math>. The ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi mathvariant="normal">Ï</mi><mn>1</mn></msub><msub><mi mathvariant="normal">Ï</mi><mn>2</mn></msub></mfrac></math> is given by</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>1</mn></math></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></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><mn>3</mn><mn>2</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></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>1</mn></math></span> </div>

answer:
<div class="solution"><p>Fundamental frequency <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>f</mtext><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi mathvariant="normal">l</mi><mo>⁡</mo></mrow></mfrac><msqrt><mfrac><mtext>T</mtext><mi mathvariant="normal">μ</mi></mfrac></msqrt></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∴       </mo><mfrac><msub><mtext>f</mtext><mn>1</mn></msub><msub><mtext>f</mtext><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">l</mi><msub><mo>⁡</mo><mn>1</mn></msub></mrow><mrow><mi mathvariant="normal">l</mi><msub><mo>⁡</mo><mn>2</mn></msub></mrow></mfrac><msqrt><mfrac><msub><mi mathvariant="normal">μ</mi><mn>2</mn></msub><msub><mi mathvariant="normal">μ</mi><mn>1</mn></msub></mfrac></msqrt></math>          (since tension is same) </p><p>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mtext>L</mtext></mrow><mtext>L</mtext></mfrac><msqrt><mfrac><mrow><mi mathvariant="normal">π</mi><msup><mtext>r</mtext><mn>2</mn></msup><mi mathvariant="normal">ρ</mi></mrow><mrow><mi mathvariant="normal">π</mi><mo>·</mo><mn>4</mn><msup><mtext>r</mtext><mn>2</mn></msup><mi mathvariant="normal">ρ</mi></mrow></mfrac></msqrt><mtext>,</mtext></math>   (since the wires are of the same material)</p><p>Hence the desired ratio is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></p></div>


tags: ['Physics', 'Waves and Sound', 'JIPMER', 'JIPMER 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
title:
two-vibrating-strings-of-the-same-material-but-lengths-l-and-2-l-have-radii-2-r-and-r-respectively-they-are-stretched-under-the-same-tension-both-the-

slug:
two-vibrating-strings-of-the-same-material-but-lengths-l-and-2-l-have-radii-2-r-and-r-respectively-they-are-stretched-under-the-same-tension-both-the-26652

question:
<div class="question">Two vibrating strings of the same material but lengths $L$ and $2 L$ have radii $2 r$ and $r$ respectively. They are stretched under the same tension. Both the strings vibrate in their fundamental modes, the one of length $L$ with frequency $v_1$ and the other with frequency $v_2$. The ratio $v_1 / v_2$ is</div>

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

answer:
<div class="solution">Fundamental frequency,<br/>$\begin{array}{ll} <br/>&amp; v=\frac{1}{2 l} \sqrt{\frac{T}{m}}=\frac{1}{2 l} \sqrt{\frac{T}{\pi r^2 \rho}} \\<br/>\text { or } &amp; v \propto \frac{1}{l r} \\<br/>\therefore &amp; \frac{v_1}{v_2}=\frac{l_2 r_2}{l_1 r_1}=\frac{2 L \times r}{L \times 2 r}=1<br/>\end{array}$</div>


tags: ['Physics', 'Waves and Sound', 'JIPMER', 'JIPMER 2007']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$180$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-vibrating-tuning-forks-produce-progressive-waves-given-by-y-1-4-sin-500-t-and-y-2-2-sin-506-t-number-of-beats-produced-per-minute-is

slug:
two-vibrating-tuning-forks-produce-progressive-waves-given-by-y-1-4-sin-500-t-and-y-2-2-sin-506-t-number-of-beats-produced-per-minute-is-10992

question:
<div class="question">Two vibrating tuning forks produce progressive waves given by $y_1=4 \sin 500 \pi t$ and $y_2=2 \sin 506 \pi t$. Number of beats produced per minute is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$360$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$180$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$60$</span> </li> </ul>

answer:
<div class="solution">To reach the solution the given wave equations must be compared with standard equation of progressive wave.<br/>So,<br/>$\begin{aligned} &amp; y_1=4 \sin 500 \pi t...(i) \\ &amp; y_2=2 \sin 506 \pi t...(ii)\end{aligned}$<br/>Comparing Eqs. (i) and (ii) with $y=a \sin \omega t...(iii)$<br/>We have, $\omega_1=500 \pi$<br/>$\Rightarrow f_1=\frac{500 \pi}{2 \pi}=250$ beats $/ \mathrm{s}$<br/>and $\omega_2=506 \pi$<br/>$\Rightarrow f_2=\frac{506 \pi}{2 \pi}=253$ beats $/ \mathrm{s}$<br/>Thus, number of beats produced<br/>$\begin{aligned} &amp; =f_2-f_1=253-250 \\ &amp; =3 \text { beats } / \mathrm{s} \\ &amp; =3 \times 60 \text { beats } / \mathrm{min} \\ &amp; =180 \text { beats } / \mathrm{min}\end{aligned}$</div>


tags: ['Physics', 'Waves and Sound', 'AP EAMCET', 'AP EAMCET 2022 (08 Jul Shift 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$1: 1$</span> </div>
source_db: MarksBatch2_P3.db
title:
two-wires-a-and-b-of-lengths-in-the-ratio-1-2-and-masses-in-the-ratio-2-1-are-stretched-by-same-tension-the-ratio-of-the-fundamental-frequencies-of-wi

slug:
two-wires-a-and-b-of-lengths-in-the-ratio-1-2-and-masses-in-the-ratio-2-1-are-stretched-by-same-tension-the-ratio-of-the-fundamental-frequencies-of-wi-64896

question:
<div class="question">Two wires $A$ and $B$ of lengths in the ratio $1: 2$ and masses in the ratio $2: 1$ are stretched by same tension. The ratio of the fundamental frequencies of wires $A$ and $B$ is</div>

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

answer:
<div class="solution">Fundamental frequency in a stretched wire is given as,<br/>$f=\frac{1}{2 L} \sqrt{\frac{T}{\mu}}$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/q70JkRF7jy_kUlUitZGh_9JtPJSmi5rQLRlqbAjxd3c.original.fullsize.png"/><br/><br/>Ratio of fundamental frequencies of wire $A$ and $B$,<br/>$\begin{aligned} \frac{f_A}{f_B} &amp; =\frac{1}{2 L_A} \sqrt{\frac{T_A}{\mu_A}} \times \frac{2 L_B}{T} \sqrt{\frac{\mu_B}{T_B}} \\ &amp; =\frac{L_B}{L_A} \cdot \sqrt{\frac{T_A}{T_B}} \sqrt{\frac{\mu_B}{\mu_A}}\end{aligned}$<br/>Here $\frac{L_B}{L_A}=\frac{2}{1}$<br/>$\begin{aligned} \frac{\mu_A}{\mu_B}=\frac{m_A / L_A}{m_B / L_B}=\frac{m_A}{m_B} \cdot \frac{L_B}{L_A}=\frac{2}{1} \times \frac{2}{1} &amp; \\ &amp; {\left[\because \frac{m_A}{m_B}=\frac{2}{1} \text { and } \frac{L_B}{L_A}=\frac{2}{1}\right] }\end{aligned}$<br/>or $\quad \frac{\mu_A}{\mu_B}=\frac{2}{1} \times \frac{2}{1}=\frac{4}{1}$<br/>$\Rightarrow \quad \frac{\mu_B}{\mu_A}=\frac{1}{4}$<br/>So, $\quad \frac{f_A}{f_B}=\frac{2}{1} \cdot \sqrt{1} \cdot \sqrt{\frac{1}{4}}=1: 1$</div>


tags: ['Physics', 'Waves and Sound', 'NEET']
source_db: MarksBatch2_P3.db
title:
two-wires-made-up-of-same-material-are-of-equal-lengths-but-their-radii-are-in-the-ratio-1-2-on-stretching-each-of-these-two-string-by-the-same-tensio

slug:
two-wires-made-up-of-same-material-are-of-equal-lengths-but-their-radii-are-in-the-ratio-1-2-on-stretching-each-of-these-two-string-by-the-same-tensio-79981

question:
<div class="question">Two wires, made up of same material, are of equal lengths but their radii are in the ratio <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>:</mo><mn>2</mn></math>. On stretching each of these two string by the same tension, the ratio between their fundamental frequencies is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>:</mo><mn>1</mn></math></span> </div>

answer:
<div class="solution">Here, <math><msub><mrow><mi>Ï</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>Ï</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>;</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><br/><math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>l</mi><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mi>Ï</mi><msub><mrow><mi>Ï</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac></msqrt><mo>;</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><mi>l</mi><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><msqrt><mfrac><mrow><msub><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><mi>Ï</mi><msub><mrow><mi>Ï</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac></msqrt></math><br/><br/><math><mfrac><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></mfrac><mo>=</mo><mn>2</mn></math></div>


tags: ['Physics', 'Waves and Sound', 'JEE Main', 'JEE Main 2017 (08 Apr Online)']
source_db: MarksBatch2_P3.db
title:
two-wires-w-1-and-w-2-have-the-same-radius-r-and-respective-densities-1-and-2-such-that-2-4-1-they-are-joined-together-at-the-point-o-as-shown-in-the-

slug:
two-wires-w-1-and-w-2-have-the-same-radius-r-and-respective-densities-1-and-2-such-that-2-4-1-they-are-joined-together-at-the-point-o-as-shown-in-the-74541

question:
<div class="question">Two wires <math><msub><mrow><mi>W</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow></msub></math> have the same radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo> </mo></math>and respective, densities <math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, such that <math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>4</mn><msub><mrow><mi>ρ</mi></mrow><mrow><mn>1</mn></mrow></msub></math> . They are joined together at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math>, as shown in the figure. The combination is used as a sonometer wire and kept under tension <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>. The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>O</mi></math> is midway between the two bridges. When a stationary wave is set up in the composite wire, the joint is found to be a node. The ratio of the number of antinodes formed in <math><msub><mrow><mi>W</mi></mrow><mrow><mn>1</mn></mrow></msub></math> to <math><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow></msub></math> is <br/><br/><img src="https://cdn-question-pool.getmarks.app/pyq/jee_main/0025c51d-f9e5-4e95-92e8-0b7c176530a0-image.png" style="width: 350px; height: 104px;"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>:</mo><mn>2</mn></math></span> </div>

answer:
<div class="solution"><p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>p</mi><mn>2</mn></msub></math> be the number of antinodes of both wires.</p><p>Fundamental frequency remains the same for both wires. <br/><br/><math><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>→</mo><mi>same</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>→</mo><mi>same</mi></math><br/><br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>l</mi><mo>→</mo><mi>same</mi></math><br/><br/><math><mi>n</mi><mo>=</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>l</mi></mrow></mfrac><mi> </mi><msqrt><mfrac><mrow><mi>T</mi></mrow><mrow><mi>π</mi><msup><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>ρ</mi></mrow></mfrac></msqrt></math></p><p>From the above equation, we have <br/><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>p</mi><mn>1</mn></msub><msqrt><msub><mi>ρ</mi><mn>1</mn></msub><mi> </mi><mi> </mi></msqrt></mfrac><mo>=</mo><mfrac><msub><mi>p</mi><mn>2</mn></msub><msqrt><msub><mi>ρ</mi><mn>2</mn></msub><mi> </mi><mi> </mi></msqrt></mfrac></math></p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>→</mo><mfrac><msub><mi>p</mi><mn>1</mn></msub><msub><mi>p</mi><mn>2</mn></msub></mfrac><mo>=</mo><mi> </mi><msqrt><mfrac><msub><mi>ρ</mi><mn>1</mn></msub><msub><mi>ρ</mi><mn>2</mn></msub></mfrac></msqrt></math><br/><br/>The ratio of antinodes is <math><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math>. </p></div>


source_db: MarksBatch2_P3.db
title:
two-wooden-blocks-of-mass-m-1-and-m-2-rest-on-a-frictionless-table-a-bullet-of-mass-m-is-fired-at-m-1-with-speed-v-which-embedded-in-it-and-the-two-to

slug:
two-wooden-blocks-of-mass-m-1-and-m-2-rest-on-a-frictionless-table-a-bullet-of-mass-m-is-fired-at-m-1-with-speed-v-which-embedded-in-it-and-the-two-to-86625

question:
<div class="question">Two wooden blocks of mass $M_1$ and $M_2$ rest on a frictionless table. $A$ bullet of mass $m$ is fired at $M_1$ with speed $v$ which embedded in it and the two together finally collide with $M_2$. Find the velocity of $M_2$ after collision. (Ignore any energy loss and treat the problem to be one dimensional)</div>

tags:
['Physics', 'Center of Mass Momentum and Collision', 'AP EAMCET', 'AP EAMCET 2022 (07 Jul Shift 2)']

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{2 m v}{M_1+M_2+m}$</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">$\frac{m v}{M_1+M_2+m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\left(M_1+M_2+m\right) v}{M_1+M_2+m}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{M_1+M_2}{M_1+M_2+m} v$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2 m v}{M_1+M_2+m}$</span> </div>

answer:
<div class="solution">( ) For an elastic collision of 2 masses $m_1$ and $m_2$ with initial velocities $u_1$ and $u_2$, final velocities of masses after collision are<br/>$$<br/>v_1=\left(\frac{m_1-m_2}{m_1+m_2}\right) u_1+\frac{2 m_2 u_2}{\left(m_1+m_2\right)}<br/>$$<br/>and<br/>$$<br/>v_2=\left(\frac{m_2-m_1}{m_1+m_2}\right) u_2+\frac{2 m_1 u_2}{m_1+m_2}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/KqsBxBKaLEI5rn2ypEvydbq6op5qYrpQUn8Pdz56c34.original.fullsize.png"/><br/><br/>These are obtained as standard results by applying conservation of energy and momentum equations. Now, in given case at first bullet is embedded in mass $M_1$. Let speed of bullet + Block system be $v_1$ after collision.<img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/2K_YzJe5tJsqgSp3Ic6TfGnL727d9vz1-7Ii6cWUUHY.original.fullsize.png"/><br/><br/>Then by conservation of momentum we have.<br/>$$<br/>\begin{aligned}<br/>&amp; m v=\left(m+M_1\right) v_1 \\<br/>&amp; \text { or } \quad v_1=\left(\frac{m v}{m+M_1}\right)<br/>\end{aligned}<br/>$$<br/><br/>Note Here we are not using standard result as first collision is perfectly inelastic.<br/>Now, bullet $+M_1$ strikes $M_2$ and it is given in problem that this collision is elastic.<img src="https://cdn-question-pool.getmarks.app/pyq/ap_eamcet/HTaodU3BVpgswBNuCO9za94dRGZf3IuALEmeiAfXu8I.original.fullsize.png"/><br/><br/>Now, by formula we have,<br/>$$<br/>\begin{aligned}<br/>v_2 &amp; =\left(\frac{\left(m+M_1\right)-M_2}{m+M_1+M_2}\right) v_1+0 \\<br/>\text { and, } v_3 &amp; =\left(\frac{M_2-\left(m+M_1\right)}{M_2+m+M_1} \times 0\right)+\left(\frac{2\left(m+M_1\right) v_1}{m+M_1+M_2}\right) \\<br/>\Rightarrow \quad v_3 &amp; =\frac{2\left(m+M_1\right)}{m+M_1+M_2} \cdot \frac{m v}{\left(m+M_1\right)} \\<br/>&amp; =\frac{2 m v}{m+M_1+M_2}<br/>\end{aligned}<br/>$$<br/><br/>Note You can remember these standard and useful results for finding final velocities in case of elastic collisions.</div>


tags: ['Biology', 'Anatomy of Flowering Plants', 'JIPMER', 'JIPMER 2014']
source_db: MarksBatch2_P3.db
title:
tyloses-an-out-growth-from-ray-or-axial-parenchyma-cell-into-the-lumen-of-a-vessel-which-partially-or-completely-blocks-the-cavity-are-present-in

slug:
tyloses-an-out-growth-from-ray-or-axial-parenchyma-cell-into-the-lumen-of-a-vessel-which-partially-or-completely-blocks-the-cavity-are-present-in-21677

question:
<div class="question">Tyloses an out growth from ray or axial parenchyma cell into the lumen of a vessel, which partially or completely blocks the cavity are present in</div>

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

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

answer:
<div class="solution"><strong>Tyloses</strong> are found in heart wood, sometimes the xylem parenchyma develop balloon-like structure to which penetrates into the adjacent xylem. Vessels, which are called tyloses, block the luman of the xylem.</div>


title: tyndall-effect-is-more-effectively-shown-by
slug: tyndall-effect-is-more-effectively-shown-by-27555
question: <div class="question">Tyndall effect is more effectively shown by:</div>
tags: ['Chemistry', 'Surface Chemistry', 'JEE Main', 'JEE Main 2021 (27 Aug Shift 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">lyophilic colloid</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">suspension</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">lyophobic colloid</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">true solution</span> </li> </ul>

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

answer:
<div class="solution"><p>Tyndall effect is due to Scattering of light.</p><table border="1" cellpadding="1" cellspacing="1" style="width: 500px;"> <tbody> <tr> <td>Property</td> <td>Lyophilic colloids</td> <td>Lyophobic colloids</td> </tr> <tr> <td>Tyndall effect</td> <td>They do not show tyndall effect</td> <td>They show tyndall effect.</td> </tr> </tbody></table></div>


