# questions

title: what-does-the-following-diagram-represent-13
slug: what-does-the-following-diagram-represent-13-91189
tags: ['Biology', 'Microbes in Human Welfare', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Both</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/modules/dbq/bisdfalmos/pblz5c8pEOYZexkrg7lgJhd2Z3TMJZa4yzyQMQoxuWMmCppDvw.jpeg"/>What does the following diagram represent?</div>

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

answer:
<div class="solution">The diagram represents secondary treatment, which is a part of sewage treatment.</div>


title: what-does-the-formation-of-turbidity-in-the-lucas-test-indicate
slug: what-does-the-formation-of-turbidity-in-the-lucas-test-indicate-62424
question: <div class="question">What does the formation of turbidity in the Lucas test indicate?</div>
tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Presence of a primary alcohol</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Presence of a secondary alcohol</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Presence of a tertiary alcohol</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Presence of an aromatic alcohol</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Presence of a tertiary alcohol</span> </div>

answer:
<div class="solution">The formation of turbidity in the Lucas test indicates the presence of a tertiary alcohol.</div>


title: what-does-the-negative-sign-in-the-expression-e-zn-2-zn-076-v-mean-1
slug: what-does-the-negative-sign-in-the-expression-e-zn-2-zn-076-v-mean-1-99890
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the negative sign in the expression E<sup>â</sup>(Zn<sup>2+</sup>/Zn) = â 0.76 V mean?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">It means that H is more reactive than zinc. When the zinc electrode will be connected to Standard Hydrogen Electrode , Zn will get reduced and H<sup>+</sup>will get oxidized.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">It means that Zn and H will both get reduced when connected to the standard hydrogen electrode.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">It means that Zn is more reactive than hydrogen. When the zinc electrode will be connected to Standard Hydrogen Electrode , Zn will get oxidized and H<sup>+</sup>will get reduced.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The negative sign has no significance.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It means that Zn is more reactive than hydrogen. When the zinc electrode will be connected to Standard Hydrogen Electrode , Zn will get oxidized and H<sup>+</sup>will get reduced.</span> </div>

answer:
<div class="solution">Correct Option is : (C)<br/>It means that Zn is more reactive than hydrogen. When the zinc electrode will be connected to Standard Hydrogen Electrode , Zn will get oxidized and H<sup>+</sup>will get reduced.</div>


title: what-does-the-negative-sign-in-the-expression-e-zn-2-zn-076-v-mean
slug: what-does-the-negative-sign-in-the-expression-e-zn-2-zn-076-v-mean-59731
tags: ['Chemistry', 'Electrochemistry']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the negative sign in the expression $E_{\mathrm{Zn}{ }^{2+} / \mathrm{Zn}}^{\ominus}=-0.76 \mathrm{~V}$ mean?</div>

answer:
<div class="solution">Greater negative reactivity of standard reduction potential shows that zinc metal is more reactive than hydrogen. When zinc electrode will be connected to standard hydrogen electrode. $\mathrm{Zn}$ will get oxidised and $\mathrm{H}^{+}$will get reduced.<br/>Thus, zinc electrode will act as the anode of the cell and hydrogen electrode will act as the cathode of the cell.</div>


title: what-does-the-positive-or-negative-sign-on-boundary-surface-diagrams-represent
slug: what-does-the-positive-or-negative-sign-on-boundary-surface-diagrams-represent-81733
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the positive or negative sign on boundary surface diagrams represent?</div>

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

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

answer:
<div class="solution">The positive and negative signs on boundary surface diagrams do not represent charge but instead indicate the sign (phase) of the orbital wave function.</div>


title: what-does-the-primary-valence-represent-in-werners-theory
slug: what-does-the-primary-valence-represent-in-werners-theory-71920
question: <div class="question">What does the primary valence represent in Werner's theory?</div>
tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The number of ligands</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The number of secondary valences</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The coordination number</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The metal's charge</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The number of secondary valences</span> </div>

answer:
<div class="solution">The primary valence in Werner's theory represents the number of secondary valences for a metal.</div>


title: what-does-the-secondary-valence-represent-in-werners-theory
slug: what-does-the-secondary-valence-represent-in-werners-theory-11911
question: <div class="question">What does the secondary valence represent in Werner's theory?</div>
tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The number of ligands</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">The number of primary valences</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">The coordination number</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The metal's charge</span> </li> </ul>

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

answer:
<div class="solution">The secondary valence in Werner's theory represents the coordination number for a metal in coordination compounds.</div>


title: what-does-the-series-1-3-2-1-3-3-3-1-represents
slug: what-does-the-series-1-3-2-1-3-3-3-1-represents-12414
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the series<br/>$1+3^{-\frac{1}{2}}+3+\frac{1}{3 \sqrt{3}}+\ldots$ represents?</div>

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

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

answer:
<div class="solution">Given series is $1+\frac{1}{\sqrt{3}}+3+\frac{1}{3 \sqrt{3}}+\ldots .$<br/>Consider $\frac{\mathrm{a}_{2}}{\mathrm{a}_{1}}=\frac{1}{\sqrt{3}}, \frac{\mathrm{a}_{3}}{\mathrm{a}_{2}}=\frac{3}{1 / \sqrt{3}}, \frac{\mathrm{a}_{4}}{\mathrm{a}_{3}}=\frac{1 / 3 \sqrt{3}}{3}$<br/>Also find $a_{2}-a_{1}=\frac{1}{\sqrt{3}}-1=\frac{1-\sqrt{3}}{\sqrt{3}}$<br/>$a_{3}-a_{2}=3-\frac{1}{\sqrt{3}}=\frac{3 \sqrt{3}-1}{\sqrt{3}}$<br/>$a_{4}-a_{3}=\frac{1}{3 \sqrt{3}}-3=\frac{1-9 \sqrt{3}}{3 \sqrt{3}}$<br/>Thus, $\frac{a_{2}}{a_{1}} \neq \frac{a_{3}}{a_{2}} \neq \frac{a_{4}}{a_{3}}$<br/>and $\mathrm{a}_{2}-\mathrm{a}_{1} \neq \mathrm{a}_{3}-\mathrm{a}_{2} \neq \mathrm{a}_{4}-\mathrm{a}_{3}$<br/>Hence, Given series is neither A.P., GP. nor H.P.</div>


title: what-does-the-shaded-region-in-the-above-diagram-represent
slug: what-does-the-shaded-region-in-the-above-diagram-represent-17663
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/nda/fuckGiR4XqKRF3TSn5yaf-nz2kfEZhbgC00Khc3D8zk.original.fullsize.png"/><br/><br/>What does the shaded region in the above diagram represent?</div>

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

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

answer:
<div class="solution">In the given Venn diagram, shaded region shows $(\mathrm{A} \cup \mathrm{B}) \cap \mathrm{C}$</div>


title: what-does-the-shaded-region-in-the-venn-diagram-given-below-represent
slug: what-does-the-shaded-region-in-the-venn-diagram-given-below-represent-37379
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2007 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the shaded region in the Venn diagram given below represent? <br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/ayJqB9EAfhjL_OXRJc_7rXElppOSv0KTMfvkiMy-tHI.original.fullsize.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$C \cup\left(\mathrm{C}^{\prime} \cap \mathrm{A} \cap \mathrm{B}\right)$</span> </div>

answer:
<div class="solution">In the given Venn diagram shaded region is $\mathrm{C} \cup\left(\mathrm{C}^{\prime} \cap \mathrm{A} \cap \mathrm{B}\right)$</div>


title: what-does-the-shaded-region-represent-in-the-figure-given-below
slug: what-does-the-shaded-region-represent-in-the-figure-given-below-51361
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2006 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the shaded region represent in the figure given below<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/-RJDYAAdeR2CNjQ9nv1L4u8qFrADyaNkCmNM5VD5vxI.original.fullsize.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(\mathrm{P} \cap \mathrm{Q}) \cup(\mathrm{P} \cap \mathrm{R})$</span> </div>

answer:
<div class="solution">The shaded region represents $(\mathrm{P} \cap \mathrm{Q}) \cup(\mathrm{P} \cap \mathrm{R})$ Let the intersecting sets $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ divide it into $7 \mathrm{regions}$ marked, a to g as shown below.<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/j-DbkNBFeUxdmte5HTyuh2x2Sq_GkC_Qv9-DXQVcJMY.original.fullsize.png"/><br/><br/>The shaded part contains regions $\mathrm{b}, \mathrm{c}$, and $\mathrm{d}$.<br/>(a) $(\mathrm{P} \cup \mathrm{Q})-(\mathrm{P} \cap \mathrm{Q})=$ regions $\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{f}, \mathrm{g},-\mathrm{b}, \mathrm{c}$<br/>$=\mathrm{a}, \mathrm{d}, \mathrm{f}_{\mathrm{g}} \mathrm{g}$, notcorrect.<br/>(b) $(\mathrm{P} \cap(\mathrm{Q} \cap \mathrm{R})=\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \cap \mathrm{c}, \mathrm{f}=\mathrm{c}$ not correct.<br/>(c) $(P \cap Q) \cap(P \cap R)=$ regions $b, c, \cap r e g i o n, c, d=c, s o$<br/>not correct<br/>(d) $(P \cap Q) \cup(P \cap R)=$ regions $b, c, \cup c, d=b, c, d s o$<br/>correct</div>


title: what-does-the-solution-of-the-differential-equation-x-d-x-d-y-y-represent
slug: what-does-the-solution-of-the-differential-equation-x-d-x-d-y-y-represent-47933
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the solution of the differential equation<br/>$x \frac{d y}{d x}=y$ represent?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Family of straight lines through the origin</span> </div>

answer:
<div class="solution">Given differential equation is<br/>$$<br/>x \frac{d y}{d x}=y<br/>$$<br/>By separating the variables, we get<br/>$$<br/>\frac{\mathrm{dy}}{\mathrm{y}}=\frac{\mathrm{dx}}{\mathrm{x}}<br/>$$<br/>Intergrate both the sides, we get<br/>$$<br/>\begin{aligned}<br/>&amp; \int \frac{\mathrm{dy}}{\mathrm{y}}=\int \frac{\mathrm{dx}}{\mathrm{x}} \\<br/>\Rightarrow &amp; \log y=\log x+\log c \\<br/>\Rightarrow &amp; y=x c<br/>\end{aligned}<br/>$$<br/>which is a family of straight lines through the origin.</div>


title: what-does-the-solution-of-the-differential-equation-x-d-y-y-d-x-0-represent
slug: what-does-the-solution-of-the-differential-equation-x-d-y-y-d-x-0-represent-68896
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the solution of the differential equation $x d y-y d x=0$ represent?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Straight line passing through the origin</span> </div>

answer:
<div class="solution">$x d y-y d x=0$<br/>$\Rightarrow x d y=y d x$<br/>$\Rightarrow \int \frac{1}{y} d y=\int \frac{1}{x} d x$<br/>$\Rightarrow \log y=\log x+\log c$<br/>$\Rightarrow \log y=\log c x$<br/>$\Rightarrow y=c x$<br/>Thus, the solution of equation $x d y-y d x=0$ represents straight line passing through the origin.</div>


title: what-does-the-solution-of-the-differential-equation-xdy-ydx-0-represent
slug: what-does-the-solution-of-the-differential-equation-xdy-ydx-0-represent-92624
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2006 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the solution of the differential equation $\mathrm{xdy}-\mathrm{ydx}=0$ represent?</div>

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

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

answer:
<div class="solution">Given that $x \mathrm{dy}-\mathrm{ydx}=0$ Dividing both the sides by $\mathrm{x}^{2}$<br/>$\Rightarrow \quad \frac{\mathrm{xdy}-\mathrm{ydx}}{\mathrm{x}^{2}}=0$<br/>$\Rightarrow \mathrm{d}\left(\frac{\mathrm{y}}{\mathrm{x}}\right)=0$<br/>$\Rightarrow \frac{y}{x}=c \Rightarrow y=c x$, where $c$ is a constant. Thus, it a straight line passing through $(0,0)$. Aliter :<br/>Given, $x d y-y d x=0$<br/>$\Rightarrow \frac{d y}{y}=\frac{d x}{x}$<br/>Integrating both the sides, $\int \frac{\mathrm{dy}}{\mathrm{y}}=\int \frac{\mathrm{dx}}{\mathrm{x}}+\log \mathrm{c}$<br/>$\log \mathrm{y}=\log \mathrm{x}+\log \mathrm{c}$ where $\mathrm{c}$ is constant<br/>$\Rightarrow \mathrm{y}=\mathrm{cx}$</div>


title: what-does-the-standard-electrode-potential-of-lithium-metal-indicate-about-its-reactivity
slug: what-does-the-standard-electrode-potential-of-lithium-metal-indicate-about-its-reactivity-89371
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the standard electrode potential of lithium metal indicate about its reactivity?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">It is highly reducing.</span> </div>

answer:
<div class="solution">Lithium has the lowest electrode potential indicating that lithium ion is the weakest oxidising agent while lithium metal is the most powerful reducing agent in an aqueous solution.</div>


title: what-does-the-symbol-m-represent-in-the-context-of-molar-conductivity
slug: what-does-the-symbol-m-represent-in-the-context-of-molar-conductivity-14012
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What does the symbol ÎÂ°<sub>m</sub> represent in the context of molar conductivity?</div>

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

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

answer:
<div class="solution">When concentration approaches zero, the molar conductivity is known as limiting molar conductivity and is represented by the symbol ÎÂ°<sub>m</sub>.</div>


title: what-does-the-term-secondary-valence-represent-in-werners-concept
slug: what-does-the-term-secondary-valence-represent-in-werners-concept-41486
question: <div class="question">What does the term "secondary valence" represent in Werner's concept?</div>
tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The number of moles of AgCl formed</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The number of groups bound directly to the metal ion</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The color of the compound</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The conductivity of the compound in solution</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The number of groups bound directly to the metal ion</span> </div>

answer:
<div class="solution">In Werner's concept, "secondary valence" represents the number of groups bound directly to the metal ion.</div>


title: what-does-the-variable-q-represent-in-the-nernst-equation-for-a-general-electrochemical-reaction
tags: ['Chemistry', 'Electrochemistry', 'NEET']
source_db: MarksBatch2_P3.db
slug:
what-does-the-variable-q-represent-in-the-nernst-equation-for-a-general-electrochemical-reaction-17508

question:
<div class="question">What does the variable 'Q' represent in the Nernst equation for a general electrochemical reaction?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The equilibrium constant of the reaction.</span> </div>

answer:
<div class="solution">'Q' represents the reaction quotient, which is the equilibrium constant of the reaction at a given point.</div>


title: what-effect-is-observed-in-emulsions-due-to-the-negatively-charged-droplets
slug: what-effect-is-observed-in-emulsions-due-to-the-negatively-charged-droplets-80725
tags: ['Chemistry', 'Surface Chemistry', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What effect is observed in emulsions due to the negatively charged droplets?</div>

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

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

answer:
<div class="solution">Emulsions with negatively charged droplets often show Brownian movement.</div>


title: what-element-serves-as-the-standard-for-the-present-system-of-atomic-masses
slug: what-element-serves-as-the-standard-for-the-present-system-of-atomic-masses-59725
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'NEET']
answer: <div class="solution">The present system of atomic masses is based on carbon-12 (12</div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What element serves as the standard for the present system of atomic masses?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Carbon-12 (12<br/>C)</span> </div>


title: what-f-x-1-f-x-1-x-is-equal-to-consider-the-function-f-x-x-1-x-1
slug: what-f-x-1-f-x-1-x-is-equal-to-consider-the-function-f-x-x-1-x-1-85734
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What $=\frac{f(x)+1}{f(x)-1}+x$ is equal to ?<br/>Consider the function $f(x)=\frac{x-1}{x+1}$.</div>

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

answer:
<div class="solution">Given $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-1}{\mathrm{x}+1}$<br/>Applying componendo and dividendo, we get<br/>$\frac{\mathrm{f}(\mathrm{x})+1}{\mathrm{f}(\mathrm{x})-1}=\frac{\mathrm{x}-1+\mathrm{x}+1}{\mathrm{x}-1-\mathrm{x}-1}$<br/>$\Rightarrow \frac{\mathrm{f}(\mathrm{x})+1}{\mathrm{f}(\mathrm{x})-1}=-\mathrm{x}$<br/>Now, $\frac{\mathrm{f}(\mathrm{x})+1}{\mathrm{f}(\mathrm{x})-1}+\mathrm{x}=-\mathrm{x}+\mathrm{x}=0$</div>


title: what-factor-determines-the-standard-electrode-potential-of-a-metal
slug: what-factor-determines-the-standard-electrode-potential-of-a-metal-52488
question: <div class="question">What factor determines the standard electrode potential of a metal?</div>
tags: ['Chemistry', 'd and f Block Elements', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Metallic lustre</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Enthalpy of atomisation</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">High tensile strength</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Low volatility</span> </li> </ul>

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

answer:
<div class="solution">The enthalpy of atomisation is an important factor in determining the standard electrode potential of a metal.</div>


tags: ['Chemistry', 'Thermodynamics', 'NEET']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2.639</span> </div>
answer: <div class="solution">Correct Option is : (A)<br/>2.639</div>
source_db: MarksBatch2_P3.db
title:
what-factor-does-the-pressure-of-the-gas-increase-by-when-it-is-compressed-to-half-its-original-volume-in-a-cylinder-with-a-movable-piston-containing-

slug:
what-factor-does-the-pressure-of-the-gas-increase-by-when-it-is-compressed-to-half-its-original-volume-in-a-cylinder-with-a-movable-piston-containing-13918

question:
<div class="question">What factor does the pressure of the gas increase by when it is compressed to half its original volume in a cylinder with a movable piston containing 3 moles of hydrogen at standard temperature and pressure, where the cylinder walls are heat insulators and the piston is insulated with a pile of sand?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">2.639</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">3.223</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4.125</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">5.335</span> </li> </ul>


title: what-forces-primarily-stabilize-the-secondary-and-tertiary-structures-of-proteins
slug: what-forces-primarily-stabilize-the-secondary-and-tertiary-structures-of-proteins-58058
tags: ['Chemistry', 'Biomolecules', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What forces primarily stabilize the secondary and tertiary structures of proteins?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Van der Waals and electrostatic forces</span> </div>

answer:
<div class="solution">The van der Waals and electrostatic forces of attraction are among the forces that stabilize the secondary and tertiary structures of proteins.</div>


tags: ['Chemistry', 'Kinetic Theory of Gases', 'NEET']
answer: <div class="solution">Correct Option is : (A)<br/>$3.8~\times ~{{10}^{-4}}$</div>
source_db: MarksBatch2_P3.db
title:
what-fraction-of-the-total-volume-at-standard-temperature-and-pressure-stp-is-occupied-by-oxygen-gas-if-the-diameter-of-an-oxygen-molecule-is-3

slug:
what-fraction-of-the-total-volume-at-standard-temperature-and-pressure-stp-is-occupied-by-oxygen-gas-if-the-diameter-of-an-oxygen-molecule-is-3-11472

question:
<div class="question">What fraction of the total volume at Standard Temperature and Pressure (STP) is occupied by oxygen gas if the diameter of an oxygen molecule is 3 Ã?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3.8~\times ~{{10}^{-4}}$</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">$3.8~\times ~{{10}^{-3}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.8~\times ~{{10}^{-4}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1.8~\times ~{{10}^{-3}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$3.8~\times ~{{10}^{-4}}$</span> </div>


title: what-happens-at-the-boundary-between-ice-and-water-when-ice-and-water-are-in-equilibrium
slug: what-happens-at-the-boundary-between-ice-and-water-when-ice-and-water-are-in-equilibrium-94964
tags: ['Chemistry', 'Chemical Equilibrium', 'NEET']
source_db: MarksBatch2_P3.db
question:
<div class="question">What happens at the boundary between ice and water when ice and water are in equilibrium?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Molecules from liquid water collide with ice and adhere to it.</span> </div>

answer:
<div class="solution">At the boundary between ice and water, molecules from liquid water collide with ice and adhere to it.</div>


title: what-happens-during-vascularization-in-plants
slug: what-happens-during-vascularization-in-plants-34926
question: <div class="question">What happens during vascularization in plants?</div>
tags: ['Biology', 'Anatomy of Flowering Plants', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Differentiation of procambium is immediately followed by the development of secondary xylem and phloem</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Differentiation of procambium followed by the development of xylem and phloem</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Differentiation of procambium, xylem and phloem is simultaneous</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Differentiation of procambium followed by the development of primary phloem and then by primary xylem</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Differentiation of procambium followed by the development of xylem and phloem</span> </div>

answer:
<div class="solution">Vascularization of plant means the development of vascular bundle in the plant, which develops from the cambium.</div>


title: what-happens-if-a-reaction-has-qckc
slug: what-happens-if-a-reaction-has-qckc-43742
tags: ['Chemistry', 'Chemical Equilibrium', 'NEET']
answer: <div class="solution">Correct Option is : 1 <br/> Net reaction goes from left to right</div>
source_db: MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/modules/dbq/cheas77dtry/ZkfnPJ2HguqlWp37X9tHNKH05KtZM37BrAd0oZSNF1MdWHv7ur.png"/>What happens if a reaction has Qc<kc> <div class="tags" data-pagefind-ignore=""> <span class="tag">Chemistry</span><span class="tag">Chemical Equilibrium</span><span class="tag">NEET</span> </div> <span class="label">Options:</span>
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Net reaction goes from left to right</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Net reaction goes from right to left</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Products and reactants are at equilibrium</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Cannot say</span> </li> </ul> <!-- /Question --> <!-- Solution --> <section class="solution-container" id="accepted-answer"> <div class="solution-header"> <span class="label">Solution:</span> <div class="badges-container" data-pagefind-ignore=""> <span class="badge upvotes-count"> <svg fill="none" height="16" viewbox="0 0 16 16" width="16" xmlns="http://www.w3.org/2000/svg"> <path d="M14.625 5.0075C14.4842 4.84795 14.3111 4.72019 14.1171 4.63269C13.9231 4.54519 13.7128 4.49996 13.5 4.5H10V3.5C10 2.83696 9.73661 2.20107 9.26777 1.73223C8.79893 1.26339 8.16304 1 7.5 1C7.40711 0.999934 7.31604 1.02574 7.237 1.07454C7.15795 1.12333 7.09407 1.19318 7.0525 1.27625L4.69125 6H2C1.73478 6 1.48043 6.10536 1.29289 6.29289C1.10536 6.48043 1 6.73478 1 7V12.5C1 12.7652 1.10536 13.0196 1.29289 13.2071C1.48043 13.3946 1.73478 13.5 2 13.5H12.75C13.1154 13.5001 13.4684 13.3668 13.7425 13.1252C14.0166 12.8835 14.1931 12.5501 14.2388 12.1875L14.9888 6.1875C15.0153 5.97626 14.9966 5.76179 14.9339 5.55833C14.8712 5.35488 14.7659 5.16711 14.625 5.0075ZM2 7H4.5V12.5H2V7Z" fill="#014A82"></path> </svg> 2760 Upvotes
</span> <span class="badge expert-verification-badge" data-pagefind-ignore=""> <svg fill="none" height="16" viewbox="0 0 16 16" width="16" xmlns="http://www.w3.org/2000/svg"> <path d="M14.1163 6.42625C13.8806 6.18 13.6369 5.92625 13.545 5.70312C13.46 5.49875 13.455 5.16 13.45 4.83187C13.4406 4.22187 13.4306 3.53062 12.95 3.05C12.4694 2.56937 11.7781 2.55937 11.1681 2.55C10.84 2.545 10.5012 2.54 10.2969 2.455C10.0744 2.36312 9.82 2.11937 9.57375 1.88375C9.1425 1.46937 8.6525 1 8 1C7.3475 1 6.85812 1.46937 6.42625 1.88375C6.18 2.11937 5.92625 2.36312 5.70312 2.455C5.5 2.54 5.16 2.545 4.83187 2.55C4.22187 2.55937 3.53062 2.56937 3.05 3.05C2.56937 3.53062 2.5625 4.22187 2.55 4.83187C2.545 5.16 2.54 5.49875 2.455 5.70312C2.36312 5.92562 2.11937 6.18 1.88375 6.42625C1.46937 6.8575 1 7.3475 1 8C1 8.6525 1.46937 9.14187 1.88375 9.57375C2.11937 9.82 2.36312 10.0738 2.455 10.2969C2.54 10.5012 2.545 10.84 2.55 11.1681C2.55937 11.7781 2.56937 12.4694 3.05 12.95C3.53062 13.4306 4.22187 13.4406 4.83187 13.45C5.16 13.455 5.49875 13.46 5.70312 13.545C5.92562 13.6369 6.18 13.8806 6.42625 14.1163C6.8575 14.5306 7.3475 15 8 15C8.6525 15 9.14187 14.5306 9.57375 14.1163C9.82 13.8806 10.0738 13.6369 10.2969 13.545C10.5012 13.46 10.84 13.455 11.1681 13.45C11.7781 13.4406 12.4694 13.4306 12.95 12.95C13.4306 12.4694 13.4406 11.7781 13.45 11.1681C13.455 10.84 13.46 10.5012 13.545 10.2969C13.6369 10.0744 13.8806 9.82 14.1163 9.57375C14.5306 9.1425 15 8.6525 15 8C15 7.3475 14.5306 6.85812 14.1163 6.42625ZM10.8538 6.85375L7.35375 10.3538C7.30731 10.4002 7.25217 10.4371 7.19147 10.4623C7.13077 10.4874 7.06571 10.5004 7 10.5004C6.93429 10.5004 6.86923 10.4874 6.80853 10.4623C6.74783 10.4371 6.69269 10.4002 6.64625 10.3538L5.14625 8.85375C5.05243 8.75993 4.99972 8.63268 4.99972 8.5C4.99972 8.36732 5.05243 8.24007 5.14625 8.14625C5.24007 8.05243 5.36732 7.99972 5.5 7.99972C5.63268 7.99972 5.75993 8.05243 5.85375 8.14625L7 9.29313L10.1462 6.14625C10.1927 6.09979 10.2479 6.06294 10.3086 6.0378C10.3692 6.01266 10.4343 5.99972 10.5 5.99972C10.5657 5.99972 10.6308 6.01266 10.6914 6.0378C10.7521 6.06294 10.8073 6.09979 10.8538 6.14625C10.9002 6.1927 10.9371 6.24786 10.9622 6.30855C10.9873 6.36925 11.0003 6.4343 11.0003 6.5C11.0003 6.5657 10.9873 6.63075 10.9622 6.69145C10.9371 6.75214 10.9002 6.8073 10.8538 6.85375Z" fill="#00672E"></path> </svg>
Verified Answer
</span> </div> </div> <div class="correct-answer">
The correct answer is:
<span class="option-value">Net reaction goes from left to right</span> </div> <div class="solution">Correct Option is : 1 <br/> Net reaction goes from left to right</div> </section> <!-- /Solution --> <!-- Call to Action --> <section class="call-to-action" data-pagefind-ignore=""> <div class="call-to-action-header"> <svg fill="none" height="24" viewbox="0 0 25 24" width="25" xmlns="http://www.w3.org/2000/svg"> <path d="M19.9999 13.5C20.0018 13.8058 19.9089 14.1047 19.734 14.3555C19.559 14.6063 19.3107 14.7968 19.023 14.9006L14.1874 16.6875L12.4062 21.5269C12.3007 21.8134 12.1098 22.0608 11.8594 22.2355C11.6089 22.4101 11.3109 22.5038 11.0055 22.5038C10.7002 22.5038 10.4022 22.4101 10.1517 22.2355C9.90124 22.0608 9.71039 21.8134 9.60491 21.5269L7.8124 16.6875L2.97303 14.9062C2.68646 14.8008 2.43915 14.6099 2.26445 14.3595C2.08976 14.109 1.99609 13.811 1.99609 13.5056C1.99609 13.2003 2.08976 12.9022 2.26445 12.6518C2.43915 12.4013 2.68646 12.2105 2.97303 12.105L7.8124 10.3125L9.59366 5.47313C9.69914 5.18656 9.88998 4.93924 10.1404 4.76455C10.3909 4.58985 10.6889 4.49619 10.9943 4.49619C11.2996 4.49619 11.5977 4.58985 11.8481 4.76455C12.0986 4.93924 12.2894 5.18656 12.3949 5.47313L14.1874 10.3125L19.0268 12.0938C19.3146 12.1986 19.5628 12.3901 19.7371 12.642C19.9114 12.8939 20.0033 13.1937 19.9999 13.5ZM14.7499 4.5H16.2499V6C16.2499 6.19891 16.3289 6.38968 16.4696 6.53033C16.6102 6.67098 16.801 6.75 16.9999 6.75C17.1988 6.75 17.3896 6.67098 17.5302 6.53033C17.6709 6.38968 17.7499 6.19891 17.7499 6V4.5H19.2499C19.4488 4.5 19.6396 4.42098 19.7802 4.28033C19.9209 4.13968 19.9999 3.94891 19.9999 3.75C19.9999 3.55109 19.9209 3.36032 19.7802 3.21967C19.6396 3.07902 19.4488 3 19.2499 3H17.7499V1.5C17.7499 1.30109 17.6709 1.11032 17.5302 0.96967C17.3896 0.829018 17.1988 0.75 16.9999 0.75C16.801 0.75 16.6102 0.829018 16.4696 0.96967C16.3289 1.11032 16.2499 1.30109 16.2499 1.5V3H14.7499C14.551 3 14.3602 3.07902 14.2196 3.21967C14.0789 3.36032 13.9999 3.55109 13.9999 3.75C13.9999 3.94891 14.0789 4.13968 14.2196 4.28033C14.3602 4.42098 14.551 4.5 14.7499 4.5ZM22.9999 7.5H22.2499V6.75C22.2499 6.55109 22.1709 6.36032 22.0302 6.21967C21.8896 6.07902 21.6988 6 21.4999 6C21.301 6 21.1102 6.07902 20.9696 6.21967C20.8289 6.36032 20.7499 6.55109 20.7499 6.75V7.5H19.9999C19.801 7.5 19.6102 7.57902 19.4696 7.71967C19.3289 7.86032 19.2499 8.05109 19.2499 8.25C19.2499 8.44891 19.3289 8.63968 19.4696 8.78033C19.6102 8.92098 19.801 9 19.9999 9H20.7499V9.75C20.7499 9.94891 20.8289 10.1397 20.9696 10.2803C21.1102 10.421 21.301 10.5 21.4999 10.5C21.6988 10.5 21.8896 10.421 22.0302 10.2803C22.1709 10.1397 22.2499 9.94891 22.2499 9.75V9H22.9999C23.1988 9 23.3896 8.92098 23.5302 8.78033C23.6709 8.63968 23.7499 8.44891 23.7499 8.25C23.7499 8.05109 23.6709 7.86032 23.5302 7.71967C23.3896 7.57902 23.1988 7.5 22.9999 7.5Z" fill="#1589EE"></path> </svg> <h3>Looking for more such questions to practice?</h3> </div> <p>
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options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Net reaction goes from left to right</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Net reaction goes from right to left</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Products and reactants are at equilibrium</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Cannot say</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Net reaction goes from left to right</span> </div>


tags: ['Chemistry', 'States of Matter', 'NEET']
answer: <div class="solution">Correct Option is : (D)<br/>It disappears completely.</div>
source_db: MarksBatch2_P3.db
title:
what-happens-to-the-boundary-between-the-liquid-phase-and-gaseous-phase-when-a-liquid-is-heated-up-to-its-critical-temperature-in-a-closed-vessel

slug:
what-happens-to-the-boundary-between-the-liquid-phase-and-gaseous-phase-when-a-liquid-is-heated-up-to-its-critical-temperature-in-a-closed-vessel-87621

question:
<div class="question">What happens to the boundary between the liquid phase and gaseous phase when a liquid is heated up to its critical temperature in a closed vessel?</div>

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


tags: ['Chemistry', 'Thermodynamics', 'NEET']
answer: <div class="solution">Correct Option is : (C)<br/>The internal energy remains constant.</div>
source_db: MarksBatch2_P3.db
title:
what-happens-to-the-change-in-internal-energy-of-the-gas-when-the-stopcock-between-cylinders-a-and-b-both-of-equal-capacity-is-suddenly-opened-while-a

slug:
what-happens-to-the-change-in-internal-energy-of-the-gas-when-the-stopcock-between-cylinders-a-and-b-both-of-equal-capacity-is-suddenly-opened-while-a-32127

question:
<div class="question">What happens to the change in internal energy of the gas when the stopcock between cylinders A and B, both of equal capacity, is suddenly opened while A contains a gas at standard temperature and pressure, and B is completely evacuated, with the entire system thermally insulated?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The internal energy remains constant.</span> </div>


title: what-happens-to-the-primary-oocyte-in-the-tertiary-follicle
slug: what-happens-to-the-primary-oocyte-in-the-tertiary-follicle-37565
question: <div class="question">What happens to the primary oocyte in the tertiary follicle?</div>
tags: ['Biology', 'Human Reproduction', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Forms the mature follicle</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">Completes its first meiotic division</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Changes into the secondary follicle</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of the above</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Completes its first meiotic division</span> </div>

answer:
<div class="solution">The primary oocyte within the tertiary follicle grows in size and completes its first meiotic division, resulting in the formation of a large haploid secondary oocyte and a tiny first polar body.</div>


tags: ['Physics', 'Semiconductors', 'JEE Main']
source_db: MarksBatch2_P3.db
title:
what-happens-to-the-reverse-current-in-a-zener-diode-when-the-reverse-voltage-is-less-than-the-zener-voltage

slug:
what-happens-to-the-reverse-current-in-a-zener-diode-when-the-reverse-voltage-is-less-than-the-zener-voltage-84556

question:
<div class="question">What happens to the reverse current in a Zener diode when the reverse voltage is less than the Zener voltage?</div>

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

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

answer:
<div class="solution">When the reverse voltage is less than the Zener voltage, the reverse current in a Zener diode is very low, similar to the leakage current in a standard diode.</div>


tags: ['Chemistry', 'Thermodynamics', 'NEET']
source_db: MarksBatch2_P3.db
title:
what-happens-to-the-temperature-of-the-gas-when-the-stopcock-is-suddenly-opened-between-cylinders-a-and-b-given-that-a-contains-a-gas-at-standard-temp

slug:
what-happens-to-the-temperature-of-the-gas-when-the-stopcock-is-suddenly-opened-between-cylinders-a-and-b-given-that-a-contains-a-gas-at-standard-temp-95691

question:
<div class="question">What happens to the temperature of the gas when the stopcock is suddenly opened between cylinders A and B, given that A contains a gas at standard temperature and pressure, B is completely evacuated, and the entire system is thermally insulated?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The temperature of the gas remains the same.</span> </div>

answer:
<div class="solution">Correct Option is : (C)<br/>The temperature of the gas remains the same.</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'NEET']
source_db: MarksBatch2_P3.db
title:
what-happens-when-a-secondary-alcohol-reacts-with-lucas-reagent-at-room-temperature-in-the-lucas-test

slug:
what-happens-when-a-secondary-alcohol-reacts-with-lucas-reagent-at-room-temperature-in-the-lucas-test-78430

question:
<div class="question">What happens when a secondary alcohol reacts with Lucas reagent at room temperature in the Lucas test?</div>

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

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

answer:
<div class="solution">Secondary alcohols may produce turbidity in the Lucas test after heating.</div>


tags: ['Chemistry', 'Waves and Sound', 'NEET']
answer: <div class="solution">Correct Option is : (D)<br/>3rd harmonic</div>
source_db: MarksBatch2_P3.db
title:
what-harmonic-mode-of-a-pipe-20-cm-long-closed-at-one-end-is-resonantly-excited-by-a-source-of-12375-hz-assume-the-speed-of-sound-in-air-to-be-330-ms

slug:
what-harmonic-mode-of-a-pipe-20-cm-long-closed-at-one-end-is-resonantly-excited-by-a-source-of-12375-hz-assume-the-speed-of-sound-in-air-to-be-330-ms-22495

question:
<div class="question">What harmonic mode of a pipe 20 cm long, closed at one end, is resonantly excited by a source of 1237.5 Hz? (Assume the speed of sound in air to be 330 m/s.)</div>

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


title: what-is-0-1-1-x-2-e-t-a-n-1-x-d-x-equal-to
slug: what-is-0-1-1-x-2-e-t-a-n-1-x-d-x-equal-to-75372
question: <div class="question">What is $\int_{0}^{1} \frac{e^{\tan ^{-1} x} d x}{1+x^{2}}$ equal to ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$e^{\frac{\pi}{4}}-1 \quad$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$e^{\frac{\pi}{4}}+1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$e-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$e$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{I}=\int_{0}^{1} \frac{\mathrm{e}^{\mathrm{tan}^{-1} \mathrm{x}}}{1+\mathrm{x}^{2}} \mathrm{dx}$<br/>Let $\tan ^{-1} \mathrm{x}=\mathrm{t}$<br/>$\frac{1}{1+\mathrm{x}^{2}} \mathrm{dx}=\mathrm{dt}$<br/>Lower limit $\rightarrow \mathrm{t}=\tan ^{-1} 0=0$<br/>upper limit $\rightarrow \mathrm{t}=\tan ^{-1},=\pi / 4$<br/>$\therefore \int_{0}^{\pi / 4} \mathrm{e}^{\mathrm{t}}-\mathrm{dt}=\left[\mathrm{e}^{\mathrm{t}}\right]_{0}^{\pi / 4}$<br/>$\mathrm{e}^{\pi / 4}-\mathrm{e}^{0} \Rightarrow \mathrm{e}^{\pi / 4}-1$<br/>$=(-\mathrm{a} \cos \theta,-\mathrm{b} \sin \theta)$</div>


title: what-is-0-1-1-x-2-t-a-n-1-x-d-x-equal-to
slug: what-is-0-1-1-x-2-t-a-n-1-x-d-x-equal-to-66437
question: <div class="question">What is $\int_{0}^{1} \frac{\tan ^{-1} x}{1+x^{2}} d x$ equal to ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi^{2}}{8}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\pi^{2}}{32}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\pi}{8}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\pi^{2}}{32}$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int_{0}^{1} \frac{\tan ^{-1} \mathrm{x}}{1+\mathrm{x}^{2}} \mathrm{dx}$<br/>Let, $\tan ^{-1} x=t$<br/>$\frac{1}{1+x^{2}} d x=d t$<br/>Also, $x=0 \rightarrow t=0$<br/>$x=1 \rightarrow t=\frac{\pi}{4}$<br/>$\therefore \mathrm{I}=\int_{0}^{\frac{\pi}{4}} \mathrm{t} \mathrm{dt}=\left.\frac{\mathrm{t}^{2}}{2}\right|_{0} ^{\frac{\pi}{4}}=\frac{\pi^{2}}{32}$</div>


title: what-is-0-1-2-s-i-n-2-x-d-x-equal-to
slug: what-is-0-1-2-s-i-n-2-x-d-x-equal-to-11591
question: <div class="question">What is $\int_{0}^{\pi} \frac{d x}{1+2 \sin ^{2} x}$ equal to?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\pi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{3}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\pi}{\sqrt{3}}$</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">$\frac{2 \pi}{\sqrt{3}}$</span> </li> </ul>

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

answer:
<div class="solution">$\begin{aligned} I=\int_{0}^{\pi} \frac{d x}{1+2 \sin ^{2} x}=2 \int_{0}^{\pi / 2} \frac{d x}{1+2 \sin ^{2} x} \\ &amp;=2 \int_{0}^{\pi / 2} \frac{\sec ^{2} x d x}{\sec ^{2} x+2 \tan ^{2} x}=2 \int_{0}^{\pi / 2} \frac{\sec ^{2} x d x}{1+3 \tan ^{2} x} \\ \text { Put } \tan x=t \Rightarrow \sec ^{2} x d x=d t \\ \text { When } x=0, \mathrm{t}=0 \\ \text { When } x=\frac{\pi}{2}, t=\infty \\ \therefore \quad I=2 \int_{0}^{\infty} \frac{d t}{1+3 t^{2}}=\frac{2}{3} \int_{0}^{\infty} \frac{d t}{t^{2}+\left(\frac{1}{\sqrt{3}}\right)^{2}} \\=\frac{2}{3} \times \sqrt{3}\left[\tan ^{-1} \frac{t}{1 / \sqrt{3}}\right]_{0}^{\infty} \\=\frac{2}{\sqrt{3}}\left[\tan ^{-1} \sqrt{3} t\right]_{0}^{\infty} \\=\frac{2}{\sqrt{3}}\left[\tan ^{-1} \infty-\tan ^{-1} 0\right] \\=\frac{2}{\sqrt{3}} \times \frac{\pi}{2}=\frac{\pi}{\sqrt{3}} \end{aligned}$</div>


title: what-is-0-1-x-1-x-9-dx-equal-to
slug: what-is-0-1-x-1-x-9-dx-equal-to-21482
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{0}^{1} \mathrm{x}(1-\mathrm{x})^{9} \mathrm{dx}$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{110}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{1}{132}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{148}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{240}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{110}$</span> </div>

answer:
<div class="solution">$\int_{0}^{1} \int_{0}^{1} x(1-x)^{9} \cdot d x$<br/>$=\int_{0}^{1}(1-x)(1-(1-x))^{9} \cdot d x\left(\because \int_{0}^{a} f(x) \cdot d x=\int_{0}^{a} f(a-x) \cdot d x\right)$<br/>$\quad=\int_{0}^{1}(1-x) \cdot x^{9} \cdot d x$<br/>$\quad=\int_{0}^{1}\left(x^{9}-x^{10}\right) \cdot d x$<br/>$\quad=\left(\frac{x^{10}}{10}-\frac{x^{11}}{11}\right)_{0}^{1}$<br/>$\quad=\frac{1}{10}-\frac{1}{11}=\frac{11-10}{110}=\frac{1}{110} .$</div>


title: what-is-0-1-x-1-x-n-d-x-equal-to
slug: what-is-0-1-x-1-x-n-d-x-equal-to-51900
question: <div class="question">What is $\int_{0}^{1} x(1-x)^{n} d x$ equal to? $\quad$</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{n(n+1)}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{1}{(n+1)(n+2)}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

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

answer:
<div class="solution">Let $I=\int_{0}^{1} x(1-x)^{n} d x$<br/>Put $1-x=t \Rightarrow d x=-d t$<br/>when $x=0$ then $t=1$ when $x=1$ then $t=0$<br/>$\begin{aligned} \therefore &amp; I=-\int_{1}^{0}(1-t) t^{n} d t=\int_{0}^{1}\left(t^{n}-t^{n+1}\right) d t \\ &amp;=\left[\frac{t^{n+1}}{n+1}-\frac{t^{n+2}}{n+2}\right]_{0}^{1} \\ &amp;=\frac{1}{n+1}-\frac{1}{n+2}=\frac{1}{(n+1)(n+2)} \end{aligned}$</div>


title: what-is-0-2-1-sin-2-x-d-x-equal-to
slug: what-is-0-2-1-sin-2-x-d-x-equal-to-99188
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2017 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{0}^{2 \pi} \sqrt{1+\sin \frac{x}{2} \mathrm{~d} x \text { equal to? }}$</div>

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

answer:
<div class="solution">$\int_{0}^{2 \pi} \sqrt{1+\sin \frac{x}{2}} \cdot d x$<br/>$\sin \frac{x}{2}=\sin 2\left(\frac{x}{4}\right)=2 \sin \frac{x}{4} \cos \frac{x}{4}$<br/>$\therefore \int_{0}^{2 \pi} \sqrt{\sin ^{2} \frac{x}{4}+\cos ^{2} \frac{x}{4}+2 \sin \frac{x}{4} \cos \frac{x}{4}}$<br/>$=\int_{0}^{2 \pi} \sqrt{\left(\sin \frac{x}{4}+\cos \frac{x}{4}\right)^{2}} \cdot d x$<br/>$=\int_{0}^{2 \pi}\left|\sin \frac{x}{4}+\cos \frac{x}{4}\right| \cdot d x$<br/>$=4\left[-\cos \frac{x}{4}+\sin \frac{x}{4}\right]_{0}^{2 \pi}$<br/>$=4\left[-\left(\cos \frac{2 \pi}{4}-\cos 0\right)+\sin \left(\frac{2 \pi}{4}-\sin 0\right)\right]$<br/>$=4[-(-1)+(1)]=4 \times 2=8 .$</div>


title: what-is-0-2-a-2-c-o-s-2-x-b-2-s-i-n-2-x-d-x-equal-to
slug: what-is-0-2-a-2-c-o-s-2-x-b-2-s-i-n-2-x-d-x-equal-to-73338
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{0}^{\pi / 2} \frac{d x}{a^{2} \cos ^{2} x+b^{2} \sin ^{2} x}$ equal to?</div>

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

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

answer:
<div class="solution">Let $\mathrm{I}=\int_{0}^{\frac{\pi}{2}} \frac{\mathrm{d} \mathrm{x}}{\mathrm{a}^{2} \cos ^{2} \mathrm{x}+\mathrm{b}^{2} \sin ^{2} \mathrm{x}}$<br/>$=\int_{0}^{\frac{\pi}{2}} \frac{\sec ^{2} \mathrm{x} \mathrm{dx}}{\mathrm{a}^{2}+\mathrm{b}^{2} \tan ^{2} \mathrm{x}}$<br/>[divide numerator and denominator by $\left.\cos ^{2} \mathrm{x}\right]$ Put tan $x=t$ $\Rightarrow \sec ^{2} x d x=d t$<br/>When $x=0$, then $t=0$ and when $x=\frac{\pi}{2}$, then $t=\infty$<br/>$\therefore \mathrm{I}=\int_{0}^{\infty} \frac{\mathrm{dt}}{\mathrm{a}^{2}+\mathrm{b}^{2} \mathrm{t}^{2}}=\frac{1}{\mathrm{~b}^{2}} \int_{0}^{\infty} \frac{\mathrm{dt}}{\left(\frac{\mathrm{a}}{\mathrm{b}}\right)^{2}+\mathrm{t}^{2}}$<br/>$=\frac{1}{\mathrm{~b}^{2}} \frac{1}{\left(\frac{\mathrm{a}}{\mathrm{b}}\right)}\left[\tan ^{-1}\left(\frac{\mathrm{bt}}{\mathrm{a}}\right)\right]_{0}^{\infty}$<br/>$\quad\left[\because \int \frac{\mathrm{dx}}{\mathrm{a}^{2}+\mathrm{x}^{2}}=\frac{1}{\mathrm{a}} \tan ^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right)+\mathrm{C}\right]$<br/>$=\frac{1}{\mathrm{ab}}\left[\tan ^{-1}(\infty)-\tan ^{-1}(0)\right]$<br/>$=\frac{1}{\mathrm{ab}}\left[\frac{\pi}{2}-0\right]=\frac{\pi}{2 \mathrm{ab}}$</div>


title: what-is-0-2-e-l-n-x-d-x-equal-to
slug: what-is-0-2-e-l-n-x-d-x-equal-to-31084
question: <div class="question">What is $\int_{0}^{2} e^{\ln x} d x$ equal to ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$\int_{0}^{2} \mathrm{e}^{\ln x} \mathrm{dx}=\int_{0}^{2} x \mathrm{dx}=\left|\frac{\mathrm{x}^{2}}{2}\right|_{0}^{2}=2$</div>


title: what-is-0-2-ln-tan-x-d-x-equal-to
slug: what-is-0-2-ln-tan-x-d-x-equal-to-88787
question: <div class="question">What is $\int_{0}^{\frac{\pi}{2}} \ln (\tan x) d x$ equal to ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\ln 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\ln 2$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">$\quad I=\int_{0}^{\frac{\pi}{2}} \ln (\tan x) d x$<br/>$I=\int_{0}^{\frac{\pi}{2}} \ln \left(\tan \left(\frac{\pi}{2}-x\right)\right) d x$<br/>$=\int_{0}^{\frac{\pi}{2}} \ln \cot x d x$<br/>Adding equations (i) and (ii)<br/>$2 \mathrm{I}=\int_{0}^{\frac{\pi}{2}} \ln (\tan \mathrm{x} \cdot \cot \mathrm{x}) \mathrm{dx}$<br/>$2 \mathrm{I}=0$<br/>$\mathrm{I}=0$</div>


title: what-is-0-2-s-i-n-3-x-c-o-s-3-x-s-i-n-3-x-d-x
slug: what-is-0-2-s-i-n-3-x-c-o-s-3-x-s-i-n-3-x-d-x-89452
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin ^{3} x+\cos ^{3} x} d x ? \quad$</div>

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

answer:
<div class="solution">Let $I=\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin ^{3} x+\cos ^{3} x} d x$<br/>...(i)<br/>$=\int_{0}^{\pi / 2} \frac{\sin ^{3}(\pi / 2-x)}{\sin ^{3}(\pi / 2-x)+\cos ^{3}(\pi / 2-x)} d x$<br/>By using the property $\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x$<br/>$\Rightarrow I \int_{0}^{\pi / 2} \frac{\cos ^{3} x}{\sin ^{3} x+\cos ^{3} x} d x\left(\begin{array}{c}\because \sin (\pi / 2-\theta)=\cos \theta \text { and } \\ \cos (\pi / 2-\theta)=\sin \theta\end{array}\right)$<br/>On adding Eqs. ( $i$ ) and ( $(i i)$, we get $2 I=\int_{0}^{\pi / 2} \frac{\sin ^{3} x d x}{\sin ^{3} x+\cos ^{3} x}+\int_{0}^{\pi / 2} \frac{\cos ^{3} x d x}{\sin ^{3} x+\cos ^{3} x}$<br/>$=\int_{0}^{\pi / 2} \frac{\sin ^{3} x+\cos ^{3} x d x}{\sin ^{3} x+\cos ^{3} x}$<br/>$2 I=\int_{0}^{\pi / 2} 1 d x \Rightarrow 2 I=[x]_{0}^{\pi / 2}=\frac{\pi}{2} \Rightarrow I=\frac{\pi}{4}$</div>


title: what-is-0-2-x-1-c-o-s-d-equal-to
slug: what-is-0-2-x-1-c-o-s-d-equal-to-35097
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2017 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{0}^{\frac{x}{2}} \frac{\mathrm{d} \theta}{1+\cos \theta}$ 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="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">$\sqrt{3}$</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">$\int_{0}^{\pi / 2} \frac{\mathrm{d} \theta}{1+\cos \theta}=\int_{0}^{\pi / 2} \frac{\mathrm{d} \theta}{2 \cos ^{2}\left(\frac{\theta}{2}\right)}=\frac{1}{2} \int_{0}^{\pi / 2} \sec ^{2} \frac{\theta}{2} \mathrm{~d} \theta$<br/>$=\frac{1}{2}\left[\frac{\tan \theta / 2}{1 / 2}\right]_{0}^{\pi / 2}$<br/>$=\tan \frac{\pi}{4}-\tan 0$<br/>$=1-0=1$</div>


title: what-is-0-2-x-2-dx-equal-to-where-is-the-greatest-integer-function
slug: what-is-0-2-x-2-dx-equal-to-where-is-the-greatest-integer-function-82981
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{0}^{\sqrt{2}}\left[\mathrm{x}^{2}\right] \mathrm{dx}$ equal to (where $[.]$ is the greatest integer function)?</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; \int_{0}^{\sqrt{2}}\left[x^{2}\right] \mathrm{d} x \\ &amp;=\int_{0}^{1}\left[x^{2}\right] d x+\int_{1}^{\sqrt{2}}\left[x^{2}\right] d x \end{aligned}$<br/>$=0+\int_{1}^{\sqrt{2}} 1 \mathrm{dx} \quad(\because[]$ is greatest integer function $)$<br/>$(\mathrm{x})_{1}^{\sqrt{2}}=\sqrt{2}-1$</div>


title: what-is-0-4-cos-x-d-x-equal-to
slug: what-is-0-4-cos-x-d-x-equal-to-23077
question: <div class="question">What is $\int_{0}^{4 \pi}|\cos x| d x$ equal to?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">8</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\begin{aligned} \int_{0}^{4 \pi}|\cos x| d x &amp;=4 \int_{0}^{\pi}|\cos x| d x \\ &amp;=4\left[\int_{0}^{\pi / 2} \cos x d x-\int_{\pi / 2}^{\pi} \cos x d x\right] \\ &amp;=4\left[(\sin x)_{0}^{\pi / 2}-(\sin x)_{\pi / 2}^{\pi}\right] \\ &amp;=4\left[\sin \frac{\pi}{2}-0-\sin \pi+\sin \frac{\pi}{2}\right] \\ &amp;=4\left[2 \sin \frac{\pi}{2}\right]=8 \end{aligned}$</div>


title: what-is-0-e-x-sin-x-dx-equal-to
slug: what-is-0-e-x-sin-x-dx-equal-to-26830
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{0}^{\pi} \mathrm{e}^{\mathrm{x}} \sin x \mathrm{dx}$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{e}^{\pi}+1}{2}$</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">$\frac{\mathrm{e}^{\pi}-1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{e}^{\pi}+1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{e}^{\pi}+1}{4}$</span> </li> </ul>

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

answer:
<div class="solution">$\int_{0}^{\pi} \mathrm{e}^{\mathrm{x}} \sin x \cdot \mathrm{d} \mathrm{x}=\mathrm{I}$<br/>$\mathrm{I}=\left(\sin \mathrm{x} \cdot \mathrm{e}^{\mathrm{x}}\right)_{0}^{\pi}-\int_{0}^{\pi} \cos \mathrm{x} \cdot \mathrm{e}^{\mathrm{x}} \cdot \mathrm{dx}$<br/>$\mathrm{I}=\left(\sin \pi \cdot \mathrm{e}^{\pi}-\sin 0 . \mathrm{e}^{0}\right)-\left\{\left[\cos x \cdot e^{\mathrm{x}}\right]_{0}^{\pi}-\int_{0}^{\pi} \sin x \cdot \mathrm{e}^{\mathrm{x}} \cdot \mathrm{dx}\right\}$<br/>$\mathrm{I}=0-\left\{\left(\cos \pi \cdot \mathrm{e}^{\pi}-\cos 0 \cdot \mathrm{e}^{0}\right)-\mathrm{I}\right\}$<br/>$\mathrm{I}=-\left[-\mathrm{e}^{\pi}-1\right]-\mathrm{I}$<br/>$\Rightarrow 2 \mathrm{I}=\mathrm{e}^{\pi}+1 \Rightarrow \mathrm{I}=\frac{\mathrm{e}^{\pi}+1}{2} .$</div>


title: what-is-1-1-2-equal-to-where-is-the-cube-root-of-unity
slug: what-is-1-1-2-equal-to-where-is-the-cube-root-of-unity-60658
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\sqrt{\frac{1+\omega^{2}}{1+\omega}}$ equal to, where $\omega$ is the cube root of unity?</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">$\omega$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\omega^{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$i\omega$, where $i=\sqrt{-1}$</span> </li> </ul>

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

answer:
<div class="solution">$\quad X=\sqrt{\frac{1+\omega^{2}}{1+\omega}} \quad\left(\because 1+\omega+\omega^{2}=0\right.$ and $\left.\omega^{3}=1\right)$<br/>$\Rightarrow X=\sqrt{\frac{-\omega}{-\omega^{2}}}=\sqrt{\frac{1}{\omega}}=\sqrt{\frac{\omega^{3}}{\omega}}=\sqrt{\omega^{2}}=\omega$</div>


title: what-is-1-1-dx-d-tan-1-x-1-dx-equal-to
slug: what-is-1-1-dx-d-tan-1-x-1-dx-equal-to-51497
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{-1}^{1}\left\{\frac{\mathrm{d}}{\mathrm{dx}}\left(\tan ^{-1} \frac{1}{\mathrm{x}}\right)\right\} \mathrm{dx}$ equal to?</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; \int_{-1}^{1}\left(\frac{d}{d x} \tan ^{-1} \frac{1}{x}\right) d x \\ &amp;=\int_{-1}^{1}\left(\frac{d}{d x} \cot ^{-1} x\right) d x \\ &amp;=\int_{-1}^{1} \frac{-1}{\left(1+x^{2}\right)} d x=-2 \int_{0}^{1} \frac{1}{1+x^{2}} d x \\ &amp;=-2\left[\tan ^{-1} x\right]_{0}^{1}=-2 \times \frac{\pi}{4}=-\frac{\pi}{2} \end{aligned}$</div>


title: what-is-1-2-n-x-d-x-equal-to
slug: what-is-1-2-n-x-d-x-equal-to-34684
question: <div class="question">What is $\int_{1}^{2} \ell n x d x$ equal to ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\ln 2$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\ln \left(\frac{4}{e}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\ln \left(\frac{e}{4}\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\ln \left(\frac{4}{e}\right)$</span> </div>

answer:
<div class="solution">$\int_{1}^{2} \ln x \mathrm{~d} \mathrm{x}=[\mathrm{x} \ln \mathrm{x}-\mathrm{x}]_{1}^{2}$<br/>$=2 \ln 2-2-\ln \mathrm{l}+1$<br/>$=\ln 4-\ln \mathrm{e}=\ln \left(\frac{4}{\mathrm{e}}\right)$</div>


title: what-is-1-3-1-x-4-d-x-equal-to
slug: what-is-1-3-1-x-4-d-x-equal-to-59501
question: <div class="question">What is $\int_{1}^{3}\left|1-x^{4}\right| d x$ equal to?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-232 / 5$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-116 / 5$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$116 / 5$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$232 / 5$</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">$232 / 5$</span> </div>

answer:
<div class="solution">$\begin{aligned} I &amp;=\int_{1}^{3}\left|1-x^{4}\right| d x \\ I &amp;=\int_{1}^{3}-\left(1-x^{4}\right) d x \quad\left(:-|x|=\left\{\begin{array}{ll}x, &amp; x \geq 0 \\ -x, &amp; x   &lt; 0\end{array}\right)\right.\\ I &amp;=\int_{1}^{3}\left(x^{4}-1\right) d x \Rightarrow I=\left[\frac{x^{5}}{5}-x\right]_{1}^{3} \\ I &amp;=\left(\frac{3^{5}}{5}-3\right)-\left(\frac{1^{5}}{5}-1\right) \Rightarrow I=\frac{232}{5} \end{aligned}$</div>


title: what-is-1-c-o-s-x-s-i-n-x-s-i-n-x-1-c-o-s-x-equal-to
slug: what-is-1-c-o-s-x-s-i-n-x-s-i-n-x-1-c-o-s-x-equal-to-84358
question: <div class="question">What is $\frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2011 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2 \tan x$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$2 \operatorname{cosec} x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2 \cos x$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \sin x$</span> </li> </ul>

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

answer:
<div class="solution">$\quad \frac{\sin x}{1+\cos x}+\frac{1+\cos x}{\sin x}=\frac{\sin ^{2} x+1+\cos ^{2} x+2 \cos x}{(1+\cos x)(\sin x)}$<br/>$=\frac{2+2 \cos x}{(1+\cos x)(\sin x)}=\frac{2(1+\cos x)}{(1+\cos x)(\sin x)}$<br/>$=\frac{2}{\sin x}=2 \operatorname{cosec} x$</div>


title: what-is-1-e-x-1-d-x-equal-to
slug: what-is-1-e-x-1-d-x-equal-to-97715
question: <div class="question">What is $\int \frac{1}{1+e^{x}} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x-\log x+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x-\log (\tan x)+c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x-\log \left(1+e^{x}\right)+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\log \left(1+e^{x}\right)+c$<br/>where $\mathrm{c}$ is a constant of integration.</span> </li> </ul>

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

answer:
<div class="solution">$\int \frac{1}{1+e^{x}} d x=\int \frac{1}{e^{x}\left(1+\frac{1}{e^{x}}\right)} d x=\int \frac{e^{-x}}{e^{-x}+1} d x$<br/>Put $1+e^{-x}=t$<br/>$-e^{-x} d x=d t$<br/>$\therefore \quad \int \frac{e^{-x}}{1+e^{-x}} d x=-\int \frac{d t}{t}=-\log t+c$<br/>$=-\log \left(1+e^{-x}\right)+c$<br/>$=-\log \left(\frac{1+e^{x}}{e^{x}}\right)+c=x-\log \left(1+e^{x}\right)+c$<br/>$\left(\because \log e^{x}=x\right)$</div>


title: what-is-1-e-x-ln-x-dx-equal-to
slug: what-is-1-e-x-ln-x-dx-equal-to-29840
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{1}^{\mathrm{e}} \mathrm{x} \ln \mathrm{x} \mathrm{dx}$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{e}^{2}+1}{4}$</span> </div>

answer:
<div class="solution">$\int_{1}^{e} x \cdot \ell n x \cdot d x=\left[\ell n x \cdot \frac{x^{2}}{2}\right]_{1}^{e}-\int_{1}^{e} \frac{1}{x} \cdot x^{2} \cdot \mathrm{dx}$<br/>$=\left(\frac{\mathrm{e}^{2}}{2}-0\right)-\frac{1}{2}\left(\frac{1}{2}\right)\left(x^{2}\right)_{1}^{e}=\frac{\mathrm{e}^{2}}{2}-\frac{1}{4}\left(\mathrm{e}^{2}-1\right)=\frac{2 \mathrm{e}^{2}-\mathrm{e}^{2}+1}{4}$<br/>$=\frac{\mathrm{e}^{2}+1}{4}$</div>


title: what-is-1-i-4-n-3-1-i-4-n-5-equal-to-where-n-is-a-natural-number
slug: what-is-1-i-4-n-3-1-i-4-n-5-equal-to-where-n-is-a-natural-number-79531
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{(1+i)^{4 n+5}}{(1-i)^{4 n+3}}$ equal to, where $n$ is a natural number</div>

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

answer:
<div class="solution">$\quad$ Given $\frac{(1+\mathrm{i})^{4 \mathrm{n}+5}}{(1-\mathrm{i})^{4 \mathrm{n}+3}}$<br/>$=\frac{(1+\mathrm{i})^{4 \mathrm{n}+3} \cdot(1+\mathrm{i})^{2}}{(1-\mathrm{i})^{4 \mathrm{n}+3}}=\left(\frac{1+\mathrm{i}}{1-\mathrm{i}}\right)^{4 \mathrm{n}+3} \cdot(1+\mathrm{i})^{2}$<br/>$=\left[\frac{(1+\mathrm{i})(1+\mathrm{i})}{(1-\mathrm{i})(1+\mathrm{i})}\right]^{4 \mathrm{n}+3} \cdot\left(1+\mathrm{i}^{2}+2 \mathrm{i}\right)$<br/>$=\left[\frac{1+\mathrm{i}^{2}+2 \mathrm{i}}{1+1}\right]^{4 \mathrm{n}+3} \cdot 2 \mathrm{i}=(\mathrm{i})^{4 \mathrm{n}+3} \cdot 2 \mathrm{i}=2(\mathrm{i})^{4 \mathrm{n}+4}$<br/>$=2 \cdot\left(\mathrm{i}^{4(\mathrm{n}+1)}\right)=2$</div>


title: what-is-1-l-o-g-x-2-l-o-g-x-d-x-equal-to
slug: what-is-1-l-o-g-x-2-l-o-g-x-d-x-equal-to-17087
question: <div class="question">What is $\int \frac{\log x}{(1+\log x)^{2}} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{(1+\log x)^{3}}+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{(1+\log x)^{2}}+c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{x}{(1+\log x)}+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{x}{(1+\log x)^{2}}+c$<br/>Where $c$ is a constant.</span> </li> </ul>

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

answer:
<div class="solution">Let $I=\int \frac{\log x}{(1+\log x)^{2}} d x$<br/>Put $\log x=t \Rightarrow \frac{1}{x} d x=d t$<br/>$\Rightarrow d x=x \cdot d t \Rightarrow d x=e^{t} \cdot d t \quad\left(\because x=e^{t}\right)$<br/>$I=\int \frac{e^{t} t}{(1+t)^{2}} d t=\int \frac{e^{t} \cdot(t+1-1)}{(1+t)^{2}} d t$<br/>$=\int \frac{e^{t}(1+t)}{(1+t)^{2}} d t-\int \frac{e^{t}}{(1+t)^{2}} d t=\int \frac{e^{t}}{1+t} d t-\int \frac{e^{t}}{(1+t)^{2}} d t$<br/>$=\frac{e^{t}}{1+t}-\int-e^{t} \frac{1}{(1+t)^{2}} d t-\int \frac{e^{t}}{(1+t)^{2}} d t$<br/>$=\frac{\mathrm{e}^{\mathrm{t}}}{1+t}+\int^{e^{t}} \frac{1}{(1+t)^{2}} d t-\int \frac{e^{t}}{(1+t)^{2}} d t$<br/>$=\frac{e^{t}}{1+t}+c=\frac{x}{1+\log x}+c$</div>


title: what-is-1-s-i-n-a-1-s-i-n-a-1-s-i-n-a-1-s-i-n-a-equal-to
slug: what-is-1-s-i-n-a-1-s-i-n-a-1-s-i-n-a-1-s-i-n-a-equal-to-79570
question: <div class="question">What is $\frac{1+\sin A}{1-\sin A}-\frac{1-\sin A}{1+\sin A}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sec A-\tan A$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \sec A \cdot \tan A$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$4 \sec A \cdot \tan A$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \operatorname{cosec} A \cdot \cot A$</span> </li> </ul>

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

answer:
<div class="solution">$\frac{1+\sin A}{1-\sin A}-\frac{1-\sin A}{1+\sin A}$<br/>$=\frac{(1+\sin \mathrm{A})^{2}-(1-\sin \mathrm{A})^{2}}{(1-\sin \mathrm{A})(1+\sin \mathrm{A})}=\frac{4 \sin \mathrm{A}}{\cos ^{2} \mathrm{~A}}$<br/>$=\frac{4 \sin \mathrm{A}}{\cos \mathrm{A}} \cdot \frac{1}{\cos \mathrm{A}}=4 \sec \mathrm{A} \cdot \tan \mathrm{A}$</div>


title: what-is-1-sin-2-1-tan-2-equal-to
slug: what-is-1-sin-2-1-tan-2-equal-to-92438
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\left(1-\sin ^{2} \theta\right)\left(1+\tan ^{2} \theta\right)$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sin ^{2} \theta$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cos ^{2} \theta$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\tan ^{2} \theta$</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">$\begin{aligned}\left(1-\sin ^{2} \theta\right)\left(1+\tan ^{2} \theta\right) &amp;=\cos ^{2} \theta \cdot \sec ^{2} \theta \\=&amp; \cos ^{2} \theta \cdot \frac{1}{\cos ^{2} \theta}=1 \end{aligned}$</div>


title: what-is-1-sin-2-equal-to
slug: what-is-1-sin-2-equal-to-77973
question: <div class="question">What is $\sqrt{1+\sin 2 \theta}$ equal to ?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\cos \theta-\sin \theta$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\cos \theta+\sin \theta$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$2 \cos \theta+\sin \theta$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\cos \theta+2 \sin \theta$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\cos \theta+\sin \theta$</span> </div>

answer:
<div class="solution">Consider, $\sqrt{1+\sin 2 \theta}$<br/>$=\sqrt{\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta}$<br/>$=\sqrt{(\sin \theta+\cos \theta)^{2}}=\sin \theta+\cos \theta$</div>


title: what-is-1-t-a-n-2-2-t-a-n-equal-to
slug: what-is-1-t-a-n-2-2-t-a-n-equal-to-51949
question: <div class="question">What is $\frac{2 \tan \theta}{1+\tan ^{2} \theta}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\cos 2 \theta$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\tan 2 \theta$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\sin 2 \theta$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\operatorname{cosec} 2 \theta$</span> </li> </ul>

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

answer:
<div class="solution">$\frac{2 \tan \theta}{1+\tan ^{2} \theta}=\frac{2 \tan \theta}{\sec ^{2} \theta}$<br/>$=2 \tan \theta \cdot \cos ^{2} \theta$<br/>$=2 \sin \theta \cdot \cos \theta=\sin 2 \theta$</div>


title: what-is-1-t-a-n-2-2-x-2-equal-to
slug: what-is-1-t-a-n-2-2-x-2-equal-to-52146
question: <div class="question">What is $\frac{2}{1+\tan ^{2} \frac{\mathrm{x}}{2}}$ equal to:</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sin x \cdot \cos x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\tan x$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sin x$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\cos x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

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

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


title: what-is-1-t-a-n-c-o-s-1-c-o-t-s-i-n-equal-to
slug: what-is-1-t-a-n-c-o-s-1-c-o-t-s-i-n-equal-to-12395
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{\cos \theta}{1-\tan \theta}+\frac{\sin \theta}{1-\cot \theta}$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sin \theta+\cos \theta$</span> </div>

answer:
<div class="solution">$\frac{\cos \theta}{1-\tan \theta}+\frac{\sin \theta}{1-\cot \theta}$<br/>$=\frac{\cos \theta}{1-\frac{\sin \theta}{\cos \theta}}+\frac{\sin \theta}{1-\frac{\cos \theta}{\sin \theta}}$<br/>$=\frac{\cos ^{2} \theta}{\cos \theta-\sin \theta}+\frac{\sin ^{2} \theta}{\sin \theta-\cos \theta}$<br/>$=\frac{\cos ^{2} \theta}{\cos \theta-\sin \theta}-\frac{\sin ^{2} \theta}{\cos \theta-\sin \theta}$<br/>$=\frac{\cos ^{2} \theta-\sin ^{2} \theta}{\cos \theta-\sin \theta}=\frac{(\cos \theta-\sin \theta)(\cos \theta+\sin \theta)}{\cos \theta-\sin \theta}$<br/>$=\cos \theta+\sin \theta$<br/>$\therefore \quad$ Option (b) is correct.</div>


title: what-is-1-tan-tan-2-tan-tan-2-sec-2-sec-2-equal-to
slug: what-is-1-tan-tan-2-tan-tan-2-sec-2-sec-2-equal-to-16393
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2019 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $(1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2}-\sec ^{2} \alpha \sec ^{2} \beta$<br/>equal to</div>

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

answer:
<div class="solution">\begin{aligned}<br/>&amp;(1+\tan \alpha \tan \beta)^{2}+(\tan \alpha-\tan \beta)^{2}-\sec ^{2} \alpha \sec ^{2} \beta \\<br/>&amp;=1+\tan ^{2} \alpha \tan ^{2} \beta+2 \tan \alpha \tan \beta+\tan ^{2} \alpha+\tan ^{2} \beta \\<br/>&amp; \quad-2 \tan \alpha \tan \beta-\sec ^{2} \alpha \sec ^{2} \beta \\<br/>&amp;=1+\tan ^{2} \alpha \tan ^{2} \beta+\tan ^{2} \alpha+\tan ^{2} \beta-\sec ^{2} \alpha \sec ^{2} \beta \\<br/>&amp;=\left(1+\tan ^{2} \alpha\right)\left(1+\tan ^{2} \beta\right)-\sec ^{2} \alpha \sec ^{2} \beta \\<br/>&amp;=\sec ^{2} \alpha \sec ^{2} \beta-\sec ^{2} \alpha \sec ^{2} \beta \\<br/>&amp;=0 \\</div>


title: what-is-2-2-2-2-cos-4-a-equal-to
slug: what-is-2-2-2-2-cos-4-a-equal-to-41685
question: <div class="question">What is $\sqrt{2+\sqrt{2+\sqrt{2+2 \cos 4 A}}}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\cos A$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cos (2 A)$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2 \cos (A / 2)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\sqrt{2 \cos A}$</span> </li> </ul>

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

answer:
<div class="solution">$\sqrt{2+\sqrt{2+\sqrt{2+2 \cos 4 A}}}$<br/>$=\sqrt{2+\sqrt{2+\sqrt{2(1+\cos 4 A)}}}$<br/>$=\sqrt{2+\sqrt{2+2 \cos 2 A}}=\sqrt{2+\sqrt{2(1+\cos 2 A)}}$<br/>$\left(\because 1+\cos 4 \mathrm{~A}=2 \cos ^{2} 2 \mathrm{~A}\right)$<br/>$=\sqrt{2+2 \cos A}=\sqrt{2(1+\cos A)}$<br/>$\left(\because 1+\cos 2 \mathrm{~A}=2 \cos ^{2} \mathrm{~A}\right)$<br/>$=2 \cos \left(\frac{A}{2}\right) \quad\left(\because 1+\cos \mathrm{A}=2 \cos ^{2}\left(\frac{\mathrm{A}}{2}\right)\right)$</div>


title: what-is-2-2-sin-x-d-x-equal-to
slug: what-is-2-2-sin-x-d-x-equal-to-64882
question: <div class="question">What is $\int_{-\pi / 2}^{\pi / 2}|\sin x| \mathrm{d} x$ equal to?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pi$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">0</span> </li> </ul>

answer:
<div class="solution">Let $\mathrm{I}=\int_{-\pi / 2}^{\pi / 2}|\sin \mathrm{x}| \mathrm{dx}$<br/>Consider, $|\sin x|=\left\{\begin{array}{l}\sin x, 0   &lt; x &lt; \frac{\pi}{2} \\ -\sin x,-\frac{\pi}{2} &lt; x &lt; 0\end{array}\right.$<br/>$I=\int_{-\pi / 2}^{0}-\sin x d x+\int_{0}^{\pi / 2} \sin x d x$<br/>$=-[-\cos x]_{-\pi / 2}^{0}+\left.(-\cos x)\right|_{0} ^{\pi / 2}$<br/>$=(\cos 0-\cos (-\pi / 2))+(-\cos \pi / 2-(-\cos 0))$<br/>$=(1-0)+(-0+1)=2 .$</div>


title: what-is-2-2-x-d-x-2-2-x-d-x-equal-to-where-is-the-greatest-integer-function
slug: what-is-2-2-x-d-x-2-2-x-d-x-equal-to-where-is-the-greatest-integer-function-27465
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2016 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is <br/>$\int_{-2}^{2} x d x-\int_{-2}^{2}[x] d x$<br/>equal to, where $[\cdot]$ is the greatest integer function?</div>

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

answer:
<div class="solution">$\int_{-2}^{2} x d x-\int_{-2}^{2}[x] d x$<br/>$=\left[\frac{x^{2}}{2}\right]_{-2}^{2}-\int_{-2}^{-1}[x] d x-\int_{-1}^{0}[x] d x-\int_{0}^{1}[x] d x-\int_{1}^{2}[x] d x$<br/>$=\frac{1}{2}[4-4]-(-2)-(-1)-0-(1)$<br/>$=2+1-1=2$</div>


title: what-is-2-2-x-sin-x-d-x-equal-to
slug: what-is-2-2-x-sin-x-d-x-equal-to-34044
question: <div class="question">What is $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x \sin x d x$ equal to?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">$-2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pi$</span> </li> </ul>

answer:
<div class="solution">$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} x \sin x d x=2 \int_{0}^{\frac{\pi}{2}} x \sin x d x$<br/>$\{\mathrm{x} \sin \mathrm{x}$ is an even function\}<br/>$=2[-x \cos x+\sin x]_{0}^{\frac{\pi}{2}}=2$</div>


title: what-is-2-8-x-5-dx-equal-to
slug: what-is-2-8-x-5-dx-equal-to-40385
question: <div class="question">What is $\int_{2}^{8}|x-5| \mathrm{dx}$ equal to?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2018 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">4</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">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> </ul>

answer:
<div class="solution">$\int_{2}^{8}|x-5| d x=2 \times \frac{1}{2} \times 3 \times 3=9$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/nda/6L7IyBT4t3O9x9mOI47Oeq3SJ612vLp5TRyznky5Wy0.original.fullsize.png"/></div>


title: what-is-2-x-1-dx-equal-to
slug: what-is-2-x-1-dx-equal-to-50334
question: <div class="question">What is $\int \frac{\mathrm{dx}}{2^{\mathrm{x}}-1}$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\ln \left(2^{x}-1\right)+c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\ln \left(1-2^{-x}\right)}{\ln 2}+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\ln \left(2^{-\mathrm{x}}-1\right)}{2 \ln 2}+\mathrm{c}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\ln \left(1+2^{-x}\right)}{\ln 2}+c$</span> </li> </ul>

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

answer:
<div class="solution">$\int \frac{\mathrm{dx}}{2^{\mathrm{x}}-1}=\int \frac{\mathrm{dx}}{\frac{1}{2^{-\mathrm{x}}}}$<br/>$=\int \frac{2^{-\mathrm{x}}}{1-2^{-\mathrm{x}}} \cdot \mathrm{dx}$<br/>Let $1-2^{-\mathrm{x}}=\mathrm{t}$<br/>$\Rightarrow 2^{-\mathrm{x}} \cdot \log 2=\frac{\mathrm{dt}}{\mathrm{dx}} \Rightarrow 2^{-\mathrm{x}}=\frac{1}{\log 2} \cdot \frac{\mathrm{dt}}{\mathrm{d} \mathrm{x}} \Rightarrow 2^{-\mathrm{x}} \cdot \mathrm{d} \mathrm{x}=\frac{\mathrm{dt}}{\log 2}$<br/>$\quad \therefore \int \frac{2^{-\mathrm{x}}}{1-2^{-\mathrm{x}}} \mathrm{dx}=\frac{1}{\log 2} \int \frac{\mathrm{dt}}{\mathrm{t}}=\frac{1}{\log 2}(\log \mathrm{t})+\mathrm{c}$<br/>$=\frac{1}{\log 2}\left(\log \left(1-2^{-\mathrm{x}}\right)\right)+\mathrm{c}$</div>


title: what-is-3-i-1-3-i-equal-to
slug: what-is-3-i-1-3-i-equal-to-90466
question: <div class="question">What is $(\sqrt{3}+i) /(1+\sqrt{3} i)$ equal to ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2009 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1+i$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$1-i$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sqrt{3}(1-i) / 2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(\sqrt{3}-i) / 2$</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">$(\sqrt{3}-i) / 2$</span> </div>

answer:
<div class="solution">$\frac{\sqrt{3}+i}{1+\sqrt{3} i}=\frac{(\sqrt{3}+i)(1-\sqrt{3} i)}{(1+\sqrt{3} i)(1-\sqrt{3} i)}$<br/>$=\frac{\sqrt{3}-3 i+i+\sqrt{3}}{1+3}=\frac{2 \sqrt{3}-2 i}{4}=\frac{(\sqrt{3}-i)}{2}$</div>


title: what-is-3-i-3-i-6-equal-to
slug: what-is-3-i-3-i-6-equal-to-40918
question: <div class="question">What is $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^{6}$ equal to ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2010 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">-1</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Consider $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)=\frac{(\sqrt{3}+i)(\sqrt{3}+i)}{(\sqrt{3}-i)(\sqrt{3}+i)}$<br/>$=\frac{(\sqrt{3}+i)^{2}}{(\sqrt{3})^{2}-(i)^{2}}=\frac{3+i^{2}+2 \sqrt{3} i}{3-i^{2}}$<br/>$=\frac{2+2 \sqrt{3} i}{4}=\frac{1+\sqrt{3} i}{2}=-\omega^{2} \quad\left(\because i^{2}=-1\right.$<br/>Now, $\left(\frac{\sqrt{3}+i}{\sqrt{3}-i}\right)^{6}=\left(-\omega^{2}\right)^{6}=\omega^{12}=1\left(\because \omega^{3}=1\right)$</div>


title: what-is-4-4-tan-3-x-d-x-equal-to
slug: what-is-4-4-tan-3-x-d-x-equal-to-60359
question: <div class="question">What is $\int_{-\pi / 4}^{\pi / 4} \tan ^{3} x d x$ equal to ?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2010 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{2}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">We know<br/>$\int_{-a}^{a} f(x) d x=\left\{\begin{array}{l}0 \text { if } f(x) \text { is odd. } \\ 2 \int_{0}^{a} f(x) d x \text { if } f(x) \text { is even }\end{array}\right.$<br/>Since, $\tan ^{3} x$ is an odd function<br/>$\therefore \quad \int_{-\pi / 4}^{\pi / 4} \tan ^{3} x d x=0$</div>


title: what-is-4-x-2-d-x-equal-to
slug: what-is-4-x-2-d-x-equal-to-97888
question: <div class="question">What is $\int \frac{d x}{\sqrt{4+x^{2}}}$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\ln \left|\sqrt{4+x^{2}}+x\right|+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\ln \left|\sqrt{4+x^{2}}-x\right|+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\sin ^{-1}\left(\frac{x}{2}\right)+c$</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">$\ln \left|\sqrt{4+x^{2}}+x\right|+c$</span> </div>

answer:
<div class="solution">$\int \frac{\mathrm{dx}}{\sqrt{4+\mathrm{x}^{2}}}=\int \frac{\mathrm{dx}}{\sqrt{2^{2}+\mathrm{x}^{2}}}=\ln \left|\sqrt{4+\mathrm{x}^{2}}+\mathrm{x}\right|+\mathrm{C}$</div>


title: what-is-6-6-x-4-s-i-n-5-x-c-o-s-3-x-d-x-is-equal-to
slug: what-is-6-6-x-4-s-i-n-5-x-c-o-s-3-x-d-x-is-equal-to-53195
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\sin ^{5} x \cos ^{3} x}{x^{4}} d x$ is equal to?</div>

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

answer:
<div class="solution">Let $f(x)=\frac{\sin ^{5} x \cos ^{3} x}{x^{4}}$<br/>$f(-x)=\frac{\sin ^{5}(-x) \cos ^{3}(-x)}{(-x)^{4}}$<br/>$=\frac{-\sin ^{5} x \cos ^{3} x}{x^{4}}=-f(x)$<br/>$\Rightarrow f(x)$ is an odd function.<br/>Hence, $\int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \frac{\sin ^{5} x \cos ^{3} x}{x^{4}} d x=0$</div>


title: what-is-09-009-0009-equal-to
slug: what-is-09-009-0009-equal-to-42863
question: <div class="question">What is $0.9+0.09+0.009+\ldots$ equal to?</div>
tags: ['Mathematics', 'Sequences and Series', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
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">$1.01$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.001$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$1.1$</span> </li> </ul>

answer:
<div class="solution">$\begin{aligned} &amp; \mathrm{S}=0.9+0.09+0.009+\ldots \\ &amp;=9(0.1+0.01+0.001+\ldots) \\ &amp;=9\left[\frac{0.1}{1-0.1}\right]=9 \times \frac{0.1}{0.9}=1 \end{aligned}$</div>


title: what-is-47-c-4-51-c-3-j-2-5-52-j-c-3-equal-to
slug: what-is-47-c-4-51-c-3-j-2-5-52-j-c-3-equal-to-42423
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is ${ }^{47} \mathrm{C}_{4}+{ }^{51} \mathrm{C}_{3}+\sum_{\mathrm{j}=2}^{5}{ }^{52-\mathrm{j}} \mathrm{C}_{3}$ equal to?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">${ }^{52} \mathrm{C}_{4}$</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">${ }^{51} \mathrm{C}_{5}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">${ }^{53} \mathrm{C}_{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">${ }^{52} \mathrm{C}_{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">${ }^{52} \mathrm{C}_{4}$</span> </div>

answer:
<div class="solution">$\begin{array}{l}{ }^{47} C_{4}+{ }^{51} C_{3}+{ }^{50} C_{3}+{ }^{49} C_{3}+{ }^{48} C_{3}+{ }^{47} C_{3} \\ <br/>= { }^{47} C_{3}+{ }^{47} C_{4}+{ }^{48} C_{3}+{ }^{49} C_{3}+{ }^{50} C_{3}+{ }^{51} C_{3}\end{array}$<br/>${ }^{48} C_{4}+{ }^{48} C_{3}+{ }^{49} C_{3}+{ }^{50} C_{3}+{ }^{51} C_{3}$<br/>$\left({ }^{n} C_{r}+{ }^{n} C_{r+1}={ }^{n+1} C_{r+1}\right)$<br/>$={ }^{49} C_{4}+{ }^{49} C_{3}+{ }^{50} C_{3}+{ }^{51} C_{3}$<br/>$={ }^{50} C_{4}+{ }^{50} C_{3}+{ }^{51} C_{3}$<br/>$={ }^{51} C_{4}+{ }^{51} C_{3}$<br/>$={ }^{52} C_{4}$</div>


title: what-is-100-200-300-equal-to-where-is-the-cube-root-of-unity
slug: what-is-100-200-300-equal-to-where-is-the-cube-root-of-unity-76860
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2016 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\omega^{100}+\omega^{200}+\omega^{300}$ equal to, where $\omega$ is the cube root of unity?</div>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>\omega^{100}+\omega^{200}+\omega^{300} &amp;=\left(\omega^{99} \cdot \omega\right)+\left(\omega^{100}\right)^{2}+\left(\omega^{3}\right)^{100} \\<br/>&amp;=\left(\omega^{99} \cdot \omega\right)+\left(\omega^{99} \cdot \omega\right)^{2}+\left(\omega^{3}\right)^{100} \\<br/>\because \omega^{3}=1 \quad \omega^{99}=\left(\omega^{3}\right)^{33}=1^{33}=1 \\<br/>\Rightarrow \omega^{100}+\omega^{200}+\omega^{300}=(1 \cdot \omega)+(1 \cdot \omega)^{2}+1^{100} &amp; \\<br/>=\omega+\omega^{2}+1=1+\omega+\omega^{2}=0<br/>\end{aligned}<br/>$$</div>


title: what-is-1001-2-equal-to
slug: what-is-1001-2-equal-to-50286
question: <div class="question">What is $(1001)_{2}$ equal to ?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(5)_{10}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(4)_{10}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(17)_{10}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(11)_{10}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(4)_{10}$</span> </div>

answer:
<div class="solution">$(1001)_{2}=\left(2^{3} \times 1+2^{2} \times 0+2^{1} \times 0+2^{0} \times 1\right)_{10}$<br/>$(8+1)_{10}=(9)_{10}$</div>


title: what-is-1111-2-1001-2-1010-2-equal-to
slug: what-is-1111-2-1001-2-1010-2-equal-to-35521
question: <div class="question">What is $(1111)_{2}+(1001)_{2}-(1010)_{2}$ equal to?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(111)_{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$(1100)_{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$(1110)_{2}$</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">$(1010)_{2}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(1110)_{2}$</span> </div>

answer:
<div class="solution">Since, $(1111)_{2}=1 \times 2^{3}+1 \times 2^{2}+1 \times 2^{1}+1 \times 2^{0}$<br/>$=8+4+2+1=15$<br/>$(1001)_{2}=1 \times 2^{3}+0 \times 2^{2}+0 \times 2^{1}+1 \times 2^{0}=8+1=9$<br/>and $(1010)_{3}=1 \times 2^{3}+0 \times 2^{2}+1 \times 2^{1}+0 \times 2^{0}=8+2=10$<br/>$(1111)_{2}+(1001)_{2}-(1010)_{2}=15+9-10=14$<br/>$(14)_{10}=(1110)_{2}$</div>


title: what-is-1000000001-2-00101-equal-to
slug: what-is-1000000001-2-00101-equal-to-15329
question: <div class="question">What is (1000000001) $_{2}-(0.0101)$, equal to?</div>
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2015 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(512.6775)_{10}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(512.6875)_{10}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(512.0909)_{10}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(512.6975)_{10}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(512.6875)_{10}$</span> </div>

answer:
<div class="solution">$\quad(1000000001)_{2}$<br/>$=1 \times 2^{0}+0 \times 2^{1}+0 \times 2^{2}+\ldots \ldots+1 \times 2^{9}$<br/>$=1+0+0+\ldots \ldots .+512$<br/>$=(513)_{00}$<br/>$(0.0101)_{2}=0 \times 2^{-1}+1 \times 2^{-2}+0 \times 2^{-3}+1 \times 2^{-4}$<br/>$=\frac{1}{4}+\frac{1}{16}=\frac{5}{16}=(0.3125)_{10}$<br/>$(1000000001)_{2}-(0.0101)_{2}=513-0.312$<br/>$=(512.6875)_{\mathrm{s}}$</div>


title: what-is-a-2-ab-ac-ab-b-2-bc-ac-bc-c-2-equal-to
slug: what-is-a-2-ab-ac-ab-b-2-bc-ac-bc-c-2-equal-to-21326
tags: ['Mathematics', 'Determinants', 'NDA', 'NDA 2012 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\left|\begin{array}{ccc}-\mathrm{a}^{2} &amp; \mathrm{ab} &amp; \mathrm{ac} \\ \mathrm{ab} &amp; -\mathrm{b}^{2} &amp; \mathrm{bc} \\ \mathrm{ac} &amp; \mathrm{bc} &amp; -\mathrm{c}^{2}\end{array}\right|$ equal to?</div>

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

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

answer:
<div class="solution">Consider $\left|\begin{array}{ccc}-\mathrm{a}^{2} &amp; \mathrm{ab} &amp; \mathrm{ac} \\ \mathrm{ab} &amp; -\mathrm{b}^{2} &amp; \mathrm{bc} \\ \mathrm{ac} &amp; \mathrm{bc} &amp; -\mathrm{c}^{2}\end{array}\right|$<br/>Take out $^{\circ} \mathrm{a}^{\prime},{ }^{\prime} \mathrm{b}^{\prime}$ and ' $\mathrm{c}$ ' common from $\mathrm{R}_{1}, \mathrm{R}_{2}$ and $\mathrm{R}_{3}$<br/>respectively.<br/>$=a b c\left|\begin{array}{ccc}-a &amp; b &amp; c \\ a &amp; -b &amp; c \\ a &amp; b &amp; -c\end{array}\right|$<br/>Now, take out 'a', 'b' and 'c' common from $\mathrm{C}_{1}, \mathrm{C}_{2}$ and $\mathrm{C}_{3}$ respectively.<br/>$=\mathrm{a}^{2} \mathrm{~b}^{2} \mathrm{c}^{2}\left|\begin{array}{ccc}-1 &amp; 1 &amp; 1 \\ 1 &amp; -1 &amp; 1 \\ 1 &amp; 1 &amp; -1\end{array}\right|$<br/>Applying $\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}+\mathrm{R}_{1}, \mathrm{R}_{3} \rightarrow \mathrm{R}_{3}+\mathrm{R}_{1}$<br/>$=a^{2} b^{2} c^{2}\left|\begin{array}{ccc}-1 &amp; 1 &amp; 1 \\ 0 &amp; 0 &amp; 2 \\ 0 &amp; 2 &amp; 0\end{array}\right|=a^{2} b^{2} c^{2}[(-1)(-4)]$<br/>$=4 a^{2} b^{2} c^{2}$</div>


title: what-is-a-2-s-i-n-2-x-b-2-c-o-s-2-x-dx-equal-to
slug: what-is-a-2-s-i-n-2-x-b-2-c-o-s-2-x-dx-equal-to-65184
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int \frac{\mathrm{dx}}{\mathrm{a}^{2} \sin ^{2} \mathrm{x}+\mathrm{b}^{2} \cos ^{2} \mathrm{x}}$ equal to?</div>

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

answer:
<div class="solution">$\mathrm{I}=\frac{1}{a^{2}} \int \frac{\sec ^{2} x}{\tan ^{2} x+\left(\frac{b}{a}\right)^{2}}$<br/>$=\frac{1}{a^{2}} \times \frac{a}{b} \tan ^{-1}\left(\frac{a \tan x}{b}\right)+c$<br/>$=c+\frac{1}{a b} \tan ^{-1}\left(\frac{a \tan x}{b}\right)$</div>


title: what-is-a-b-a-b-equal-to
slug: what-is-a-b-a-b-equal-to-36836
question: <div class="question">What is $(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b})$ equal to?</div>
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\overrightarrow{0}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$2(\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}})$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$|\vec{a}|^{2}-|\vec{b}|^{2}$</span> </li> </ul>

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

answer:
<div class="solution">$\begin{aligned} &amp;(\vec{a}-\vec{b}) \times(\vec{a}+\vec{b}) \\ &amp;=\vec{a} \times \vec{a}+\vec{a} \times \vec{b}-\vec{b} \times \vec{a}-\vec{b} \times \vec{b} \\ &amp;=0+\vec{a} \times \vec{b}+\vec{a} \times \vec{b}-0 \\ &amp;=2(\vec{a} \times \vec{b}) \end{aligned}$</div>


title: what-is-a-b-x-l-o-g-x-d-x-equal-to
slug: what-is-a-b-x-l-o-g-x-d-x-equal-to-37543
question: <div class="question">What is $\int_{a}^{b} \frac{\log x}{x} d x$ equal to?</div>
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2017 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$(1 / 2) \log (a b) \cdot \log (b / a)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\log \mathrm{b} / \log \mathrm{a}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\log (\mathrm{b} / \mathrm{a})$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(1 / 2) \log [(a+b) / a b]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(1 / 2) \log (a b) \cdot \log (b / a)$</span> </div>

answer:
<div class="solution">Let the given integer be<br/>$$<br/>\mathrm{I}=\int_{\mathrm{a}}^{b} \frac{\log \mathrm{x}}{\mathrm{x}} \mathrm{dx}<br/>$$<br/>Put $\log x=\operatorname{tand} \frac{\mathrm{dx}}{\mathrm{x}}=\mathrm{dt}$ when $\mathrm{x}=\mathrm{a}, \mathrm{t}=\log \mathrm{a}$ and if<br/>$x=b, t=\log b$<br/>$\therefore \quad I=\int_{\log a}^{\log b} t d t=\left[\frac{t^{2}}{2}\right]_{a}^{b}$<br/>$\quad=\frac{1}{2}\left[(\log b)^{2}-(\log a)^{2}\right]$<br/>$\quad=\frac{1}{2}[(\log b+\log a)(\log b-\log a)]$<br/>$\quad=\frac{1}{2} \log (a b) \log \left(\frac{b}{a}\right)$</div>


title: what-is-a-keystone-species-1
slug: what-is-a-keystone-species-1-44716
question: <div class="question">What is a keystone species?</div>
tags: ['Biology', 'Organisms and Populations', 'AIIMS', 'AIIMS 2008']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">A species which makes up only a small proportion of the total biomass of a community, yet has a huge impact on the community's organization and survival.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">A common species that has plenty of biomass, yet has a fairly low impact on the community's organization.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">A rare species that has minimal impact on the biomass and on other species in the community.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">A dominant a large proportion of the biomass and which affects many other species.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A species which makes up only a small proportion of the total biomass of a community, yet has a huge impact on the community's organization and survival.</span> </div>

answer:
<div class="solution">Keystone species are those species which has significant and disproportionately large influence on the community structure and characteristics. It has often considerably low abundance and biomass as compared to dominant species. Removal of such species causes serious disruption in structure and function of community.</div>


title: what-is-a-keystone-species
slug: what-is-a-keystone-species-55879
question: <div class="question">What is a keystone species?</div>
tags: ['Biology', 'Biodiversity and its Conservation', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">A species which adds upto only a small proportion of the total biomass of a community, yet has a huge impact on the communityâs organization and survival.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">A common species that has plenty of biomass, yet has a fairly low impact on the community’s organization</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">A rare species that has minimal impact on the biomass and on other species in the community</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">A dominant species that constitutes a large proportion of the biomass and which affects many other species.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A species which adds upto only a small proportion of the total biomass of a community, yet has a huge impact on the communityâs organization and survival.</span> </div>

answer:
<div class="solution">Species having a much greater influence on community characteristics, relative to their low abundance or biomass are called keystone species, e. g., in tropical forests, figs are keystone species. Removal of these species causes serious disruption in the functioning of the community.</div>


title: what-is-a-thermochemical-equation
slug: what-is-a-thermochemical-equation-37409
question: <div class="question">What is a thermochemical equation?</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">An equation that describes a chemical reaction.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">An equation that specifies the physical state of a substance.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">A balanced chemical equation  together with the value of its â<sub>r</sub>H.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">An equation that represents the temperature dependence of a reaction.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">A balanced chemical equation  together with the value of its â<sub>r</sub>H.</span> </div>

answer:
<div class="solution">A thermochemical  equation is a balanced chemical equation that includes the value of the  standard enthalpy change (â<sub>r</sub>H) for the reaction.</div>


title: what-is-a-tonoplast-1
slug: what-is-a-tonoplast-1-27159
question: <div class="question">What is a tonoplast?</div>
tags: ['Biology', 'Cell - The Unit of Life', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Outer membrane of mitochondria</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Inner membrane of chloroplast</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Membrane boundary of the vacuole of plant cells</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Cell membrane of a plant cell.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Membrane boundary of the vacuole of plant cells</span> </div>

answer:
<div class="solution">The vacuole is the membrane-bound space found in the cytoplasm. The vacuole is bound by a single membrane called tonoplast.</div>


title: what-is-a-vector-of-unit-length-orthogonal-to-both-the-vectors-i-j-k-and-2-i-3-j-k
slug: what-is-a-vector-of-unit-length-orthogonal-to-both-the-vectors-i-j-k-and-2-i-3-j-k-60988
tags: ['Mathematics', 'Vector Algebra', 'NDA', 'NDA 2016 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is a vector of unit length orthogonal to both the vectors $\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}} ?$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{-4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}}}{\sqrt{26}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{-4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{26}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{-3 \hat{i}+2 \hat{j}-\hat{k}}{\sqrt{14}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{-3 \hat{i}+2 \hat{j}+\hat{k}}{\sqrt{14}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{-4 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+\hat{\mathrm{k}}}{\sqrt{26}}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \vec{A}=\hat{i}+\hat{j}+\hat{k} \\ \vec{B}=2 \hat{i}+3 \hat{j}-\hat{k} \\ \vec{A} \times \vec{B}=\left|\begin{array}{lll}\hat{i} &amp; \hat{j} &amp; \hat{k} \\ 1 &amp; 1 &amp; 1 \\ 2 &amp; 3 &amp; -1\end{array}\right| \\=&amp; \hat{i}(-1-3)-\hat{j}(-1-2)+\hat{k}(3-2) \\ &amp;=-4 \hat{i}+3 \hat{j}+\hat{k} \end{aligned}$<br/>Vector of unit length orthogonal to both the vectors $\vec{A}$ and $\vec{B}$<br/>$=\frac{\vec{A} \times \vec{B}}{|\vec{A} \times \vec{B}|}$<br/>$=\frac{-4 i+3 j+k}{\sqrt{16+9+1}}=\frac{-4 \hat{i}+3 \hat{j}+\hat{k}}{\sqrt{26}}$</div>


title: what-is-a-x-e-x-dx-equal-to
slug: what-is-a-x-e-x-dx-equal-to-15711
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int \mathrm{a}^{\mathrm{x}} \mathrm{e}^{\mathrm{x}} \mathrm{dx}$ equal to ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{\mathrm{a}^{\mathrm{x}} \mathrm{e}^{\mathrm{x}}}{\ell \mathrm{n}(\mathrm{ae})}+\mathrm{c}$</span> </div>

answer:
<div class="solution">Let $I=\int \underset{I}{a^{x}} e^{x} d x$<br/>$I=a^{x} \int e^{x} d x-\int a^{x} \ell n a e^{x} d x$<br/>$\mathrm{I}=\mathrm{a}^{\mathrm{x}} \mathrm{e}^{\mathrm{x}}-\ell \mathrm{n} \mathrm{a} \int \mathrm{a}^{\mathrm{x}} \mathrm{e}^{\mathrm{x}} \mathrm{dx}$<br/>$\mathrm{I}=\mathrm{a}^{\mathrm{x}} \mathrm{e}^{\mathrm{x}}-\ln \mathrm{a} .(\mathrm{I})$<br/>$\Rightarrow(1+\ln \mathrm{a}) \mathrm{I}=\mathrm{a}^{\mathrm{x}} \mathrm{e}^{\mathrm{x}} .$<br/>$\Rightarrow \mathrm{I}=\frac{\mathrm{a}^{\mathrm{x}} \mathrm{e}^{\mathrm{x}}}{\ell \mathrm{n}(\mathrm{ae})}+\mathrm{c}(\because \ell \mathrm{n} \mathrm{e}=1)$</div>


title: what-is-an-active-site-in-an-enzyme
slug: what-is-an-active-site-in-an-enzyme-72873
question: <div class="question">What is an active site in an enzyme?</div>
tags: ['Biology', 'Biomolecules (B)', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The substrate</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The pocket or crevice into which the substrate fits</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The amino acid sequence</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The secondary structure</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The pocket or crevice into which the substrate fits</span> </div>

answer:
<div class="solution">An active site of an enzyme is a crevice or pocket into which the substrate fits.</div>


title: what-is-an-endangered-species-give-an-example-of-an-endangered-plant-and-animal-species-each
slug: what-is-an-endangered-species-give-an-example-of-an-endangered-plant-and-animal-species-each-17350
tags: ['Biology', 'Biodiversity and its Conservation']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
question:
<div class="question">What is an endangered species? Give an example of an endangered plant and animal species each.</div>

answer:
<div class="solution">An endangered species is a population of organisms, which is facing a high risk of becoming extinct in near future due to<br/>(i) Its number is very low.<br/>(ii) It is threatened by the continuously changing changing environment.<br/>(iii) It is facing predator threat.<br/>An endangered plant species is Venus fly trap.<br/>An endangered animal species is Siberian tiger.</div>


title: what-is-an-example-of-competitive-release
slug: what-is-an-example-of-competitive-release-16264
question: <div class="question">What is an example of "competitive release"?</div>
tags: ['Biology', 'Organisms and Populations', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Species expand their range dramatically when a competing species is removed</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Competition occurs between unrelated species for the same resource</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Interference competition reduces the feeding efficiency of one species</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Competitive exclusion occurs only when resources are abundant</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Species expand their range dramatically when a competing species is removed</span> </div>

answer:
<div class="solution">Competitive release is when a species whose distribution is restricted to a small geographical area because of the presence of a competitively superior species, is found to expand its distributional range dramatically when the competing species is experimentally removed.</div>


title: what-is-an-incomplete-ecosystem-explain-with-the-help-of-suitable-example
slug: what-is-an-incomplete-ecosystem-explain-with-the-help-of-suitable-example-68319
tags: ['Biology', 'Ecosystem']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
question:
<div class="question">What is an incomplete ecosystem? Explain with the help of suitable example.</div>

answer:
<div class="solution">An ecosystem is comprised of biotic and a biotic components. Abiotic components include light, air, water, temperature, humidity etc., while biotic factor comprises all living organisms. An is limited absent ecosystem incomplete is any component (either abiotic or biotic). For example the profundal and benthic zone in an aquatic ecosystem.</div>


title: what-is-another-term-for-the-central-atomion-in-a-coordination-entity-1
slug: what-is-another-term-for-the-central-atomion-in-a-coordination-entity-1-51989
question: <div class="question">What is another term for the central atom/ion in a coordination entity?</div>
tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Primary valence</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Secondary valence</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Lewis acid</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Coordination entity</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Coordination entity</span> </div>

answer:
<div class="solution">The central atom/ion in a coordination entity is also referred to as a coordination entity.</div>


title: what-is-another-term-for-the-central-atomion-in-a-coordination-entity
slug: what-is-another-term-for-the-central-atomion-in-a-coordination-entity-93163
question: <div class="question">What is another term for the central atom/ion in a coordination entity?</div>
tags: ['Chemistry', 'Coordination Compounds', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">Primary valence</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Secondary valence</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Lewis acid</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Coordination entity</span> </li> </ul>

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

answer:
<div class="solution">The central atom/ion in a coordination entity is also referred to as a Lewis acid.</div>


title: what-is-are-the-critical-points-s-of-the-function-f-x-x-23-5-2-x-on-the-interval-1-2
slug: what-is-are-the-critical-points-s-of-the-function-f-x-x-23-5-2-x-on-the-interval-1-2-56094
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2007 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1 only</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is /are the critical points $(\mathrm{s})$ of the function $\mathrm{f}(\mathrm{x})=$ $\mathrm{x}^{2 / 3}(5-2 \mathrm{x})$ on the interval $[-1,2] ?$</div>

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

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\mathrm{x}^{2 / 3}(5-2 \mathrm{x})$<br/>or, $f(x)=5 x^{2 / 3}-2 x^{5 / 3}$<br/>differentiating both the sides.<br/>$\mathrm{f}^{\prime}(\mathrm{x})=5 \times \frac{2}{3} \mathrm{x}^{-1 / 3}-2 \times \frac{5}{3} \mathrm{x}^{2 / 3}$<br/>or, $\mathrm{f}^{\prime}(\mathrm{x})=\frac{10}{3}\left(\mathrm{x}^{-1 / 3}-\mathrm{x}^{2 / 3}\right)$<br/>To get the critical value. $\mathrm{f}^{\prime}(\mathrm{x})=0$<br/>so, $\mathrm{x}^{-1 / 3}-\mathrm{x}^{2 / 3}=0 \Rightarrow \mathrm{x}^{-1 / 3}(1-\mathrm{x})=0$<br/>$\Rightarrow 1-x=0$ as $x^{-1 / 3} \neq 0$<br/>or, $\mathrm{x}=1$ is the only value in the interval $[-1,2]$</div>


title: what-is-arg-bi-where-b-0
slug: what-is-arg-bi-where-b-0-76831
question: <div class="question">What is arg (bi) where $\mathrm{b}&gt;0$ ?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2008 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{\pi}{2}$</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">$\pi$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3 \pi}{2}$</span> </li> </ul>

answer:
<div class="solution">$\arg ($ bi $) \tan ^{-1}\left(\frac{\mathrm{b}}{0}\right)=\tan ^{-1}(\infty)=\frac{\pi}{2}(\because \mathrm{b}&gt;0)$</div>


title: what-is-bond-dissociation-enthalpy
slug: what-is-bond-dissociation-enthalpy-57490
question: <div class="question">What is bond dissociation enthalpy?</div>
tags: ['Chemistry', 'Thermodynamics (C)', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The enthalpy changes when two atoms combine to form a molecule.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">The enthalpy changes when one mole of covalent bonds in a gaseous covalent compound is broken to form products in the gas phase.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">The enthalpy change when a solid substance sublimes at a constant temperature and under standard pressure.</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The enthalpy change when a substance is heated at constant volume.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">The enthalpy changes when one mole of covalent bonds in a gaseous covalent compound is broken to form products in the gas phase.</span> </div>

answer:
<div class="solution">Bond dissociation enthalpy specifically refers to the energy required to break one mole of covalent bonds in a gaseous covalent compound to form products in the gas phase.</div>


title: what-is-c-n-r-2-c-n-r-1-c-n-r-2-equal-to
slug: what-is-c-n-r-2-c-n-r-1-c-n-r-2-equal-to-96998
tags: ['Mathematics', 'Binomial Theorem', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\mathrm{C}(\mathrm{n}, \mathrm{r})+2 \mathrm{C}(\mathrm{n}, \mathrm{r}-1)+\mathrm{C}(\mathrm{n}, \mathrm{r}-2)$ equal to?</div>

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

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

answer:
<div class="solution">$C(n, r)+2 C(n, r-1)+C(n, r-2)$<br/>$={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+2\left({ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}\right)+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-2}$<br/>$={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-2}$<br/>$\quad\left(\because{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}-1}={ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}}\right)$<br/>$={ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}}+{ }^{\mathrm{n}+1} \mathrm{C}_{\mathrm{r}-1}$<br/>$={ }^{\mathrm{n}+2} \mathrm{C}_{\mathrm{r}}$<br/>$=\mathrm{C}(\mathrm{n}+2, \mathrm{r})$</div>


title: what-is-c-o-s-2-x-1-s-i-n-2-x-1-dx-equal-to
slug: what-is-c-o-s-2-x-1-s-i-n-2-x-1-dx-equal-to-25738
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2012 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int\left(\frac{1}{\cos ^{2} x}-\frac{1}{\sin ^{2} x}\right) \mathrm{dx}$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2 \operatorname{cosec} 2 x+c$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int\left(\frac{1}{\cos ^{2} \mathrm{x}}-\frac{1}{\sin ^{2} \mathrm{x}}\right) \mathrm{d} \mathrm{x}$<br/>$=\int\left(\sec ^{2} \mathrm{x}-\operatorname{cosec}^{2} \mathrm{x}\right) \mathrm{dx}$<br/>$=\int \sec ^{2} \mathrm{x} \mathrm{d} \mathrm{x}-\int_{\mathrm{c}} \operatorname{cosec}^{2} \mathrm{x} \mathrm{d} \mathrm{x}$<br/>$=\tan \mathrm{x}+\cot \mathrm{x}+\mathrm{c}$<br/>$=\tan \mathrm{x}+\frac{1}{\tan \mathrm{x}}+\mathrm{c}=\frac{\tan ^{2} \mathrm{x}+1}{\tan \mathrm{x}}+\mathrm{c}=\frac{\sec ^{2} \mathrm{x}}{\tan \mathrm{x}}+\mathrm{c}$<br/>$=\frac{2}{2 \sin \mathrm{x} \cos \mathrm{x}}+\mathrm{c}=\frac{2}{\sin 2 \mathrm{x}}+\mathrm{c}=2 \operatorname{cosec} 2 \mathrm{x}+\mathrm{c}$</div>


title: what-is-c-o-s-2-x-a-b-s-i-n-x-d-x-equal-to
slug: what-is-c-o-s-2-x-a-b-s-i-n-x-d-x-equal-to-63411
question: <div class="question">What is $\int \frac{a+b \sin x}{\cos ^{2} x} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2009 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a \sec x+b \tan x+c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$a \tan x+b \sec x+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a \cot x+b \operatorname{cosec} x+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a \operatorname{cosec} x+b \cot x+c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a \tan x+b \sec x+c$</span> </div>

answer:
<div class="solution">Consider $\int \frac{a+b \sin x}{\cos ^{2} x} d x=\int\left(\frac{a}{\cos ^{2} x}+\frac{b \sin x}{\cos ^{2} x}\right) d x$<br/>$=\int\left(a \sec ^{2} x+b \tan x \sec x\right) d x=a \tan x+b \sec x+c$</div>


title: what-is-c-o-s-2-x-e-x-e-x-1-x-d-x-equal-to
slug: what-is-c-o-s-2-x-e-x-e-x-1-x-d-x-equal-to-12301
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int \frac{e^{x}(1+x)}{\cos ^{2}\left(x e^{x}\right)} d x$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\tan \left(x e^{x}\right)+c$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int \frac{\mathrm{e}^{\mathrm{x}}(1+\mathrm{x})}{\cos ^{2}\left(\mathrm{xe}^{\mathrm{x}}\right)} \mathrm{dx}$<br/>Put, $x e^{x}=t \Rightarrow e^{x}(1+x) d x=d t$<br/>$\therefore \quad \mathrm{I}=\int \frac{\mathrm{dt}}{\cos ^{2} \mathrm{t}}=\int \mathrm{sec}^{2} \mathrm{tdt}=\tan \mathrm{t}+\mathrm{c}$<br/>where ' $\mathrm{c}$ ' is a constant of integration. $=\tan \left(\mathrm{xe}^{\mathrm{x}}\right)+\mathrm{c}$</div>


title: what-is-c-o-s-5-x-c-o-s-3-x-s-i-n-5-x-s-i-n-3-x-equal-to
slug: what-is-c-o-s-5-x-c-o-s-3-x-s-i-n-5-x-s-i-n-3-x-equal-to-66643
question: <div class="question">What is $\frac{\sin 5 x-\sin 3 x}{\cos 5 x+\cos 3 x}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sin x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\cos x$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\tan x$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\cot \mathrm{x}$</span> </li> </ul>

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

answer:
<div class="solution">$\begin{aligned} &amp; \frac{\sin 5 x-\sin 3 x}{\cos 5 x+\cos 3 x} \\ &amp; \sin c-\sin d=2 \cos \left(\frac{\mathrm{c}+\mathrm{d}}{2}\right) \sin \left(\frac{\mathrm{c}-\mathrm{d}}{2}\right) \\ &amp; \cos c-\cos d=2 \cos \left(\frac{\mathrm{c}+\mathrm{d}}{2}\right) \cos \left(\frac{\mathrm{c}-\mathrm{d}}{2}\right) \end{aligned}$<br/>$$<br/>\begin{array}{l}<br/>\begin{array}{l}<br/>\therefore \frac{\sin 5 x-\sin 3 x}{\cos 5 x+\cos 3 x}=\frac{2 \cos \left(\frac{5 x+3 x}{2}\right) \sin \left(\frac{5 x-3 x}{2}\right)}{2 \cos \left(\frac{5 x+3 x}{2}\right) \cos \left(\frac{5 x-3 x}{2}\right)} \\</div>


title: what-is-c-o-s-s-i-n-1-equal-to
slug: what-is-c-o-s-s-i-n-1-equal-to-85594
question: <div class="question">What is $\frac{\sin \theta+1}{\cos \theta}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2010 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\sin \theta+\cos \theta-1}{\sin \theta+\cos \theta+1}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sin \theta+\cos \theta+1}{\sin \theta+\cos \theta-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\sin \theta-\cos \theta-1}{\sin \theta+\cos \theta+1}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}$</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">$\frac{\sin \theta-\cos \theta+1}{\sin \theta+\cos \theta-1}$</span> </div>

answer:
<div class="solution">Consider $\frac{1+\sin \theta}{\cos \theta}=\frac{1+\frac{2 \tan \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}}{\frac{1-\tan ^{2} \frac{\theta}{2}}{1+\tan ^{2} \frac{\theta}{2}}}$<br/>$\left(\because \sin 2 \theta=\frac{2 \tan \theta}{1+\tan ^{2} \theta}\right.$ and $\left.\cos 2 \theta=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}\right)$<br/>$=\frac{\left(1+\tan \frac{\theta}{2}\right)^{2}}{\left(1-\tan \frac{\theta}{2}\right)\left(1+\tan \frac{\theta}{2}\right)}$<br/>$=\frac{1+\tan \frac{\theta}{2}}{1-\tan \frac{\theta}{2}}=\frac{\cos \frac{\theta}{2}+\sin \frac{\theta}{2}}{\cos \frac{\theta}{2}-\sin \frac{\theta}{2}}$<br/>Multiplied and divide by $2 \sin \frac{\theta}{2}$ $=\frac{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}+2 \sin ^{2} \frac{\theta}{2}}{2 \sin \frac{\theta}{2} \cos \frac{\theta}{2}-2 \sin ^{2} \frac{\theta}{2}}$<br/>$=\frac{\sin \theta+1-\cos \theta}{\sin \theta-1+\cos \theta}$<br/>$(\because \sin 2 \theta=2 \sin \theta \cos \theta$ and<br/>$\left.\cos 2 \theta=1-2 \sin ^{2} \theta\right)$</div>


title: what-is-c-o-t-22-6-c-o-t-31-6-c-o-t-22-4-c-o-t-13-4-equal-to
slug: what-is-c-o-t-22-6-c-o-t-31-6-c-o-t-22-4-c-o-t-13-4-equal-to-80911
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{\cot 224^{\circ}-\cot 134^{\circ}}{\cot 226^{\circ}+\cot 316^{\circ}}$ equal to ?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\operatorname{cosec} 88^{\circ}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\operatorname{cosec} 2^{\circ}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\operatorname{cosec} 44^{\circ}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\operatorname{cosec} 46^{\circ}$</span> </li> </ul>

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

answer:
<div class="solution">$\frac{\cot 224^{\circ}-\cot 134^{\circ}}{\cot 226^{\circ}+\cot 316^{\circ}}$<br/>$=\frac{\cot \left(180^{\circ}+44^{\circ}\right)-\cot \left(180^{\circ}-46^{\circ}\right)}{\cot \left(180+46^{\circ}\right)+\cot \left(270^{\circ}+46\right)^{\circ}}$<br/>$=\frac{\cot 44^{\circ}+\cot 46^{\circ}}{\cot 46^{\circ}-\tan 46^{\circ}}=\frac{\tan 46^{\circ}+\tan 44^{\circ}}{\tan 44^{\circ}-\tan 46^{\circ}}$<br/>$=\frac{\sin \left(46^{\circ}+44^{\circ}\right)}{\sin \left(44^{\circ}-46^{\circ}\right)}=-\operatorname{cosec} 2^{\circ}$</div>


tags: ['Biology', 'Biodiversity and its Conservation', 'NEET']
source_db: MarksBatch2_P3.db
title:
what-is-causing-the-decline-of-commercially-important-species-of-marine-fish-populations-around-the-world

slug:
what-is-causing-the-decline-of-commercially-important-species-of-marine-fish-populations-around-the-world-95814

question:
<div class="question">What is causing the decline of commercially important species of marine fish populations around the world?</div>

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

answer:
<div class="solution">Many marine fish populations around the world are over harvested, endangering the continued existence of some commercially important species due to over:exploitation.</div>


title: what-is-common-to-lantana-eichhomia-and-african-catfish
slug: what-is-common-to-lantana-eichhomia-and-african-catfish-21481
question: <div class="question">What is common to Lantana, Eichhomia and African catfish?</div>
tags: ['Biology', 'Biodiversity and its Conservation', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">All are endangered species of India.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">All are key stone species.</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">All are mammals found in India.</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">All the species are neither threatened nor indigenous species of India.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">All the species are neither threatened nor indigenous species of India.</span> </div>

answer:
<div class="solution">Lantana, Eichhomia and African catfish, all the species are neither threatened nor indigenous species of India.</div>


title: what-is-common-to-lantana-eichhornia-and-african-catfish-1
slug: what-is-common-to-lantana-eichhornia-and-african-catfish-1-82733
question: <div class="question">What is common to Lantana, Eichhornia and African catfish?</div>
tags: ['Biology', 'Biodiversity and its Conservation']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><br/>All are endangered species of India<br/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><br/>All are key stone species<br/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><br/>All are mammals found in India<br/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><br/> All the species are neither threatened nor indigenous species of India.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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"><br/> All the species are neither threatened nor indigenous species of India.</span> </div>

answer:
<div class="solution">All the three above mentioned species are neither threatened nor indigenous species of India, Lantana Eichhornia (water hyacinth) and African catfish (Clarias gariepinus) all are alien (exotic species) that are invasive and also have an harmful impact resulting in extinction of the indigenous species.</div>


title: what-is-common-to-lantana-eichhornia-and-african-catfish
slug: what-is-common-to-lantana-eichhornia-and-african-catfish-13405
question: <div class="question">What is common to <em>Lantana, Eichhornia</em> and African catfish?</div>
tags: ['Biology', 'Biodiversity and its Conservation', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">All are endangered species of India</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">All are key stone species</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">All are mammals found in India</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">All the species are not indigenous species of India</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">All the species are not indigenous species of India</span> </div>

answer:
<div class="solution">Non-native or alien species are often introduced inadvertently for their economic and other uses. They often become invasive and drive away from the local species. These species are considered to be the second major cause of extinction of species (the first being habitat destruction). Water Hyacinth (<em>Eichhornia crassipes</em>) was introduced in Indian water to reduce pollution. It has clogged water bodies including wetlands at many places, resulting in the death of several aquatic plants and animals. <em>Lantana camara</em> has replaced many species in forest of U.P. and M.P. African catfish, <em>Clarias gariepinus</em>, has been illegally introduced for aquaculture in India. It is threatening native Catfishes (<em>e.g., Clarias bacterachus</em>) of Indian rivers.</div>


title: what-is-correct-regarding-ecological-succession
slug: what-is-correct-regarding-ecological-succession-87833
question: <div class="question">What is correct regarding ecological succession?</div>
tags: ['Biology', 'Ecosystem', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">The decrease in biodiversity at each transitional level.</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">Primary succession is faster than secondary succession.</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">Change to environmental condition extreme to mesic.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">The climax community is most stable with the environment as the niche and food web complexity is less.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Change to environmental condition extreme to mesic.</span> </div>

answer:
<div class="solution">Both hydrarch and xerarch successions lead to medium water conditions (mesic) â neither too dry (xeric) nor too wet (hydric).</div>


title: what-is-cos-2-0-cos-10-0-cos-14-0-equal-to
slug: what-is-cos-2-0-cos-10-0-cos-14-0-equal-to-99856
question: <div class="question">What is $\cos 20^{\circ}+\cos 100^{\circ}+\cos 140^{\circ}$ equal to?</div>
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2014 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1 / 2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$\cos 20^{\circ}+\cos 100^{\circ}+\cos 140^{\circ}$<br/>$=\left(\cos 140^{\circ}+\cos 20^{\circ}\right)+\cos 100^{\circ}$<br/>$=2 \cos \left(\frac{160^{\circ}}{2}\right) \cdot \cos \left(\frac{120^{\circ}}{2}\right)+\cos 100^{\circ}$<br/>$=2 \cdot \cos 80^{\circ} \cdot \frac{1}{2}+\cos 100^{\circ}$<br/>$=2 \cos \left(\frac{180^{\circ}}{2}\right) \cdot \cos \left(\frac{20^{\circ}}{2}\right)$<br/>$=2 \cos 90^{\circ} \cdot \cos 10^{\circ}$<br/>$=2 \times 0 \times \cos 10^{\circ}=0$</div>


title: what-is-cosec-s-i-n-s-e-c-c-o-s-equal-to
slug: what-is-cosec-s-i-n-s-e-c-c-o-s-equal-to-49678
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'NDA', 'NDA 2012 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{\sin \theta}{\operatorname{cosec} \theta}+\frac{\cos \theta}{\sec \theta}$ equal to?</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">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">$\frac{\sin \theta}{\operatorname{cosec} \theta}+\frac{\cos \theta}{\sec \theta}=\frac{\sin \theta}{\frac{1}{\sin \theta}}+\frac{\cos \theta}{\frac{1}{\cos \theta}}=\sin ^{2} \theta+\cos ^{2} \theta=1$</div>


title: what-is-dna
slug: what-is-dna-25989
question: <div class="question">What is DNA?</div>
tags: ['Biology', 'Molecular Basis of Inheritance', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Deoxyribonucleic acid</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Ribonucleic acid</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Polymer of ribonucleotides</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Polymer of deoxyribonucleotides</span> </li> </ul>

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

answer:
<div class="solution">DNA stands for Deoxyribonucleic Acid and it is the most abundant genetic material found in living systems.</div>


title: what-is-dx-d-1-s-i-n-2-x-equal-to-where-4-x-2
slug: what-is-dx-d-1-s-i-n-2-x-equal-to-where-4-x-2-54682
tags: ['Mathematics', 'Application of Derivatives', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{\mathrm{d} \sqrt{1-\sin 2 \mathrm{x}}}{\mathrm{dx}}$ equal to, where $\frac{\pi}{4}   &lt; \mathrm{x} &lt; \frac{\pi}{2}$ ?</div>

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

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

answer:
<div class="solution">For $\frac{\pi}{4}   &lt; x &lt; \frac{\pi}{2}$<br/>$\quad \sqrt{1-\sin 2 x}=\left(\sin ^{2} x+\cos ^{2} x+2 \sin x \cdot \cos x\right)^{1 / 2}$<br/>$=(\sin x-\cos x)$<br/>$\therefore \quad$ Differentiation $=\cos x+\sin x$</div>


title: what-is-dy-2-d-2-x-equal-to
slug: what-is-dy-2-d-2-x-equal-to-39124
question: <div class="question">What is $\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{dy}^{2}}$ equal to?</div>
tags: ['Mathematics', 'Differential Equations', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$-\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\left(\frac{d y}{d x}\right)^{-3}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}\left(\frac{d y}{d x}\right)^{-2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}$</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">$\left(\frac{d^{2} y}{d x^{2}}\right)^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-\left(\frac{d^{2} y}{d x^{2}}\right)\left(\frac{d y}{d x}\right)^{-3}$</span> </div>

answer:
<div class="solution">Let $\mathrm{y}=\mathrm{f}(\mathrm{x}) \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{f}^{1}(\mathrm{x}) \Rightarrow \frac{\mathrm{dx}}{\mathrm{dy}}=\frac{1}{\mathrm{f}^{\prime}(\mathrm{x})}$<br/>$\frac{d^{2} x}{d y^{2}}=\frac{\frac{-d}{d y} f^{\prime}(x)}{\left(f^{\prime}(x)\right)^{2}}$<br/>$\frac{-d}{d y} f^{\prime}(x)$<br/>$\frac{d y}{\left(f^{\prime}(x)\right)^{2}}=\frac{\frac{-d}{d y}\left(\frac{d y}{d x}\right) \cdot \frac{d x}{d x}}{\left(f^{\prime}(x)\right)^{2}}=\frac{\frac{-d}{d x}\left(f^{\prime}(x)\right) \cdot \frac{d x}{d y}}{\left(\frac{d y}{d x}\right)^{2}}$<br/>$\Rightarrow \frac{-f^{\prime}(x) \cdot \frac{d x}{d y}}{\left(\frac{d y}{d x}\right)^{2}}=\frac{-\left(\frac{d^{2} y}{d x^{2}}\right)}{\left(\frac{d y}{d x}\right)^{3}}=\frac{-d^{2} y}{d x^{2}}\left(\frac{d y}{d x}\right)^{-3}$</div>


title: what-is-e-1-e-2-x-l-n-x-dx-equal-to
slug: what-is-e-1-e-2-x-l-n-x-dx-equal-to-63258
tags: ['Mathematics', 'Definite Integration', 'NDA', 'NDA 2017 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int_{e^{-1}}^{\mathrm{e}^{2}}\left|\frac{\ln x}{\mathrm{x}}\right| \mathrm{d} \mathrm{x}$ equal to?</div>

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

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

answer:
<div class="solution">$\int_{x-1}^{e^{2}}\left|\frac{\log x}{x}\right| d x=\int_{s^{-1}}^{e^{0}} \frac{-\log x}{x} d x+\int_{\varepsilon^{0}}^{e^{2}} \frac{\log x}{x} d x$<br/>$=\frac{-1}{2}\left[(\log \mathrm{x})^{2}\right]_{\mathrm{e}}^{\mathrm{e}^{0}}-\frac{1}{2}\left[(\log \mathrm{x})^{2}\right]_{e^{0}}^{\mathrm{c}^{2}}$<br/>$=\frac{-1}{2}\left[0-\left(\log \mathrm{e}^{-1}\right)^{2}\right]+\frac{1}{2}\left[\left(\log \mathrm{e}^{2}\right)^{2}-0\right]$<br/>$=\frac{-1}{2}(-1)+\frac{1}{2}(2)^{2}$<br/>$=\frac{1}{2}+\frac{4}{2}=\frac{5}{2}$</div>


title: what-is-e-e-x-e-x-d-x-equal-to
slug: what-is-e-e-x-e-x-d-x-equal-to-76472
question: <div class="question">What is $\int e^{e^{x}} e^{x} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2013 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$e^{e^{x}}+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 e^{e^{x}}+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$e^{e^{x}} e^{x}+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 e^{e^{x}} e^{x}+c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$e^{e^{x}}+c$</span> </div>

answer:
<div class="solution">$\mathrm{I}=\int \mathrm{e}^{\mathrm{e}^{\mathrm{x}}} \mathrm{e}^{\mathrm{x}} \mathrm{dx}$<br/>Let $\mathrm{e}^{x}=\mathrm{y} \Rightarrow \mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dy}$<br/>$\mathrm{dx}=\frac{\mathrm{dy}}{\mathrm{e}^{\mathrm{x}}}$<br/>$\mathrm{I}=\int \mathrm{e}^{\mathrm{y}} \mathrm{e}^{\mathrm{x}} \frac{\mathrm{dy}}{\mathrm{e}^{\mathrm{x}}}=\int \mathrm{e}^{\mathrm{y}} \mathrm{dy}=\mathrm{e}^{y}+\mathrm{c}$<br/>$\mathrm{I}=\mathrm{e}^{\mathrm{e}^{\mathrm{x}}}+\mathrm{c}$</div>


title: what-is-e-in-t-a-n-x-dx-equal-to
slug: what-is-e-in-t-a-n-x-dx-equal-to-30198
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2018 (Phase 2)']
answer: <div class="solution">$\int e^{\ln (\tan x)} d x=\int \tan x d x$<br/>$=\ell n|\sec x|+c$</div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int \mathrm{e}^{\operatorname{In}(\tan x)} \mathrm{dx}$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\operatorname{In}|\sec x|+c$</span> </div>


title: what-is-e-l-n-x-dx-equal-to
slug: what-is-e-l-n-x-dx-equal-to-23787
question: <div class="question">What is $\int \mathrm{e}^{\ln \mathrm{x}} \mathrm{dx}$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{xe}^{\mathrm{fnx}}+\mathrm{c}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-x e^{-i n x}+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x+c$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{x^{2}}{2}+c$<br/>where $\mathrm{c}$ is constant of integration.</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\frac{x^{2}}{2}+c$<br/>where $\mathrm{c}$ is constant of integration.</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int \mathrm{e}^{\ln \mathrm{x}} \mathrm{dx}=\int \mathrm{x} \mathrm{dx}=\frac{\mathrm{x}^{2}}{2}+\mathrm{c}$</div>


title: what-is-e-l-n-x-sin-x-d-x-equal-to
slug: what-is-e-l-n-x-sin-x-d-x-equal-to-20682
question: <div class="question">What is $\int e^{\ln x} \sin x d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2010 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$e^{\ln x}(\sin x-\cos x)+c$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(\sin x-x \cos x)+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(x \sin x+\cos x)+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$(\sin x+x \cos x)-c$<br/>Where 'c' is a constant of integration.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(\sin x-x \cos x)+c$</span> </div>

answer:
<div class="solution">Let $\mathrm{I}=\int e^{\ln x} \sin x d x=\int x \sin x d x\left(\because e^{\log a}=a\right)$<br/>$=-x \cos x+\int 1 \cdot \cos x d x=(\sin x-x \cos x)+c$</div>


title: what-is-e-s-i-n-x-c-o-s-2-x-x-c-o-s-3-x-s-i-n-x-dx-equal-to
slug: what-is-e-s-i-n-x-c-o-s-2-x-x-c-o-s-3-x-s-i-n-x-dx-equal-to-75142
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2016 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int \mathrm{e}^{\sin \mathrm{x}} \frac{\mathrm{x} \cos ^{3} \mathrm{x}-\sin \mathrm{x}}{\cos ^{2} \mathrm{x}} \mathrm{dx}$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(x-\sec x) e^{\sin x}+c$</span> </div>

answer:
<div class="solution">Let us differentiate all the options one by one to get the expression in the question whose integral is to be found.<br/>Here $x e^{\sin x}$ is the common term in all the options. So, let us differentiate it first.<br/>Let $l=x e^{\sin x}$<br/>$\Rightarrow \frac{d l}{d x}=e^{\sin x}[x \cos x+1]$<br/>$\Rightarrow \frac{d l}{d x}=\frac{e^{\sin x}}{\cos ^{2} x}\left[x \cos ^{3} x+\cos ^{2} x\right]$<br/>Let $m=\sec x e^{\sin x}$<br/>$\Rightarrow \frac{d m}{d x}=\sec x e^{\sin x} \cdot \cos x+e^{\sin x} \sec x \tan x$<br/>$\Rightarrow \frac{d m}{d x}=e^{\sin x}\left[1+\frac{\sin x}{\cos ^{2} x}\right]$<br/>$\Rightarrow \frac{d m}{d x}=\frac{e^{\sin x}}{\cos ^{2} x}\left[\cos ^{2} x+\sin x\right]$<br/>Differentiation of option (a) is<br/>$=\frac{e^{\sin x}}{\cos ^{2} x}\left[x \cos ^{3} x+\cos ^{2} x+\cos ^{2} x+\sin x\right]$<br/>$=\frac{e^{\sin x}}{\cos ^{2} x}\left[x \cos ^{3} x+2 \cos ^{2} x+\sin x\right]$<br/>Differentiation of option (b) is<br/>$=\frac{e^{\sin x}}{\cos ^{2} x}\left[x \cos ^{3} x+\cos ^{2} x-\cos ^{2} x-\sin x\right]$<br/>$=\frac{e^{\sin x}}{\cos ^{2} x}\left[x \cos ^{3} x-\sin x\right]$<br/>$\therefore$ Option (b) is correct.</div>


title: what-is-e-x-1-1-d-x-equal-to
slug: what-is-e-x-1-1-d-x-equal-to-70798
question: <div class="question">What is $\int\left(e^{x}+1\right)^{-1} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2008 (Phase 2)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\ln \left(e^{x}+1\right)+c$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\ln \left(e^{-x}+1\right)+c$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-\ln \left(e^{-x}+1\right)+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\left(e^{x}+1\right)+c$</span> </li> </ul>

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

answer:
<div class="solution">Let $\mathrm{I}=\int\left(e^{x}+1\right)^{-1} d x=\int \frac{1}{e^{x}+1} d x=\int \frac{e^{-x}}{1+e^{-x}} d x$<br/>Let $1+e^{-x}=t \Rightarrow-e^{-x} d x=d t$<br/>$\therefore  I=-\int \frac{1}{t} d t=-\log t+c=-\log \left(1+e^{-x}\right)+c$</div>


title: what-is-e-x-l-n-a-d-x-equal-to
slug: what-is-e-x-l-n-a-d-x-equal-to-48827
question: <div class="question">What is $\int e^{x \ln (a)} d x$ equal to?</div>
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2019 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{a^{x}}{\ln (a)}+c$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{e^{x}}{\ln (a)}+c$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{e^{x}}{\ln (a e)}+c$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{a e^{x}}{\ln (a)}+c$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{a^{x}}{\ln (a)}+c$</span> </div>

answer:
<div class="solution">$\int e^{x \ln (a)} \cdot d x=\int e^{x} \cdot d x=\frac{a^{x}}{\ln (a)}+c$</div>


title: what-is-e-x-x-2-x-1-d-x-equal-to
slug: what-is-e-x-x-2-x-1-d-x-equal-to-42737
tags: ['Mathematics', 'Indefinite Integration', 'NDA', 'NDA 2011 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\int e^{x}\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x$ equal to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$e^{x}(\sqrt{x})+C$</span> </div>

answer:
<div class="solution">Let $I=\int e^{x}\left(\sqrt{x}+\frac{1}{2 \sqrt{x}}\right) d x$<br/>$=\int e^{x} \sqrt{x} d x+\int e^{x} \cdot \frac{1}{2 \sqrt{x}} d x$<br/>$=e^{x} \cdot \sqrt{x}-\int e^{x} \cdot \frac{1}{2 \sqrt{x}} d x+\int e^{x} \cdot \frac{1}{2 \sqrt{x}} d x$<br/>$=e^{x} \cdot \sqrt{x}+C$<br/>where $C$ is constant of integration.</div>


tags: ['Mathematics', 'Straight Lines', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
title:
what-is-equation-of-straight-line-pass-through-the-point-of-intersection-of-the-line-2-x-3-y-1-and-3-x-2-y-1-and-parallel-the-4-x-5-y-6-0

slug:
what-is-equation-of-straight-line-pass-through-the-point-of-intersection-of-the-line-2-x-3-y-1-and-3-x-2-y-1-and-parallel-the-4-x-5-y-6-0-30159

question:
<div class="question">What is equation of straight line pass through the point of<br/>intersection of the line $\frac{x}{2}+\frac{y}{3}=1$ and $\frac{x}{3}+\frac{y}{2}=1$, and parallel the $4 \mathrm{x}+5 \mathrm{y}-6=0 ?$</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$20 \mathrm{x}+25 \mathrm{y}-54=0$</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">$25 x+20 y-54=0$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4 x+5 y-54=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 x+5 y-45=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$20 \mathrm{x}+25 \mathrm{y}-54=0$</span> </div>

answer:
<div class="solution">Given equatiosns: $\frac{x}{2}+\frac{y}{3}=1$ and $\frac{x}{3}+\frac{y}{2}=1$<br/>Point of intersection $=\left(\frac{6}{5}, \frac{6}{5}\right)$<br/>Let equation of line be $4 x+5 y+k=0$<br/>Putting $\left(\frac{6}{5}, \frac{6}{5}\right), k=-\frac{54}{5}$<br/>Equation of line is $20 x+25 y-54=0$</div>


title: what-is-f-2-x-equal-to-consider-the-function-f-x-x-1-x-1
slug: what-is-f-2-x-equal-to-consider-the-function-f-x-x-1-x-1-59063
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $f(2 x)$ equal to ?<br/>Consider the function $f(x)=\frac{x-1}{x+1}$.</div>

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

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

answer:
<div class="solution">$\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-1}{\mathrm{x}+1} \Rightarrow x=\frac{1+\mathrm{f}(\mathrm{x})}{1-\mathrm{f}(\mathrm{x})}$<br/>$\Rightarrow \mathrm{f}(2 \mathrm{x})=\frac{2 \mathrm{x}-1}{2 \mathrm{x}+1} \Rightarrow \mathrm{f}(2 \mathrm{x})=\frac{2\left[\frac{[1+\mathrm{f}(\mathrm{x})]}{1-\mathrm{f}(\mathrm{x})}\right]-1}{2\left[\frac{[1+\mathrm{f}(\mathrm{x})]}{1-\mathrm{f}(\mathrm{x})}\right]+1}$<br/>$\Rightarrow \mathrm{f}(2 \mathrm{x})=\frac{3 \mathrm{f}(\mathrm{x})+1}{\mathrm{f}(\mathrm{x})+3}$</div>


title: what-is-f-f-x-equal-to-consider-the-function-f-x-x-1-x-1
slug: what-is-f-f-x-equal-to-consider-the-function-f-x-x-1-x-1-68206
tags: ['Mathematics', 'Functions', 'NDA', 'NDA 2014 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $f(f(x))$ equal to ?<br/>Consider the function $f(x)=\frac{x-1}{x+1}$.</div>

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

answer:
<div class="solution">Given $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}-1}{\mathrm{x}+1}$<br/>$\Rightarrow \mathrm{f}[\mathrm{f}(\mathrm{x})]=\frac{\mathrm{f}(\mathrm{x})-1}{\mathrm{f}(\mathrm{x})+1}$<br/>$\Rightarrow \mathrm{f}[\mathrm{f}(\mathrm{x})]=-\frac{1}{\mathrm{x}} \quad\left[\because \mathrm{x}=-\left\{\frac{\mathrm{f}(\mathrm{x})+1}{-\mathrm{f}(\mathrm{x})-1}\right\}\right]$</div>


title: what-is-hinsberg-reagent
slug: what-is-hinsberg-reagent-92423
question: <div class="question">What is Hinsberg reagent?</div>
tags: ['Chemistry', 'Amines']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
answer:
<div class="solution">Beneze sulphonyl chloride $\left(\mathrm{C}_6 \mathrm{H}_5 \mathrm{SO}_2 \mathrm{Cl}\right)$ is as Hinsberg reagent. It is used to distinguish between primary, secondary and tertiary amine.</div>


title: what-is-i-1000-i-1001-i-1002-i-1003-equal-to-where-i-1
slug: what-is-i-1000-i-1001-i-1002-i-1003-equal-to-where-i-1-34943
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2018 (Phase 1)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\mathrm{i}^{1000}+\mathrm{i}^{1001}+\mathrm{i}^{1002}+\mathrm{i}^{1003}$ equal to (where<br/>$\mathrm{i}=\sqrt{-1}$ ?</div>

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

answer:
<div class="solution">$\mathrm{i}^{1000}+\mathrm{i}^{1001}+\mathrm{i}^{1002}+\mathrm{i}^{1003}$<br/>$=\mathrm{i}=\sqrt{-1}$<br/>$=\mathrm{i}^{1000}\left(1+\mathrm{i}+\mathrm{i}^{2}+\mathrm{i}^{3}\right)$<br/>$\Rightarrow \mathrm{i}^{2}=-1$<br/>$\begin{array}{ll}= &amp; \mathrm{i}^{1000}(\not{\chi}+\dot{\lambda}-\not{\lambda}-\dot{\lambda}) \quad \begin{array}{l}\mathrm{i}^{3}=\mathrm{i}^{2} . \mathrm{i} \\ =-\mathrm{i}\end{array} \\ =0\end{array}$</div>


title: what-is-i-where-i-1-equal-to
slug: what-is-i-where-i-1-equal-to-32847
question: <div class="question">What is $\sqrt{-\mathrm{i}}$ where $\mathrm{i}=\sqrt{-1}$ equal to?</div>
tags: ['Mathematics', 'Complex Number', 'NDA', 'NDA 2013 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\pm \frac{1-\mathrm{i}}{\sqrt{2}}$</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">$\pm \frac{1+\mathrm{i}}{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\pm \frac{1-\mathrm{i}}{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\pm \frac{1+\mathrm{i}}{2}$</span> </li> </ul>

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

answer:
<div class="solution">Consider $\sqrt{-\mathrm{i}}=\sqrt{\mathrm{e}^{-\mathrm{i} \pi / 2}}=\pm \mathrm{e}^{-\mathrm{i} \pi / 4}=\pm\left(\frac{1-\mathrm{i}}{\sqrt{2}}\right)$</div>


title: what-is-imbricate-aestivation
slug: what-is-imbricate-aestivation-87807
question: <div class="question">What is imbricate aestivation?</div>
tags: ['Biology', 'Morphology of Flowering Plants', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">When the margins of sepals or petals overlap one another but not in any particular direction</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">When sepals or petals overlap one another in a particular direction</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">When sepals or petals just touch one another at the margin, without overlapping</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">When one margin of the appendage overlaps that of the next one and so on</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">When the margins of sepals or petals overlap one another but not in any particular direction</span> </div>

answer:
<div class="solution">Imbricate aestivation refers to a situation where the margins of the sepals or petals overlap one another, but not in any particular direction.</div>


title: what-is-in-situ-conservation
slug: what-is-in-situ-conservation-22108
question: <div class="question">What is in situ conservation?</div>
tags: ['Biology', 'Biodiversity and its Conservation', 'NEET']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">Protecting endangered species in their natural habitat</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">Protective maintenance in zoological parks and botanical gardens</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">In vitro fertilization</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Cryopreservation of gametes</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">Protecting endangered species in their natural habitat</span> </div>

answer:
<div class="solution">In situ conservation is the protection of endangered species in their natural habitat.</div>


title: what-is-l-o-g-2-n-1-l-o-g-3-n-1-l-o-g-4-n-1-l-o-g-100-n-1-equal-to-n-1
slug: what-is-l-o-g-2-n-1-l-o-g-3-n-1-l-o-g-4-n-1-l-o-g-100-n-1-equal-to-n-1-28384
tags: ['Mathematics', 'Sets and Relations', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\frac{1}{\log _{2} \mathrm{~N}}+\frac{1}{\log _{3} \mathrm{~N}}+\frac{1}{\log _{4} \mathrm{~N}}+\ldots .+\frac{1}{\log _{100} \mathrm{~N}}$ equal<br/>to $(\mathrm{N} \neq 1)$ ?</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\log _{1001} \mathrm{~N}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{\log _{991} N}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{99}{\log _{1001} \mathrm{~N}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{99}{\log _{991} \mathrm{~N}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\log _{1001} \mathrm{~N}}$</span> </div>

answer:
<div class="solution">$\frac{1}{\log _{2} \mathrm{~N}}+\frac{1}{\log _{3} \mathrm{~N}}+\frac{1}{\log _{4} \mathrm{~N}}+\ldots .+\frac{1}{\log _{100} \mathrm{~N}}$<br/>$=\frac{\frac{1}{\log N}}{\log _{2}}+\frac{1}{\frac{\log N}{\log _{3}}}+\frac{1}{\frac{\log N}{\log _{4}}}+\ldots .+\frac{1}{\frac{\log N}{\log _{1} \infty}}$<br/>$\left(\because \log _{a} b=\frac{\log b}{\log a}\right)$<br/>$=\frac{\log _{2}}{\log N}+\frac{\log _{3}}{\log N}+\frac{\log _{4}}{\log N}+\ldots .+\frac{\log _{100}}{\log N}$<br/>$=\frac{\log _{2}+\log _{3}+\log _{4}+\ldots+\log _{100}}{\log \mathrm{N}}$<br/>$=\frac{\log (2.3 .4 \ldots .100)}{\log \mathrm{N}}$<br/>$=\frac{\log 100 !}{\log N}=\frac{1}{\frac{\log \mathrm{N}}{\log 100 !}}=\frac{1}{\log _{100 !} \mathrm{N}}$</div>


title: what-is-lanthanoid-contraction-what-are-the-consequences-of-lanthanoid-contraction
slug: what-is-lanthanoid-contraction-what-are-the-consequences-of-lanthanoid-contraction-43264
tags: ['Chemistry', 'd and f Block Elements']
options: None
correct_option: None
source_db: MarksBatch2_P3.db
question:
<div class="question">What is lanthanoid contraction? What are the consequences of lanthanoid contraction?</div>

answer:
<div class="solution">Lanthanoid Contraction : In the lanthanoids, the electrons are getting filled in the \(4 f\)-subshell. On moving from left to right, the nuclear charge increases and this increase is expected to be compensated by the increase in the magnitude of shielding effect by the \(4 f\) - electrons However, the \(f\)-electrons have very poor shielding effect. Consequently, the atomic and ionic radii decrease from left to right and this is knwon as lanthanoid contraction.<br/>Consequences of lanthanoid Contraction<br/>(a) Separation of Lanthanoids: All the lanthanoids have quite similar properties and due to this reason they are difficult to separate.<br/>(b) Variation in basic strength of hydroxides:<br/>Due to lanthanoid contraction, size of \(\mathrm{M}^{3+}\) ions decreases and thus there is a corresponding increase in the covalent character in \(\mathrm{M}-\mathrm{OH}\) bond. Thus basic character of oxides and hydroxides decreases from \(\mathrm{La}(\mathrm{OH})_3\) to \(\mathrm{Lu}(\mathrm{OH})_3\).<br/>(c) Similarity in the atomic sizes of the elements of second and third transition series present in the same group. The difference in the value of atomic radii of \(\mathrm{Y}\) and La is quite large as compared to the difference in the value of \(\mathrm{Zr}\) and \(\mathrm{Hf}\). This is because of the lanthanoid contraction.<br/>(d) Variation in standard reduciton potential: Due to lanthanoid contraction there is a small but steady increase in the standard reduction potential \(\left(\mathrm{E}^{\circ}\right)\) for the reduction process.<br/>\(\mathrm{M}^{3+}(a q)+3 \mathrm{e}^{-} \longrightarrow \mathrm{M}(a q)\)<br/>(e) Variation in physical properties like melting point, boiling point, hardness etc.</div>


title: what-is-lim-0-1-c-o-s-equal-to
slug: what-is-lim-0-1-c-o-s-equal-to-82694
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2018 (Phase 2)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{\theta \rightarrow 0} \frac{\sqrt{1-\cos \theta}}{\theta}$ equal to?</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp; \lim _{\theta \rightarrow 0} \frac{\sqrt{1-\cos \theta}}{\theta}=\lim _{\theta \rightarrow 0} \frac{\sqrt{2} \sin (\theta / 2)}{\theta} \\ &amp;=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}} \end{aligned}$</div>


title: what-is-lim-h-0-2-h-2-x-3-h-2-x-equal-to
slug: what-is-lim-h-0-2-h-2-x-3-h-2-x-equal-to-38164
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2018 (Phase 1)']
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{h \rightarrow 0} \frac{\sqrt{2 x+3 h}-\sqrt{2 x}}{2 h}$ equal to?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{3}{4 \sqrt{2 x}}$</span> </div>

answer:
<div class="solution">$\lim _{h \rightarrow 0} \frac{\sqrt{2 x+3 h}-\sqrt{2 x}}{2 h}$<br/>Rationalise the numerator.<br/>$\lim _{h \rightarrow 0}\left(\frac{\sqrt{2 x+3 h}-\sqrt{2 x}}{2 h} \times \frac{\sqrt{2 x+3 h}+\sqrt{2 x}}{\sqrt{2 x+3 h}+\sqrt{2 x}}\right)$<br/>$=\lim _{h \rightarrow 0} \frac{2 x+3 h-2 x}{2 h(\sqrt{2 x+3 h}+\sqrt{2 x})}$<br/>$=\frac{3}{2(\sqrt{2 x+0}+\sqrt{2 x})}=\frac{3}{4 \sqrt{2 x}}$</div>


title: what-is-lim-n-1-2-2-2-3-2-n-2-1-2-3-n-equal-to
slug: what-is-lim-n-1-2-2-2-3-2-n-2-1-2-3-n-equal-to-88486
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2014 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots+n}{1^{2}+2^{2}+3^{2}+\ldots n^{2}}$ equal to ?</div>

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

answer:
<div class="solution">$\lim _{n \rightarrow \infty} \frac{1+2+3+\ldots+n}{1^{2}+2^{2}+3^{2}+\ldots .+n^{2}}=\lim _{n \rightarrow \infty} \frac{\frac{n(n+1)}{2}}{\frac{n(n+1)(2 n+1)}{6}}$<br/>$\therefore \lim _{n \rightarrow \infty} \frac{3}{2 n+1}=0$<br/>Note: $1+2+3+\ldots .+n=\frac{n(n+1)}{2}$<br/>$1^{2}+2^{2}+3^{2}+\ldots .+n^{2}=\frac{n(n+1)(2 n+1)}{6}$</div>


title: what-is-lim-x-0-1-c-o-s-x-x-equal-to
slug: what-is-lim-x-0-1-c-o-s-x-x-equal-to-18304
question: <div class="question">What is $\lim _{x \rightarrow 0} \frac{x}{\sqrt{1-\cos x}}$ equal to ?</div>
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2014 (Phase 1)']
source_db: MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{2}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\sqrt{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{\sqrt{2}}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">Limit does not exist</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Limit does not exist</span> </div>

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{x}{\sqrt{1-\cos x}}=\lim _{x \rightarrow 0} \frac{x}{\sqrt{1-\left(1-2 \sin ^{2} \frac{x}{2}\right)}}$<br/>$=\lim _{x \rightarrow 0} \frac{x}{\sqrt{2 \sin ^{2} \frac{x}{2}}}=\frac{1}{\sqrt{2}} \lim _{x \rightarrow 0} \frac{x}{\left|\sin \frac{x}{2}\right|}$<br/>L.H.L $=f(0-0)=\lim _{h \rightarrow 0} \frac{x}{\left|\sin \frac{x}{2}\right|}$<br/>$=-\frac{1}{\sqrt{2}} \lim _{x \rightarrow 0} \frac{2\left(\frac{h}{2}\right)}{\sin \frac{\mathrm{h}}{2}}$<br/>$=-\frac{1}{\sqrt{2}} \times 2 \times 1 \quad\left(\because \lim _{\theta \rightarrow 0} \frac{\theta}{\sin \theta}=1\right)$<br/>$=-\sqrt{2}$<br/>RHL $=\mathrm{f}(0+0)=\lim _{h \rightarrow 0} \mathrm{f}(0+\mathrm{h})$<br/>$=\frac{1}{\sqrt{2}} \lim _{\mathrm{h} \rightarrow 0} \frac{2\left(\frac{\mathrm{h}}{2}\right)}{\sin \frac{\mathrm{h}}{2}}=\frac{1}{\sqrt{2}} \times 2 \times 1$<br/>$=\mathrm{LHL} \neq \mathrm{RHL}=\sqrt{2}$<br/>Therefore limit does not exist.</div>


title: what-is-lim-x-0-2-x-s-i-n-4-x-s-i-n-2-x-4-x-equal-to
slug: what-is-lim-x-0-2-x-s-i-n-4-x-s-i-n-2-x-4-x-equal-to-75967
tags: ['Mathematics', 'Limits', 'NDA', 'NDA 2013 (Phase 2)']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: MarksBatch2_P3.db
question:
<div class="question">What is $\lim _{x \rightarrow 0} \frac{\sin 2 x+4 x}{2 x+\sin 4 x}$ equal to ?</div>

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

answer:
<div class="solution">$\lim _{x \rightarrow 0} \frac{\sin 2 x+4 x}{2 x+\sin 4 x}=\lim _{x \rightarrow 0} \frac{\frac{\sin 2 x}{x}+4}{2+\frac{\sin 4 x}{x}}$<br/>$=\lim _{x \rightarrow 0} \frac{2\left(\frac{\sin 2 x}{2 x}\right)+4}{2+4\left(\frac{\sin 4 x}{4 x}\right)}$<br/>Applying limit, we get $\frac{2+4}{2+4}=1$</div>


