# questions

tags: ['Chemistry', 'Solutions', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-measured-freezing-point-depression-for-a-01-m-aqueous-ch-3-cooh-solution-is-01-9-c-the-acid-dissociation-constant-k-a-at-this-concentration-will-b

slug:
the-measured-freezing-point-depression-for-a-01-m-aqueous-ch-3-cooh-solution-is-01-9-c-the-acid-dissociation-constant-k-a-at-this-concentration-will-b-46207

question:
<div class="question">The measured freezing point depression for a $0.1 \mathrm{m}$ aqueous $\mathrm{CH}_{3} \mathrm{COOH}$ solution is $0.19^{\circ} \mathrm{C}$. The acid dissociation constant $K_{a}$ at this concentration will be (Given, $K_{f}$ the molal cryoscopic constant $=1.86 \mathrm{K}$ $\mathrm{kg} \mathrm{mol}^{-1}$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.76 \times 10^{-5}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4 \times 10^{-5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$8 \times 10^{-5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 \times 10^{-5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4 \times 10^{-5}$</span> </div>

answer:
<div class="solution">$\begin{aligned} \Delta T_{f} &amp;=i \times K_{t} \times m \\ \therefore &amp; i=\frac{\Delta T_{f}}{K_{t} \times m}=\frac{0.19}{1.86 \times 0.1}=1.02 \end{aligned}$
<br/>Again from, $\alpha=\frac{i-1}{n-1}=\frac{1.02-1}{2-1}$ $=0.02=2.0 \times 10^{-2}$
<br/>$$
<br/>\begin{aligned}
<br/>\text { for } \mathrm{CH}_{3} \mathrm{COOH} &amp; \rightleftharpoons \mathrm{CH}_{3} \mathrm{COO}^{-}+\mathrm{H}^{+} \\
<br/>K \mathrm{a} &amp;=\mathrm{C} \alpha^{2} \\
<br/>&amp;=0.1 \times\left(2 \times 10^{-2}\right)^{2} \\
<br/>&amp;=4 \times 10^{-5}
<br/>\end{aligned}
<br/>$$</div>


title: the-melting-points-of-i-becl-2-ii-cacl-2-and-iii-hgcl-2-follows-the-order
slug: the-melting-points-of-i-becl-2-ii-cacl-2-and-iii-hgcl-2-follows-the-order-26082
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The melting points of (i) BeCl$_{2}$ (ii) $\mathrm{CaCl}_{2}$ and (iii) $\mathrm{HgCl}_{2}$ follows the order</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$iii    &lt; i &lt; ii$</span> </div>

answer:
<div class="solution">Key Point A compound having more ionic character, has more melting point.
<br/>Be and Ca belong to the same group and ionic character increases down the group.
<br/>Thus, $\mathrm{BeCl}_{2}$ is less ionic than $\mathrm{CaCl}_{2}$. This means that melting point of $\mathrm{CaCl}_{2}$ is higher than that of $\mathrm{BeCl}_{2}$
<br/>Also, $\mathrm{Hg}$ is a transition metal, its compound is less ionic than $\mathrm{BeCl}_{2}$. Therefore, order of melting point will be
<br/>or
<br/>$$
<br/>\begin{array}{c}
<br/>\mathrm{CaCl}_{2}&gt;\mathrm{BeCl}_{2}&gt;\mathrm{HgCl}_{2} \\
<br/>(\mathrm{ii})&gt;(\mathrm{i})&gt;(\mathrm{iii})
<br/>\end{array}
<br/>$$Hence, option (b) is the correct answer.</div>


title: the-metal-which-can-be-used-to-obtain-metallic-cu-from-aqueous-cuso-4-solution-is
slug: the-metal-which-can-be-used-to-obtain-metallic-cu-from-aqueous-cuso-4-solution-is-23931
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Fe</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The metal which can be used to obtain metallic Cu from aqueous CuSO $_{4}$ solution is</div>

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


title: the-metalpair-that-can-produce-nascent-hydrogen-in-alkaline-medium-is
slug: the-metalpair-that-can-produce-nascent-hydrogen-in-alkaline-medium-is-77059
question: <div class="question">The metal-pair that can produce nascent hydrogen in alkaline medium is</div>
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2022']
answer: <div class="solution">Fact</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{Zn}, \mathrm{Al}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\mathrm{Fe}, \mathrm{Ni}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">Al, Mg</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{Mg}, \mathrm{Zn}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Zn}, \mathrm{Al}$</span> </div>


title: the-minimum-value-of-2-s-i-n-x-2-c-o-s-x-is
slug: the-minimum-value-of-2-s-i-n-x-2-c-o-s-x-is-77969
question: <div class="question">The minimum value of $2^{\sin x}+2^{\cos x}$ is</div>
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2^{1-1 / \sqrt{2}}$</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">B</span> <span class="option-data">$2^{1+1 / \sqrt{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{\sqrt{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

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

answer:
<div class="solution">We know that
<br/>$\mathrm{AM} \geq \mathrm{GM}$
<br/>$\therefore \quad \frac{2^{\sin x}+2^{\cos x}}{2} \geq \sqrt{2^{\sin x} 2^{\cos x}}$
<br/>$\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2 \sqrt{2^{\sin x+\cos x}}$
<br/>$\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2 \times 2^{\frac{\sin x+\cos x}{2}}$
<br/>$\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2^{1+\frac{\sin x+\cos x}{2}}$
<br/>But $\sin x+\cos x=\sqrt{2} \sin \left(x+\frac{\pi}{4}\right) \geq-\sqrt{2}$
<br/>$\therefore \quad 2^{\sin x}+2^{\cos x} \geq 2^{1-\frac{\sqrt{2}}{2}}$
<br/>$\Rightarrow 2^{\sin x}+2^{\cos x} \geq 2^{1-\frac{1}{\sqrt{2}}}, \forall x \in R$
<br/>Hence, minimum value is $2^{1-\frac{1}{\sqrt{2}}}$.</div>


title: the-minimum-value-of-cos-sin-s-i-n-2-2-for-0-2-is
slug: the-minimum-value-of-cos-sin-s-i-n-2-2-for-0-2-is-58220
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The minimum value of $\cos \theta+\sin \theta+\frac{2}{\sin 2 \theta}$
<br/>for $\theta \in(0, \pi / 2),$ is</div>

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

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

answer:
<div class="solution">Here, $\cos \theta+\sin \theta+\frac{2}{\sin 2 \theta}, \theta \in\left(0, \frac{\pi}{2}\right)$
<br/>For minimum value, $\sin 2 \theta$ must be maximum
<br/>$\therefore \quad 2 \theta=\frac{\pi}{2} \Rightarrow \theta=\frac{\pi}{4}$
<br/>Hence, $\cos \frac{\pi}{4}+\sin \frac{\pi}{4}+\frac{2}{\sin \frac{\pi}{2}}=\sqrt{2}+2$</div>


title: the-minimum-value-of-f-x-e-x-4-x-3-x-2-is
slug: the-minimum-value-of-f-x-e-x-4-x-3-x-2-is-98939
question: <div class="question">The minimum value of $f(x)=e^{\left(x^4-x^3+x^2\right)}$ is</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">e</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-\mathrm{e}$</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">-1</span> </li> </ul>

answer:
<div class="solution">Hints: $f(x)=e^{\left(x^4-x^3+x^2\right)}, f^{\prime}(x)=e^{x^4-x^3+x^2}$<br/>$e^{x^4-x^3+x^2}\left(4 x^3-3 x^2+2 x\right) x\left(4 x^2-3 x+2\right)$<br/>$\Rightarrow f(x)$ is decreasing for $x &lt; 0$, increasing for $x&gt;0$<br/>$\therefore$ Minimum is at $\mathrm{x}=0 \quad \therefore \mathrm{f}(0)=\mathrm{e}^0=1$</div>


title: the-minimum-value-of-the-function-f-x-2-x-1-x-2-is
slug: the-minimum-value-of-the-function-f-x-2-x-1-x-2-is-61356
question: <div class="question">The minimum value of the function $f(x)=2|x-1|+|x-2|$ is</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: 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">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">Given,
<br/>$$
<br/>\begin{array}{l}
<br/>\quad f(x)=2|x-1|+|x-2| \\
<br/>\Rightarrow \quad f(x)=\left\{\begin{array}{c}
<br/>-2(x-1)-(x-2), x     &lt; 1 \\
<br/>2(x-1)-(x-2), 1 \leq x &lt; 2 \\
<br/>2(x-1)+(x-2), x \geq 2 \\
<br/>-3 x+4, \quad x &lt; 1 \\
<br/>x_{1} \\
<br/>3 x-4, \quad 1 \leq x &lt; 2 \\
<br/>x \geq 2
<br/>\end{array}\right.
<br/>\end{array}
<br/>$$
<br/>$f^{\prime}(x)=\left\{\begin{array}{cc}-3 &amp; x &lt; 1 \\ 1 &amp; 1 \leq x &lt; 2 \\ 3 &amp; x \geq 2\end{array}\right.$
<br/>So, $f(x)$ will be minimum at $x=1$ and the minimum value is 1 .</div>


title: the-modulus-of-3-i-1-i-5-4-i-is
slug: the-modulus-of-3-i-1-i-5-4-i-is-39782
question: <div class="question">The modulus of $\frac{1-i}{3+i}+\frac{4 i}{5}$ is</div>
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\sqrt{5}$ unit</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{\sqrt{11}}{5}$ unit</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{\sqrt{5}}{5}$ unit</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{\sqrt{12}}{5}$ unit</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\text { Hints: } \frac{1-i}{3+i}+\frac{4 i}{5}=\frac{5-5 i+4 i(3+i)}{5(3+i)}=\frac{5-5 i+12 i-4}{5(3+i)}=\frac{1+7 i}{5(3+i)}=\frac{(1+7 i)(3-i)}{5(9+1)}<br/>$$<br/><br/>$$<br/>=\frac{3+21 i-i+7}{5 \times 10}=\frac{10+20 i}{5 \times 10}=\frac{1+2 i}{5}<br/>$$<br/><br/>$$<br/>\therefore \text { Modulus }=\sqrt{\left(\frac{1}{5}\right)^2+\left(\frac{2}{5}\right)^2}=\sqrt{\frac{1}{25}+\frac{4}{25}}=\sqrt{\frac{1}{5}}=\frac{\sqrt{5}}{5} \text { unit }<br/>$$</div>


tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-molar-solubility-in-mol-l-1-of-a-sparingly-soluble-salt-m-x-4-is-s-the-corresponding-solubility-product-is-k-sp-s-in-terms-of-k-ip-s-given-by-the-

slug:
the-molar-solubility-in-mol-l-1-of-a-sparingly-soluble-salt-m-x-4-is-s-the-corresponding-solubility-product-is-k-sp-s-in-terms-of-k-ip-s-given-by-the-92850

question:
<div class="question">The molar solubility (in mol $\mathrm{L}^{-1}$ ) of a sparingly soluble salt $M X_{4}$ is ${ }^{\prime} S$. The corresponding solubility product is $K_{\mathrm{sp}} . S$ in terms of $'K_{\text {ip }}$' s given by the relation</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$S=\left(\frac{K_{\mathrm{s} \rho}}{128}\right)^{1 / 4}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$S=\left(\frac{K_{s p}}{256}\right)^{1 / 5}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$S=\left(256 K_{s p}\right)^{1 / 5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$S=\left(128 K_{\mathrm{sp}}\right)^{1 / 4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$S=\left(\frac{K_{s p}}{256}\right)^{1 / 5}$</span> </div>

answer:
<div class="solution">$\because$ For a compound $A_{x}B_{y}$
<br/>$$
<br/>K_{s p}=x^{x} \cdot y^{y} \cdot S^{x+y} . \text { (where, } S=\text { solubility })
<br/>$$
<br/>For $M x_{4}$
<br/>$$
<br/>\begin{array}{l}
<br/>K_{\mathrm{sp}}=1 \times(4)^{4} \cdot \mathrm{S}^{1+4} \\
<br/>K_{\mathrm{sp}}=256 \times \mathrm{S}^{5}
<br/>\end{array}
<br/>$$
<br/>$\therefore S=\left(\frac{K_{s}}{256}\right)^{1 / 5}$</div>


title: the-molarity-of-a-naoh-solution-by-dissolving-4-g-of-it-in-250-ml-water-is
slug: the-molarity-of-a-naoh-solution-by-dissolving-4-g-of-it-in-250-ml-water-is-87049
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2010']
answer: <div class="solution">Hints : Molarity $=\frac{4 / 40}{250 / 1000}=0.4$</div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The molarity of a $\mathrm{NaOH}$ solution by dissolving $4 \mathrm{~g}$ of it in $250 \mathrm{ml}$ water is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$0.4 \mathrm{M}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.8 \mathrm{M}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$0.2 \mathrm{M}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.1 \mathrm{M}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.4 \mathrm{M}$</span> </div>


tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2021']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">9</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-mole-fraction-of-a-solute-in-a-binary-solution-is-01-at-298-k-molarity-of-this-solution-is-same-as-as-its-molality-density-of-this-solution-at-298

slug:
the-mole-fraction-of-a-solute-in-a-binary-solution-is-01-at-298-k-molarity-of-this-solution-is-same-as-as-its-molality-density-of-this-solution-at-298-65490

question:
<div class="question">The mole fraction of a solute in a binary solution is $0.1 .$ At $298 \mathrm{~K}$, molarity of this solution is same as as its molality. Density of this solution at $298 \mathrm{~K}$ is $2.0 \mathrm{~g} \mathrm{~cm}^{-3}$. The ratio of molecular weights of the solute and the solvent $\left(\mathrm{M}_{\text {solute }} / \mathrm{M}_{\text {solvent }}\right)$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</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><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{1}{9}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$4.5$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{1}{4.5}$</span> </li> </ul>

answer:
<div class="solution">$\mathrm{X}_{2}=0.1 \Rightarrow \frac{\mathrm{n}_{2}}{\mathrm{n}_{2}+\mathrm{n}_{1}}=0.1$ $\ldots(1)$
<br/>$\mathrm{M}=\frac{\mathrm{W}_{2} \times 1000}{\mathrm{M}_{2} \times \mathrm{W}} \times \mathrm{d}$ $\ldots(2)$
<br/>$\mathrm{m}=\frac{\mathrm{W}_{2} \times 1000}{\mathrm{M}_{2} \times \mathrm{W}_{1}(\mathrm{~g})}$ $\ldots(3)$
<br/>$\mathrm{d}=2 \mathrm{~g} / \mathrm{cc}$ $\ldots(4)$
<br/>Given M = m (numerically)
<br/>$\begin{array}{l}
<br/>\Rightarrow \frac{\mathrm{W}_{2} \times 1000}{\mathrm{M}_{2} \times\left(\mathrm{W}_{2}+\mathrm{W}_{1}\right)} \times \mathrm{d}=\frac{\mathrm{W}_{2} \times 1000}{\mathrm{M}_{2} \times \mathrm{W}_{1}(\mathrm{~g})} \\
<br/>\Rightarrow \mathrm{W}_{2}+\mathrm{W}_{1}=\mathrm{W}_{1} \times \mathrm{d}=2 \mathrm{~W}_{1} \\ 
<br/>\Rightarrow \mathrm{W}_{2}=\mathrm{W}_{1} 
<br/>\end{array}$ 
<br/>Now, $\frac{n_{2}}{n_{2}+n_{1}}=0.1 \Rightarrow \frac{\frac{w_{2}}{M_{2}}}{\frac{W_{2}}{M_{2}}+\frac{W_{1}}{M_{1}}}=0.1 \Rightarrow \frac{\frac{W_{2}}{M_{2}}}{W_{2}\left(\frac{1}{M_{2}}+\frac{1}{M_{1}}\right)}=0.1$
<br/>$\Rightarrow \frac{\frac{1}{\mathrm{M}_{2}}}{\frac{\mathrm{M}_{1}+\mathrm{M}_{2}}{\mathrm{M}_{1} \mathrm{M}_{2}}}=0.1 \Rightarrow \frac{\mathrm{M}_{1}}{\mathrm{M}_{1}+\mathrm{M}_{2}}=\frac{1}{10} \Rightarrow \frac{\mathrm{M}_{1}}{\mathrm{M}_{1}+\mathrm{M}_{2}-\mathrm{M}_{1}}=\frac{1}{10-1} \Rightarrow \frac{\mathrm{M}_{1}}{\mathrm{M}_{2}}=\frac{1}{9} \Rightarrow \frac{\mathrm{M}_{2}}{\mathrm{M}_{1}}=9$</div>


title: the-mole-fraction-of-ethanol-in-water-is-008-its-molality-is
slug: the-mole-fraction-of-ethanol-in-water-is-008-its-molality-is-98207
question: <div class="question">The mole fraction of ethanol in water is $0.08$. Its molality is</div>
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$6.32 \mathrm{~mol} \mathrm{~kg}^{-1}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$4.83 \mathrm{~mol} \mathrm{~kg}^{-1}$</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">$3.82 \mathrm{~mol} \mathrm{~kg}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.84 \mathrm{~mol} \mathrm{~kg}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$4.83 \mathrm{~mol} \mathrm{~kg}^{-1}$</span> </div>

answer:
<div class="solution">Hint:
<br/>$\begin{aligned} \text { Molality } &amp;=\frac{\mathrm{x}_{\mathrm{EtOH}}}{\mathrm{x}_{\mathrm{H}_{2} \mathrm{O}}} \times \frac{1000}{\mathrm{MW}_{\text {solvent }}} \\ &amp;=\frac{0.08}{0.92} \times \frac{1000}{18} \\ &amp;=4.83 \mathrm{~mol} \mathrm{~kg}^{-1} \end{aligned}$</div>


title: the-molecular-shapes-of-mathrmsf4-mathrmcf4-and-mathrmxef4-are
slug: the-molecular-shapes-of-mathrmsf4-mathrmcf4-and-mathrmxef4-are-68245
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The molecular shapes of \(\mathrm{SF}_4, \mathrm{CF}_4\) and \(\mathrm{XeF}_4\) are</div>

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

answer:
<div class="solution">Hint : <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/AFK8bX8-BtNFjahzubgwl8Viok0xDVdxGeQKo2ducKA.original.fullsize.png"/><br/></div>


title: the-moleculemolecules-that-hashave-delocalised-lone-pairs-of-electrons-isare
slug: the-moleculemolecules-that-hashave-delocalised-lone-pairs-of-electrons-isare-84352
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The molecule/molecules that has/have delocalised lone pair(s) of electrons is/are
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_q68.png"/></div>

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

answer:
<div class="solution">The molecule in which a lone pair of electrons undergoes conjugation with an alternate double bond is said to be delocalised. The molecule that has delocalised lone-pair of electrons is,
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_che_a68.png"/> <br/>So, option (d) is correct.</div>


title: the-monomer-used-to-produce-orlon-is
slug: the-monomer-used-to-produce-orlon-is-45597
question: <div class="question">The monomer used to produce orlon is</div>
tags: ['Chemistry', 'Polymers', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{CH}_2=\mathrm{CHF}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\mathrm{CH}_2=\mathrm{CCl}_2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{CH}_2=\mathrm{CHCl}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\mathrm{CH}_2=\mathrm{CH}-\mathrm{CN}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\mathrm{CH}_2=\mathrm{CH}-\mathrm{CN}$</span> </div>

answer:
<div class="solution">Orlon or PAN<br/>Monomer $\Rightarrow \mathrm{CH}_2=\mathrm{CH}-\mathrm{CN}$</div>


title: the-most-contributing-tautomeric-enol-form-of-mecoch-2-co-2-et-is
slug: the-most-contributing-tautomeric-enol-form-of-mecoch-2-co-2-et-is-83535
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The most contributing tautomeric enol form of $\mathrm{MeCOCH}_{2} \mathrm{CO}_{2} \mathrm{Et}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{MeC}(\mathrm{OH})=\mathrm{CHCO}_{2} \mathrm{Et}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_che_a33.png"/></div>


title: the-most-likely-protonation-site-in-the-following-molecule-is
slug: the-most-likely-protonation-site-in-the-following-molecule-is-52996
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The most likely protonation site in the following molecule is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q48.png"/></div>

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

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

answer:
<div class="solution">Addition of a proton at $\mathrm{C}^{-1}$ results in the formation of tropylium carbocation (an aromatic species with more stability due to delocalisation of $\pi$ electrons). thus, it is the most reactive site towards protonation. Addition of proton at any other carbon atom interupt in the delocalisation of it electrons by disturbing planarity of molecules and herce makes it less stable. Thus, these all are lest reactive towards protonation.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a48.png"/></div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-normal-to-a-curve-at-p-x-y-meets-the-x-axis-at-g-if-the-distance-of-g-from-the-origin-is-twice-the-abscissa-of-p-then-the-curve-is

slug:
the-normal-to-a-curve-at-p-x-y-meets-the-x-axis-at-g-if-the-distance-of-g-from-the-origin-is-twice-the-abscissa-of-p-then-the-curve-is-76908

question:
<div class="question">The normal to a curve at $P(x, y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$ then the curve is</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/_nsXV8hGlbJaBEFzeQPE6b_hpezSaxs2anZJdgi76RM.original.fullsize.png"/><br/> <br/>equation of normal
<br/>$Y-y=\left(\frac{\frac{-1}{d y}}{d x}\right) \cdot(X-x) \quad Y=0 \quad \therefore \quad y \frac{d y}{d x}+x=X \quad \therefore \quad a / c \text { to } Q \quad y \cdot \frac{d y}{d x}+x=2 x$
<br/>$\mathrm{y} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{x} \Rightarrow \int \mathrm{y} \mathrm{dy}=\int \mathrm{xdx} \quad \frac{\mathrm{y}^{2}}{2}=\frac{\mathrm{x}^{2}}{2}+\mathrm{c} \quad \frac{\mathrm{y}^{2}-\mathrm{x}^{2}}{2}=\mathrm{c}$ a hyperbola</div>


title: the-normal-to-the-curve-y-x-2-x-1-drawn-at-the-points-with-the-abscissa-x-1-0-x-2-1-and-x-3-52
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
slug:
the-normal-to-the-curve-y-x-2-x-1-drawn-at-the-points-with-the-abscissa-x-1-0-x-2-1-and-x-3-52-79482

question:
<div class="question">The normal to the curve $y=x^{2}-x+1$, drawn at the points with the abscissa $x_{1}=0, x_{2}=-1$ and $x_{3}=5 / 2$</div>

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

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

answer:
<div class="solution">Given equation of curve.
<br/>$$
<br/>\begin{aligned}
<br/>y &amp;=x^{2}-x+1 \\
<br/>\Rightarrow \quad &amp; \frac{d y}{d x}=2 x-1
<br/>\end{aligned}
<br/>$$
<br/>Slope of normal $=\frac{1}{1-2 x}$
<br/>Now, at $x_{1}=0, y_{1}=1$
<br/>$\therefore$ Slope of normal at $(0,1)=1$
<br/>$\therefore$ Equation of normal, $y-1=1(x-0)$ $\Rightarrow \quad x-y+1=0$
<br/>At $x_{2}=-1, y_{2}=3$
<br/>Slope of normal at $(-1,3)=\frac{1}{3}$ Equation of normal, $y-3=\frac{1}{3}(x+1)$
<br/>$\Rightarrow \quad 3 y-9=x+1$
<br/>$x-3 y+10=0$
<br/>At $x_{3}=\frac{5}{2}, y_{3}=\frac{19}{4}$
<br/>Slope of normal at $\left(\frac{2}{5}, \frac{19}{4}\right)=-\frac{1}{4}$
<br/>Equation of normal, $y-\frac{19}{4}=-\frac{1}{4}\left(x-\frac{5}{2}\right)$
<br/>$\begin{array}{lr}\Rightarrow &amp; x+4 y=\frac{43}{2} \\ \Rightarrow &amp; 2 x+8 y=43\end{array}$
<br/>Here, intersecting point of Eqs.(i) and (ii) is 
<br/>(ii) is $\left(\frac{7}{2}, \frac{9}{2}\right)$ and normal (iii) passes through it. Hence, normals are concurrent.</div>


title: the-normality-of-30-volume-mathrmh2-mathrmo2-is
slug: the-normality-of-30-volume-mathrmh2-mathrmo2-is-71380
question: <div class="question">The normality of 30 volume \(\mathrm{H}_2 \mathrm{O}_2\) is</div>
tags: ['Chemistry', 'Some Basic Concepts of Chemistry', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(2.678 \mathrm{~N}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(5.336 \mathrm{~N}\)</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">C</span> <span class="option-data">\(8.034 \mathrm{~N}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(6.685 \mathrm{~N}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(5.336 \mathrm{~N}\)</span> </div>

answer:
<div class="solution">Hints: Volume strength \(=5.6 \times\) normality<br/>\(\begin{aligned}<br/>&amp; 30=5.6 \times \mathrm{N} \\<br/>&amp; \Rightarrow \mathrm{N}=\frac{30}{5.6}=5.3<br/>\end{aligned}\)</div>


title: the-nucleus-29-64-c-u-accepts-an-orbital-clectron-to-yield
slug: the-nucleus-29-64-c-u-accepts-an-orbital-clectron-to-yield-79672
question: <div class="question">The nucleus $^{64}_{29}Cu$ accepts an orbital clectron to yield.</div>
tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{65}{28} \mathrm{Ni}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{64}{30} \mathrm{Zn}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{64}{28} \mathrm{Ni}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\frac{65}{30} \mathrm{zn}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{64}{28} \mathrm{Ni}$</span> </div>

answer:
<div class="solution">The nucleus $^{64}_{29}Cu$ accepts an orbital electron to yield $^{64}_{28}Ni$. The atomic number of $\mathrm{Cu}$ is 29 , which is equal to the number of electrons and also equal to the number of protons. When $^{64}_{29}Cu$ accepts an orbital electron then electrons subtract from the atomic number, i.e.
<br/>$29-1=28$
<br/>$$
<br/>^{64}_{29}Cu+{ }_{-1} e^{0} \longrightarrow{ }_{28}^{64} \mathrm{Ni}
<br/>$$</div>


title: the-number-101-100-1-is-divisible-by
slug: the-number-101-100-1-is-divisible-by-76103
question: <div class="question">The number $(101)^{100}-1$ is divisible by</div>
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$10^{4}$</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">$10^{6}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$10^{8}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$10^{12}$</span> </li> </ul>

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

answer:
<div class="solution">$(101)^{100}-1=(1+100)^{100}-1$
<br/>$=\left(1+{ }^{100} C_{1} \cdot 100+{ }^{100} C_{2} 100^{2}+\ldots \ldots\right)-1$
<br/>$={ }^{100} C_{1} 100+{ }^{100} C_{2}(100)^{2}+$
<br/>$\quad{ }^{100} C_{3}(100)^{3}+\ldots \ldots+{ }^{100} C_{100}(100)^{100}$
<br/>$=10^{4}\left(1+{ }^{100} C_{2}+{ }^{100} C_{3} 10^{2}+\ldots .\right.$
<br/>$\quad+{ }^{100} C_{100}(100)^{98}$
<br/>$=10^{4}(1+$ an integer multiple of 10$)$</div>


title: the-number-1011001-is-divisible-by
slug: the-number-1011001-is-divisible-by-87087
question: <div class="question">The number \((101)^{100}-1\) is divisible by</div>
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(10^4\)</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">\(10^6\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(10^8\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(10^{12}\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(10^4\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}\text {Hints: } <br/>&amp; (101)^{100}-1={ }^{100} \mathrm{C}_1 100+{ }^{100} \mathrm{C}_2 100^2+{ }^{100} \mathrm{C}_3 100^3+\ldots \ldots +{ }^{100} \mathrm{C}_{100} 100^{100} \\<br/>&amp; =100^2\left[1+{ }^{100} \mathrm{C}_2+{ }^{100} \mathrm{C}_3 100+\ldots \ldots .\right] \\<br/>&amp; =\left(10^4\right)<br/>\end{aligned}\)</div>


title: the-number-of-acidic-protons-in-h-3-po-3-are
slug: the-number-of-acidic-protons-in-h-3-po-3-are-21188
question: <div class="question">The number of acidic protons in $\mathrm{H}_{3} \mathrm{PO}_{3}$ are</div>
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: 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">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">3</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2012_che_a21.png"/> <br/> <br/>There are two acidic protons in $\mathrm{H}_{3} \mathrm{PO}_{3}$.</div>


tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-number-of-alkene-s-which-can-produce-2-butanol-by-the-successive-treatment-of-i-b-2-h-6-in-tetrahydrofuran-solvent-and-ii-alkaline-h-2-o-2-solutio

slug:
the-number-of-alkene-s-which-can-produce-2-butanol-by-the-successive-treatment-of-i-b-2-h-6-in-tetrahydrofuran-solvent-and-ii-alkaline-h-2-o-2-solutio-10105

question:
<div class="question">The number of alkene (s) which can produce 2 -butanol by the successive treatment of
<br/>(i) $\mathrm{B}_{2} \mathrm{H}_{6}$ in tetrahydrofuran solvent and
<br/>(ii) alkaline $\mathrm{H}_{2} \mathrm{O}_{2}$ solution is</div>

answer:
<div class="solution">The number of alkene(s) that can produce 2-butanol by the successive treatment of $\mathrm{B}_{2} \mathrm{H}_{6}$ in tetrahydrofuran solvent and alkaline $\mathrm{H}_{2} \mathrm{O}_{2}$ solution is $2 .$ The alkenes are $c i s$ -but -2 - ene and trans-but-2-ene. The hydration product is obtained in accordance with opposite Markownikoff's rule. In this reaction, addition takes place through the initial formation of $\pi$ -complex which changes into a cyclic four centre transition state with the addition of boron atom to the less hindered carbon atom.
<br/>$\mathrm{H}_{3} \mathrm{C}-\mathrm{HC}=\mathrm{CH}-\mathrm{CH}_{3} \frac{(\mathrm{i}) \mathrm{B}_{2} \mathrm{H}_{6}}{(\mathrm{ii}) \mathrm{H}_{2} \mathrm{O}_{2} \mathrm{OH}} \mathrm{CH}_{3}-\mathrm{CH}-\mathrm{CH}_{2} \mathrm{CH}_{3}$</div>


title: the-number-of-all-numbers-having-5-digits-with-distinct-digits-is
slug: the-number-of-all-numbers-having-5-digits-with-distinct-digits-is-19249
question: <div class="question">The number of all numbers having 5 digits, with distinct digits is</div>
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">99999</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$9 \times{ }^{9} P_{4}$</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">${ }^{10} P_{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">${ }^{9} P_{4}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$9 \times{ }^{9} P_{4}$</span> </div>

answer:
<div class="solution">Total number of 5-digit numbers having all the digits distinct $={ }^{10} \mathrm{P}_{5}-{ }^{9} \mathrm{P}_{4}$
<br/>$$
<br/>\begin{array}{l}
<br/>=\frac{10 !}{5 !}-\frac{91}{5 !}=\frac{10 \times 9 !}{5 !}-\frac{91}{51} \\
<br/>=\frac{91}{51}(10-1)=9 \times \frac{9!}{(9-4) !} \\
<br/>=9 \times 9 P_{4}
<br/>\end{array}
<br/>$$</div>


tags: ['Chemistry', 'Biomolecules', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
the-number-of-amino-acids-and-number-of-peptide-bonds-in-a-linear-tetrapeptide-made-of-different-amino-acids-are-respectively-1

slug:
the-number-of-amino-acids-and-number-of-peptide-bonds-in-a-linear-tetrapeptide-made-of-different-amino-acids-are-respectively-1-98424

question:
<div class="question">The number of amino acids and number of peptide bonds in a linear tetrapeptide (made of different amino acids) are respectively</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">4 and 3</span> </div>

answer:
<div class="solution">Combination of four amino acid units gives linear tetrapeptide, in which three peptide bonds are present.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a36.png"/></div>


tags: ['Chemistry', 'Biomolecules', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
the-number-of-amino-acids-and-number-of-peptide-bonds-in-a-linear-tetrapeptide-made-of-different-amino-acids-are-respectively

slug:
the-number-of-amino-acids-and-number-of-peptide-bonds-in-a-linear-tetrapeptide-made-of-different-amino-acids-are-respectively-97493

question:
<div class="question">The number of amino acids and number of peptide bonds in a linear tetrapeptide (made of different amino acids) are respectively</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">4 and 3</span> </div>

answer:
<div class="solution">Combination of four amino acid units gives linear tetrapeptide, in which three peptide bonds are present.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a36.png"/></div>


title: the-number-of-and-bonds-between-two-carbon-atoms-in-calcium-carbide-are
slug: the-number-of-and-bonds-between-two-carbon-atoms-in-calcium-carbide-are-85707
tags: ['Chemistry', 's Block Elements', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of $\sigma$ and $\pi$ bonds between two carbon atoms in calcium carbide are</div>

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

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

answer:
<div class="solution">Calcium carbide is calcium acetylide, having structure
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_che_a20.png"/> <br/> <br/>Thus, it has one sigma ($\sigma$) and two pi ($\pi$) bonds.</div>


title: the-number-of-atoms-in-body-centred-and-face-centred-cubic-unit-cell-respectively-are
slug: the-number-of-atoms-in-body-centred-and-face-centred-cubic-unit-cell-respectively-are-84817
tags: ['Chemistry', 'Solid State', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of atoms in body centred and face centred cubic unit cell respectively are</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2 and 4</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/mrKvVvscJAox1KyrIHg3wHYCHQA1SdDV-FI0RNyzsaQ.original.fullsize.png"/><br/><br/>$Z_{\mathrm{BCC}}=\frac{1}{8} \times 8+1=2 \quad Z_{\mathrm{FCC}}=\frac{1}{8} \times 8+\frac{1}{2} \times 6=1+3=4$</div>


tags: ['Physics', 'Nuclear Physics', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-number-of-atoms-of-a-radioactive-substance-of-halflife-t-is-n-0-at-t-0-the-time-necessary-to-decay-from-n-0-2-atoms-to-n-0-10-atoms-will-be

slug:
the-number-of-atoms-of-a-radioactive-substance-of-halflife-t-is-n-0-at-t-0-the-time-necessary-to-decay-from-n-0-2-atoms-to-n-0-10-atoms-will-be-13327

question:
<div class="question">The number of atoms of a radioactive substance of half-life $T$ is $N_{0}$ at $t=0 .$ The time necessary to decay from $N_{0} / 2$ atoms to $N_{0} / 10$ atoms will be</div>

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

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

answer:
<div class="solution">We have, $\frac{N(t)}{N_{0}}=e^{-\lambda t}$ or $N(t)=N_{0} e^{-\lambda t}$
<br/>$\therefore$ For the given condition. $\frac{N_{0}}{2}=N_{0} e^{-\lambda t_{1}}$ and $\frac{N_{0}}{10}=N_{0} e^{-\lambda t_{2}}$
<br/>$\Rightarrow \quad \frac{1}{2}=e^{-\lambda t_{1}}$ and $\frac{1}{10}=e^{-\lambda t_{2}}$
<br/>or $\quad e^{\lambda t_{1}}=2$ and $e^{\lambda t_{2}}=10$
<br/>Taking log on both sides, $\lambda t_{1}=\log 2 \Rightarrow t_{1}=\frac{\log 2}{\lambda}$
<br/>or $\quad t_{1}=\frac{\log 2 \times \mathrm{T}}{\log 2}$
<br/>and $\quad \lambda t_{2}=\log 10$
<br/>$\Rightarrow \quad t_{2}=\frac{\log 10}{\lambda}$
<br/>of $\quad t_{2}=\frac{\log 10 \times \mathrm{T}}{\log 2}$
<br/>$\therefore \quad\left(t_{2}-t_{1}\right)=T\left[\frac{\log 10}{\log 2}-1\right]=T\left[\frac{\log 10-\log 2}{\log 2}\right]$
<br/>$=T\left[\frac{\log 5}{\log 2}\right]$
<br/>$\Rightarrow\left(t_{2}-t_{1}\right)=T \left[\frac{\log 5}{\log 2}\right]$</div>


title: the-number-of-complex-numbers-p-such-that-p-1-and-imaginary-part-of-p-4-is-0-is
slug: the-number-of-complex-numbers-p-such-that-p-1-and-imaginary-part-of-p-4-is-0-is-80574
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of complex numbers $\mathrm{p}$ such that $|\mathrm{p}|=1$ and imaginary part of $\mathrm{p}^{4}$ is 0, is</div>

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

answer:
<div class="solution">Hint: 
<br/>Let $p=x+i y, p^{2}=\left(x^{2}-y^{2}\right)+2 i x y, p^{4}=\left(x^{2}-y^{2}\right)^{2}-4 x^{2} y^{2}+4 i x y\left(x^{2}-y^{2}\right)$
<br/>Now, $x y\left(x^{2}-y^{2}\right)=0, \quad x=\pm y \Rightarrow y^{2}=\frac{1}{2} \Rightarrow y=\pm \frac{1}{\sqrt{2}}$
<br/>Four complex numbers.</div>


title: the-number-of-debroglie-wavelengths-contained-in-the-second-bohr-orbit-of-hydrogen-atom-is
slug: the-number-of-debroglie-wavelengths-contained-in-the-second-bohr-orbit-of-hydrogen-atom-is-85602
tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of de-Broglie wavelengths contained in the second Bohr orbit of hydrogen atom is</div>

answer:
<div class="solution">The number of de-Broglie wavelength contained in the second Bohr orbit of Hydrogen atom is 2</div>


title: the-number-of-diagonals-in-a-polygon-is-20-the-number-of-sides-of-the-polygon-is
slug: the-number-of-diagonals-in-a-polygon-is-20-the-number-of-sides-of-the-polygon-is-68043
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">8</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of diagonals in a polygon is 20 . The number of sides of the polygon is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">6</span> </li><li class="correct"> <span class="option-label">C</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">D</span> <span class="option-data">10</span> </li> </ul>

answer:
<div class="solution">Hints: \({ }^{\mathrm{n}} \mathrm{C}_2-\mathrm{n}=20\)<br/>\(\mathrm{n}=8\)</div>


title: the-number-of-diagonals-in-a-regular-polygon-of-100-sides-is
slug: the-number-of-diagonals-in-a-regular-polygon-of-100-sides-is-70786
question: <div class="question">The number of diagonals in a regular polygon of 100 sides is</div>
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4850</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4950</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">4850</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">4750</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">4650</span> </li> </ul>

answer:
<div class="solution">The number of diagonals in a regular polygon of $n$ sides is given by the following formula ${ }^{n} C_{2}-n$
<br/>$\therefore$ The number of diagonals in a regular polygon of 100 sides
<br/>$$
<br/>\begin{array}{l}
<br/>={ }^{100} C_{2}-100=\frac{100 \cdot 99}{2}-100 \\
<br/>=\frac{9900-200}{2}=\frac{9700}{2}=4850
<br/>\end{array}
<br/>$$</div>


title: the-number-of-digits-in-2-0-301-given-lo-g-10-2-03010-is
slug: the-number-of-digits-in-2-0-301-given-lo-g-10-2-03010-is-45206
question: <div class="question">The number of digits in $20^{301}$ (given, $\log _{10} 2=0.3010$ is</div>
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">392</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">602</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">301</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">392</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">391</span> </li> </ul>

answer:
<div class="solution">Let $y=20^{301}$
<br/>Number of digits $=$ Integral part of
<br/>$\left(301 \log _{10} 20\right)+1$
<br/>Integral part of $\left.\left[301 \log _{10} 10+\log _{10} 2\right)\right]+1$
<br/>$=$ Integral part of $[301(1+0.3010)]+1$
<br/>$=$ Integral part of $[301 \times 1.3010]+1$
<br/>Integral part of $[391.601]+1$
<br/>$=391+1=392$</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-number-of-distinct-real-roots-of-sin-x-cos-x-cos-x-cos-x-sin-x-cos-x-cos-x-cos-x-sin-x-0-in-the-interval-4-x-4-is-1

slug:
the-number-of-distinct-real-roots-of-sin-x-cos-x-cos-x-cos-x-sin-x-cos-x-cos-x-cos-x-sin-x-0-in-the-interval-4-x-4-is-1-37001

question:
<div class="question">The number of distinct real roots of $\left|\begin{array}{lll}\sin x &amp; \cos x &amp; \cos x \\ \cos x &amp; \sin x &amp; \cos x \\ \cos x &amp; \cos x &amp; \sin x\end{array}\right|=0$ in the interval $-\frac{\pi}{4} \leq x \leq \frac{\pi}{4}$ is</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">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">&gt;2</span> </li> </ul>

answer:
<div class="solution">Given, $\left|\begin{array}{ccc}\sin x &amp; \cos x &amp; \cos x \\ \cos x &amp; \sin x &amp; \cos x \\ \cos x &amp; \cos x &amp; \sin x\end{array}\right|=0$
<br/>$\sin x\left(\sin ^{2} x-\cos ^{2} x\right)-\cos x\left(\sin x \cos x-\cos ^{2} x \right ) + \cos x \left(\cos ^{2} x-\sin x \cos x\right)=0$
<br/>$\Rightarrow \sin x(\sin x-\cos x)(\sin x+\cos x)$ $-\cos ^{2} x(\sin x-\cos x)-\cos ^{2} x$ $(\sin x-\cos x)=0$
<br/>$\Rightarrow(\sin x-\cos x)\left(\sin ^{2} x+\sin x \cos x-2 \cos ^{2} x\right)=0$
<br/>$\Rightarrow(\sin x-\cos x)\left(\sin ^{2} x+2 \sin x\right.\left.\cos x-\sin x \cos x-2 \cos ^{2} x\right)=0$
<br/>$\Rightarrow(\sin x-\cos x)^{2}(\sin x+2 \cos x)=0$
<br/>$\Rightarrow \sin x-\cos x=0$ or $\sin x+2 \cos x=0$
<br/>$\Rightarrow \quad \tan x=1$ or $\tan x=-2$
<br/>But $\quad x \in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$
<br/>$\Rightarrow$ Only one solution i.e. $x=\frac{\pi}{4}$</div>


title: the-number-of-electron-in-2-coulomb-of-charge-is
slug: the-number-of-electron-in-2-coulomb-of-charge-is-19519
question: <div class="question">The number of electron in 2 coulomb of charge is</div>
tags: ['Physics', 'Electrostatics', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">$n=\frac{2}{1.6 \times 10^{-19}}=12.5 \times 10^{18}$</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$5 \times 10^{29}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$12.5 \times 10^{18}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.6 \times 10^{19}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$9 \times 10^{11}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$12.5 \times 10^{18}$</span> </div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-number-of-integer-values-of-m-for-which-the-x-coordinate-of-the-point-of-intersection-of-the-lines-3-x-4-y-9-and-y-m-x-1-is-also-an-integer-is

slug:
the-number-of-integer-values-of-m-for-which-the-x-coordinate-of-the-point-of-intersection-of-the-lines-3-x-4-y-9-and-y-m-x-1-is-also-an-integer-is-86080

question:
<div class="question">The number of integer values of $m$, for which the $x$ -coordinate of the point of intersection of the lines $3 x+4 y=9$ and $y=m x+1$ is also an integer, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">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">1</span> </li> </ul>

answer:
<div class="solution">Given lines are, $3 x+4 y=9$
<br/>$$
<br/>y=m x+1
<br/>$$
<br/>Solving Eqs. (i) and (ii), we get
<br/>$$
<br/>x=\frac{5}{3+4 m}
<br/>$$
<br/>If $m=-1,$ then $x=-5$ and $m=-2,$ then $x=-1$ are the only 
<br/>possibilities.</div>


title: the-number-of-irrational-terms-in-the-binomial-expansion-of-3-15-7-13-100-is
slug: the-number-of-irrational-terms-in-the-binomial-expansion-of-3-15-7-13-100-is-82663
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">94</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of irrational terms in the binomial expansion of $\left(3^{1 / 5}+7^{1 / 3}\right)^{100}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">90</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">88</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">93</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">94</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">General term of $\left(3^{1 / 5}+7^{1 / 3}\right)^{100}$ is given by $T_{\gamma+1}={ }^{100} C_{r}\left(3^{1 / 5}\right)^{100-r}\left(7^{1 / 3}\right)^{r}$
<br/>$={ }^{100} C_{1} \cdot 3^{\frac{100-r}{5}} \cdot 7^{\frac{r}{3}}$
<br/>For a rational term. $\frac{100-r}{5}$ and $\frac{r}{3}$ must be integer.
<br/>Hence, $r=0,15,30,45,60,75,90$
<br/>$\therefore$ There are seven rational terms. Hence, number of irrational terms
<br/>$$
<br/>=101-7=94
<br/>$$</div>


title: the-number-of-irrational-terms-in-the-expansion-of-3-8-1-5-4-1-84-is
slug: the-number-of-irrational-terms-in-the-expansion-of-3-8-1-5-4-1-84-is-23418
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">74</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of irrational terms in the expansion of $\left(3^{\frac{1}{8}}+5^{\frac{1}{4}}\right)^{84}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">73</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">74</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">75</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">76</span> </li> </ul>

answer:
<div class="solution">Given, $\left(3^{\frac{1}{8}}+5^{\frac{1}{4}}\right)^{84}$
<br/>$\begin{aligned} \text { here, } T_{r+1} &amp;={ }^{84} C_{r}\left(3^{\frac{1}{8}}\right)^{84-r}\left(5^{\frac{1}{4}}\right)^{r} \\ &amp;={ }^{84} C_{r}\left(3^{\frac{84-r}{8}}\right) \cdot 5^{\frac{r}{4}} \end{aligned}$
<br/>since, $T_{r}=1$ is rational for $r=4,12,20,28,36,44$ 52,60,68,76,84
<br/>no. of rational terms $=11$
<br/>Here, total number of terms $=85$ Number of irrational terms $=85-11$
<br/>$$
<br/>=74
<br/>$$</div>


title: the-number-of-lines-which-pass-through-the-point-23-and-are-at-a-distance-8-from-the-point-12-is
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P3.db
slug:
the-number-of-lines-which-pass-through-the-point-23-and-are-at-a-distance-8-from-the-point-12-is-84937

question:
<div class="question">The number of lines which pass through the point (2,-3) and are at a distance 8 from the point (-1,2) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">infinite</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">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">The distance from the point (-1,2) of the line which passes through (2,-3)
<br/>$$
<br/>\begin{array}{l}
<br/>=\sqrt{(2+1)^{2}+(-3-2)^{2}} \\
<br/>=\sqrt{9+25}=\sqrt{34}     &lt; 8
<br/>\end{array}
<br/>$$
<br/>Given that, the distance between the point (-1,2) and the line is 8 But the maximum distance of the line passing through (2,-3) from (-1,2) is $\sqrt{34}$. So, there is no such line possible.</div>


tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
title:
the-number-of-lone-pairs-of-electrons-on-the-central-atoms-of-h-2-o-sncl-2-pcl-3-an-xef-2-respectively-are

slug:
the-number-of-lone-pairs-of-electrons-on-the-central-atoms-of-h-2-o-sncl-2-pcl-3-an-xef-2-respectively-are-43736

question:
<div class="question">The number of lone pairs of electrons on the central atoms of $\mathrm{H}_{2} \mathrm{O}, \mathrm{SnCl}_{2}, \mathrm{PCl}_{3}$ an $\mathrm{XeF}_{2}$
<br/>respectively, are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">2,1,1,3</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2,2,1,3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3.1,1,2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2,1,2,3</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">2,1,1,3</span> </div>

answer:
<div class="solution">The number of lone pairs of electrons on the central atoms of $\mathrm{H}_{2} \mathrm{O}, \mathrm{SnCl}_{2}, \mathrm{PCl}_{3}$ and $\mathrm{XeF}_{2}$ are 2 .
<br/>1,1 and 3 respectively.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_a47.png"/></div>


title: the-number-of-onto-functions-from-the-set-1-2-11-to-the-set-1-2-10-is
slug: the-number-of-onto-functions-from-the-set-1-2-11-to-the-set-1-2-10-is-76472
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of onto functions from the set $\{1,2, \ldots, 11\}$ to the set $\{1,2, \ldots, 10\}$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$10 \times 11 !$</span> </div>

answer:
<div class="solution">Let $\quad A=\{1,2, \ldots, 11\}$
<br/>$\begin{array}{ll}\therefore &amp; n(A)=11 \text { and } B=\{1,2, \ldots, 10\} \\ \therefore &amp; n(B)=10\end{array}$
<br/>$\therefore \text{Hence number of onto function}$ $=^{n(A)} C_{n(B)} \times n(B) ! \times n(B)$
<br/>$={ }^{11} C_{10} \times 10 ! \times 10$
<br/>$=(11 \times 10 !) \times 10=11 ! \times 10$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">576</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-number-of-permutations-by-taking-all-letters-and-keeping-the-vowels-of-the-word-combine-in-the-odd-places-is

slug:
the-number-of-permutations-by-taking-all-letters-and-keeping-the-vowels-of-the-word-combine-in-the-odd-places-is-83979

question:
<div class="question">The number of permutations by taking all letters and keeping the vowels of the word COMBINE in the odd places is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">96</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">144</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">512</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">576</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Hints: Vowels : O, I, E<br/>No. of Odd place : 4<br/>No of ways $={ }^4 \mathrm{P}_3 \times 4 !=576$</div>


title: the-number-of-points-at-which-the-function-f-x-max-a-x-a-x-b-x-0-a-b-cannot-be-differentiable-is
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P3.db
slug:
the-number-of-points-at-which-the-function-f-x-max-a-x-a-x-b-x-0-a-b-cannot-be-differentiable-is-45182

question:
<div class="question">The number of points at which the function $f(x)=\max \{a-x, a+x, b\},-\infty    &lt; x &lt; \infty$ $0 &lt; a &lt; b$ cannot be differentiable, is</div>

answer:
<div class="solution">Graph of $f(x)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_mat_a60.png"/> <br/> <br/>There are two sharp turns. Hence, $f(x)$ cannot be differentiable at two points.</div>


title: the-number-of-points-of-intersection-of-2-y-1-and-y-sin-x-in-2-x-2-is
slug: the-number-of-points-of-intersection-of-2-y-1-and-y-sin-x-in-2-x-2-is-99002
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">$$<br/>\text { The number of points of intersection of } 2 y=1 \text { and } y=\sin x \text {, in }-2 \pi \leq x \leq 2 \pi \text { is }<br/>$$</div>

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

answer:
<div class="solution">Hints : $y=\frac{1}{2}=\sin \mathrm{x}$<br/>$-2 \pi \leq x \leq 2 \pi$<br/>$$<br/>x=\frac{\pi}{6}, \frac{5 \pi}{6},-\frac{7 \pi}{6},-\frac{11 \pi}{6}<br/>$$<br/>No. of sol $^{\mathrm{n}} 4$</div>


title: the-number-of-points-on-the-line-x-y-4-which-are-unit-distance-apart-from-the-line-2-x-2-y-5-is
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P3.db
slug:
the-number-of-points-on-the-line-x-y-4-which-are-unit-distance-apart-from-the-line-2-x-2-y-5-is-75959

question:
<div class="question">The number of points on the line $x+y=4$ which are unit distance apart from the line $2 x+2 y=5$ is</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">Infinity</span> </li> </ul>

answer:
<div class="solution">Hints : $x+y=4$<br/>$$<br/>\begin{aligned}<br/>&amp; x+y=\frac{5}{2} \\<br/>&amp; \mathrm{PQ}=\frac{4-5 / 2}{\sqrt{2}}=\frac{3}{2 \sqrt{2}}=\frac{3 \sqrt{2}}{4}<br/>\end{aligned}<br/>$$</div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-number-of-possible-organobromine-compounds-which-can-be-obtained-in-the-allylic-bromination-of-1-butene-with-nbromosuccinimide-is

slug:
the-number-of-possible-organobromine-compounds-which-can-be-obtained-in-the-allylic-bromination-of-1-butene-with-nbromosuccinimide-is-32849

question:
<div class="question">The number of possible organobromine compounds which can be obtained in the allylic bromination of 1 -butene with N-bromosuccinimide is</div>

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

answer:
<div class="solution">NBS (N-Bromosuccinimide) does allylic bromination through free radical substitution.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_72.png"/> <br/>\\(	herefore\) Total 4 isomers are possible.</div>


title: the-number-of-real-roots-of-equation-l-o-g-e-x-e-x-0-is
slug: the-number-of-real-roots-of-equation-l-o-g-e-x-e-x-0-is-27332
question: <div class="question">The number of real roots of equation $log_ex+e x=0$ is</div>
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: 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">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">We have, $\log _{e} x+e x=0$
<br/>$\log _{e} x=-e x$
<br/>Since, both the graphs intersect at only one since.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_mat_a113.png"/> <br/>Hence, the number of real roots of equation is one</div>


title: the-number-of-real-solutions-of-the-equation-sin-x-x-cos-x-x-2-0-is
slug: the-number-of-real-solutions-of-the-equation-sin-x-x-cos-x-x-2-0-is-62716
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">3</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of real solutions of the equation $(\sin x-x)\left(\cos x-x^{2}\right)=0$ is</div>

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

answer:
<div class="solution">Given, $(\sin x-x)\left(\cos x-x^{2}\right)=0$
<br/>$\Rightarrow \quad \sin x=x$ or $\cos x=x^{2}$
<br/>Now, if $\sin x=x,$ then only one solution ie $x=0$ is possible. Also, if $\cos x=x^{2}$, then two solutions are possible. Hence, there are total 3 solutions.</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-number-of-real-values-of-for-which-the-system-of-equations-x-3-y-5-z-x-5-x-y-3-z-y-3-x-5-y-z-z-has-infinite-number-of-solutions-is

slug:
the-number-of-real-values-of-for-which-the-system-of-equations-x-3-y-5-z-x-5-x-y-3-z-y-3-x-5-y-z-z-has-infinite-number-of-solutions-is-76014

question:
<div class="question">The number of real values of $\alpha$ for which the system of equations
<br/>$
<br/>\begin{array}{l}
<br/>x+3 y+5 z=\alpha x \\
<br/>5 x+y+3 z=\alpha y
<br/>\end{array}
<br/>$
<br/>$3 x+5 y+z=\alpha z$
<br/>has infinite number of solutions is</div>

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

answer:
<div class="solution">The system of equations are
<br/>$$
<br/>\begin{array}{l}
<br/>x+3 y+5 z=\alpha x \\
<br/>5 x+y+3 z=\alpha y \\
<br/>3 x+5 y+z=\alpha z \\
<br/>(1-\alpha) x+3 y+5 z=0 \\
<br/>5 x+(1-\alpha) y+3 z=0 \\
<br/>3 x+5 y+(1-\alpha) z=0
<br/>\end{array}
<br/>$$
<br/>For infinite number of solutions, we must have $\left|\begin{array}{ccc}1-\alpha &amp; 3 &amp; 5 \\ 5 &amp; 1-\alpha &amp; 3 \\ 3 &amp; 5 &amp; 1-\alpha\end{array}\right|=0$
<br/>$\Rightarrow \quad\left|\begin{array}{ccc}9-\alpha &amp; 3 &amp; 5 \\ 9-\alpha &amp; 1-\alpha &amp; 3 \\ 9-\alpha &amp; 5 &amp; 1-\alpha\end{array}\right|=0,$
<br/>$C_{1} \rightarrow C_{2}+C_{2}+C_{3}$
<br/>Taking $(9-\alpha)$ common from the first column, we get $\begin{array}{l}\Rightarrow \quad(9-\alpha)\left|\begin{array}{ccc}1 &amp; 3 &amp; 5 \\ 1 &amp; 1-\alpha &amp; 3 \\ 1 &amp; 5 &amp; 1-\alpha\end{array}\right|=0 \\ \Rightarrow &amp; (9-\alpha)\left|\begin{array}{ccc}1 &amp; 3 &amp; 5 \\ 0 &amp; -\alpha-2 &amp; -2 \\ 0 &amp; 2 &amp; -\alpha-4\end{array} \mid=0\right. \\ \Rightarrow &amp; (9-\alpha)\left|\begin{array}{cc}\alpha+2 &amp; 2 \\ -2 &amp; \alpha+4\end{array}\right|=0 \\ \Rightarrow &amp; (9-\alpha)\left(\alpha^{2}+6 \alpha+8+4\right)=0 \\ (\alpha-9)\left(\alpha^{2}+6 \alpha+12\right)=0\end{array}$
<br/>$\Rightarrow \alpha=9$ is the only real value of $\alpha$.</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
title:
the-number-of-selection-of-n-objects-from-2-n-objects-of-which-n-are-identical-and-the-rest-are-different-is

slug:
the-number-of-selection-of-n-objects-from-2-n-objects-of-which-n-are-identical-and-the-rest-are-different-is-31081

question:
<div class="question">The number of selection of $n$ objects from $2 n$ objects of which $n$ are identical and the rest are different, is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2^{n}$</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">$2^{n-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2^{n}-1$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2^{n-1}+1$</span> </li> </ul>

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

answer:
<div class="solution">Number of ways of selection of $n$ objects from $2 n$ objects, where as $n$ objects are identical in out of $2 n$ objects. $n$ identical and no different object $=1$ ways
<br/>$$
<br/>={ }^{n} C_{0}
<br/>$$
<br/>$n-1$ identical and 1 different object $=1 \times{ }^{n} C_{1}$
<br/>$n-2$ identical and 2 different object $=1 \times{ }^{n} C_{2}$
<br/>0 identical and $n$ different objects $=1 \times{ }^{n} C_{n}$
<br/>$$
<br/>={ }^{n} C_{0}+{ }^{n} C_{1}+{ }^{n} C_{2}+\ldots \ldots+{ }^{n} C_{n}=2^{n}
<br/>$$</div>


title: the-number-of-solutions-of-2-sin-xcos-x3-is
slug: the-number-of-solutions-of-2-sin-xcos-x3-is-26642
question: <div class="question">The number of solutions of \(2 \sin x+\cos x=3\) is</div>
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2011']
answer: <div class="solution">Hints: \(\sqrt{5} &lt; 3\) No solution</div>
source_db: 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">2</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">infinite</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">No solution</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">No solution</span> </div>


title: the-number-of-solutions-of-the-equation-2-1-lo-g-3-x-5-x-1-lo-g-9-x-5-2-1-is
slug: the-number-of-solutions-of-the-equation-2-1-lo-g-3-x-5-x-1-lo-g-9-x-5-2-1-is-88451
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of solutions of the equation $\frac{1}{2} \log _{\sqrt{3}}\left(\frac{x+1}{x+5}\right)+\log _{9}(x+5)^{2}=1$ is</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">infinite</span> </li> </ul>

answer:
<div class="solution">Given equation is $\frac{1}{2} \log _{\sqrt{3}}\left(\frac{x+1}{x+5}\right)+\log _{9}(x+5)^{2}=1$
<br/>$\Rightarrow \quad \frac{1}{2} \cdot\left(\frac{1}{1 / 2}\right) \log _{3}\left(\frac{x+1}{x+5}\right)+\frac{1}{2} \log _{3}(x+5)^{2}=1$
<br/>$\left(\because \log _{a^{n}} b=\frac{1}{n} \log _{a} b\right)$
<br/>$\Rightarrow \quad \frac{2}{2} \log _{3}\left(\frac{x+1}{x+5}\right)+\frac{1}{2} \cdot 2 \log _{3}(x+5)=\log _{3} 3$
<br/>$\Rightarrow \quad \log _{3}\left\{\left(\frac{x+1}{x+5} \cdot(x+5)\right)\right\}=\log _{3} 3$
<br/>$\left(\because \log m+\log n=\log m n\right.$ and $\log _{n} n=1$
<br/>$\Rightarrow \quad(x+1)=3 \Rightarrow x=2$
<br/>So, only one solution is possible.</div>


title: the-number-of-solutions-of-the-equation-lo-g-2-x-2-2-x-1-1-is
slug: the-number-of-solutions-of-the-equation-lo-g-2-x-2-2-x-1-1-is-11660
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of solutions of the equation $\log _{2}\left(x^{2}+2 x-1\right)=1$ is</div>

answer:
<div class="solution">Given $\log _{2}\left(x^{2}+2 x-1\right)=1$
<br/>$\Rightarrow \quad \log _{2}\left(x^{2}+2 x-1\right)=\log _{2} 2$
<br/>$\Rightarrow$ $x^{2}+2 x-1=2$
<br/>$\Rightarrow \quad x^{2}+2 x-3=0$
<br/>$\Rightarrow \quad x^{2}+3 x-x-3=0$
<br/>$\Rightarrow \quad x(x+3)-1(x+3)=0$
<br/>$=(x+3)(x-1)=0$
<br/>$x=1,-3$
<br/>Since, $x=1$ and $x=-3$ are satisfy the given equation therefore the number of solutions of the equation are two.</div>


title: the-number-of-solutions-of-the-equation-x-1-x-1-4-x-1-isare
slug: the-number-of-solutions-of-the-equation-x-1-x-1-4-x-1-isare-90295
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of solution(s) of the equation $\sqrt{x+1}-\sqrt{x-1}=\sqrt{4 x-1}$ is/are</div>

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

answer:
<div class="solution">Given equation is
<br/>$$
<br/>\sqrt{x+1}-\sqrt{x-1}=\sqrt{4 x-1}
<br/>$$
<br/>On squaring both sides, we get
<br/>$$
<br/>(x+1)+(x-1)-2 \sqrt{x^{2}-1}=4 x-1
<br/>$$
<br/>$\Rightarrow \quad 2 x-2 \sqrt{x^{2}-1}=4 x-1$
<br/>$\Rightarrow \quad-2 \sqrt{x^{2}-1}=2 x-1$
<br/>Again, squaring both sides, we get
<br/>$$
<br/>4\left(x^{2}-1\right)=4 x^{2}+1-4 x
<br/>$$
<br/>$\Rightarrow \quad-4=+1-4 x$
<br/>$\Rightarrow \quad 4 x=5$
<br/>$\Rightarrow$
<br/>$$
<br/>x=\frac{5}{4}
<br/>$$
<br/>But, when we put $x=\frac{5}{4}$ in Eq. (i), we get
<br/>$$
<br/>\sqrt{\frac{5}{4}+1}-\sqrt{\frac{5}{4}-1}=\sqrt{4 \times \frac{5}{4}-1}
<br/>$$
<br/>$\Rightarrow$
<br/>$\sqrt{\frac{9}{4}}-\sqrt{\frac{1}{4}}=\sqrt{5-1} \Rightarrow \frac{3}{2}-\frac{1}{2}=2$
<br/>$\Rightarrow 1=2,$ which is not true.
<br/>Hence, no value of $x$ satisfy the given equation.</div>


title: the-number-of-solutions-of-the-equation-x-y-z-10-where-x-y-and-z-are-positive-integers
slug: the-number-of-solutions-of-the-equation-x-y-z-10-where-x-y-and-z-are-positive-integers-41927
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">36</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of solutions of the equation $x+y+z=10$ where $x, y$ and $z$ are positive integers</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">36</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">55</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">72</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">45</span> </li> </ul>

answer:
<div class="solution">Given equation, is $x+y+z=10$
<br/>where, $x, y$ and $z$ are positive integers.
<br/>$\therefore$ Required number of solutions $={ }^{(10-1)} \mathrm{C}_{(3-1)}$
<br/>$$
<br/>={ }^{9} C_{2}=\frac{9 \times 8}{2}=36
<br/>$$</div>


title: the-number-of-unpaired-electron-in-mn-2-ion-is
slug: the-number-of-unpaired-electron-in-mn-2-ion-is-78630
question: <div class="question">The number of unpaired electron in $\mathrm{Mn}^{2+}$ ion is</div>
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: 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="correct"> <span class="option-label">C</span> <span class="option-data">5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">6</span> </li> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/_DzU_p0ANxtmZr01GUFVcB7uZBC4UcUF8SxUAnyOjPU.original.fullsize.png"/></div>


title: the-number-of-unpaired-electrons-in-ni-atomic-number-28-are
slug: the-number-of-unpaired-electrons-in-ni-atomic-number-28-are-38670
question: <div class="question">The number of unpaired electrons in Ni (atomic number $=28$ ) are</div>
tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: 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">4</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">8</span> </li> </ul>

answer:
<div class="solution">The number of unpaired electrons in $\mathrm{Ni}$ (atomic number $=28$ ) are 2 . It can be easily concluded from the electronic configuration of $\mathrm{kL}$. Electronic configuration of $$
<br/>\mathrm{Ni}=\mathrm{l} s^{2} 2 \mathrm{s}^{2} 2 p^{6} 3 \mathrm{s}^{2} 3 p^{6} 4 \mathrm{s}^{2} \mathrm{W}^{8}
<br/>$$
<br/>Orbitals of $\mathrm{Ni}$ in $3 d=$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a53.png"/></div>


title: the-number-of-unpaired-electrons-in-nicl-4-2-ni-co-4-and-cu-nh-3-4-2-respectively-are-1
slug: the-number-of-unpaired-electrons-in-nicl-4-2-ni-co-4-and-cu-nh-3-4-2-respectively-are-1-40305
tags: ['Chemistry', 'Coordination Compounds', 'JEE Main']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2,0,1</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of unpaired electrons in
<br/>$\quad\left[\mathrm{NiCl}_{4}\right]^{2-}, \mathrm{Ni}(\mathrm{CO})_{4}$ and $\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)_{4}\right]^{2}$
<br/>respectively are</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2,2,1</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">2,0,1</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">0,2,1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2,2,0</span> </li> </ul>

answer:
<div class="solution">Configuration for metal (M) in various complex.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a52.png"/> <br/> <br/>$\mathrm{Ni}^{2 *}$ has 2 unpaired electrons<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a52a.png"/> <br/> <br/>$\mathrm{Ni}^{0}$ has zero unpaired electrons with
<br/>$s p^{3}$ -hybridisation.
<br/>$\mathrm{Cu}^{2}$ "has one unpaired electron with $d s p^{2}$ -hybridisation.<img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a52b.png"/> <br/> <br/>$\therefore$ (b) is the correct answer, i.e.
<br/>Number of unpaired electron (2,0,1)</div>


title: the-number-of-unpaired-electrons-in-the-uranium-92-u-atom-is
slug: the-number-of-unpaired-electrons-in-the-uranium-92-u-atom-is-67249
tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of unpaired electrons in the uranium $\left({ }_{92} \mathrm{U}\right)$ atom is:</div>

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

answer:
<div class="solution">Hint: 
<br/>$_{92} \mathrm{U}:\left[\mathrm{R}_{\mathrm{n}}\right]^{\mathrm{g} 8} 5 \mathrm{f}^{3} 6 \mathrm{~d}^{1} 7 \mathrm{~s}^{2}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-number-of-values-of-k-for-which-the-equation-x-2-3-x-k-0-has-two-distinct-roots-lying-in-the-interval-0-1-are

slug:
the-number-of-values-of-k-for-which-the-equation-x-2-3-x-k-0-has-two-distinct-roots-lying-in-the-interval-0-1-are-19010

question:
<div class="question">The number of values of $k,$ for which the equation $x^{2}-3 x+k=0$ has two distinct roots lying in the interval $(0,1),$ are</div>

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

answer:
<div class="solution">Given equation is
<br/>$$
<br/>x^{2}-3 x+k=0
<br/>$$
<br/>As.
<br/>$$
<br/>-\frac{b}{2 a}=\frac{3}{2} \notin (0,1)
<br/>$$
<br/>Since, both roots cannot lie between 0 and
<br/>So, no value of $k$ is possible.</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-number-of-ways-in-which-the-letters-of-the-word-arrange-can-be-permuted-such-that-the-rs-occur-together-is

slug:
the-number-of-ways-in-which-the-letters-of-the-word-arrange-can-be-permuted-such-that-the-rs-occur-together-is-18209

question:
<div class="question">The number of ways in which the letters of the word ARRANGE can be permuted such that the R's occur together, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{7 !}{2!2 !}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{7!}{2!}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\frac{6 !}{2!}$</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">$5 ! \times 2!$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{6 !}{2!}$</span> </div>

answer:
<div class="solution">Given word is 'ARRANGE'.
<br/>If R's occur together, number of letters in word = 6 
<br/>To arrange R's together number of ways $=\frac{2 !}{2 !}=1$
<br/>$\therefore$ Number of ways to permuted 'ARRANGE' = $\frac{6!}{2 !}$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
the-number-of-ways-in-which-the-letters-of-the-word-vertical-can-be-arranged-without-changing-the-order-of-the-vowels-is

slug:
the-number-of-ways-in-which-the-letters-of-the-word-vertical-can-be-arranged-without-changing-the-order-of-the-vowels-is-22029

question:
<div class="question">The number of ways in which the letters of the word 'VERTICAL' can be arranged without changing the order of the vowels is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{8 !}{3 !}\)</span> </div>

answer:
<div class="solution">Hint : Vowels 'EIA' to be kept in order, so out of 8 places 3 places can be chosen in \({ }^8 \mathrm{C}_3\) ways. Remaining 5 letters can be arranged in 5 ! ways.<br/>\(\therefore\) Total number of ways \(={ }^8 \mathrm{C}_3 \times 5 !=\frac{8 !}{3 !}\)</div>


title: the-number-of-words-that-can-be-written-using-all-the-letters-of-the-word-irrational-is
slug: the-number-of-words-that-can-be-written-using-all-the-letters-of-the-word-irrational-is-60807
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The number of words that can be written using all the letters of the word "IRRATIONAL' is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{10 !}{(2 !)^{3}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{10 !}{(21)^{2}}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{10 !}{2 !}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">10 !</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{10 !}{(2 !)^{3}}$</span> </div>

answer:
<div class="solution">These are 10 letters in the word IRRATIONAL in which these are $2^{4} I, 2 \cdot R^{\prime}$ and $2 \cdot A^{\prime}$
<br/>$\therefore$ The total number of word $s=\frac{10 !}{2 ! 2 ! 2 !}=\frac{10 !}{(21)^{3}}$</div>


title: the-number-of-zeros-at-the-end-of-100-is
slug: the-number-of-zeros-at-the-end-of-100-is-70545
question: <div class="question">The number of zeros at the end of 100! is</div>
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">24</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">21</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">22</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">23</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">24</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\mathrm{E}_5(100 !)=\left[\frac{100}{5}\right]+\left[\frac{100}{25}\right]+\left[\frac{100}{125}\right]=20+4+0=24$</div>


title: the-optically-active-molecule-is
slug: the-optically-active-molecule-is-53402
question: <div class="question">The optically active molecule is</div>
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q19_opta.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q19_optb.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q19_optc.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q19_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_q19_optc.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a19.png"/> <br/> <br/>Only molecule (c) is optically active. Other moelcules are meso, due to presence of plane of symmetry.</div>


tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-orange-solid-on-heating-gives-a-colourless-gas-and-a-green-solid-which-can-be-reduced-to-metal-by-aluminium-powder-the-orange-and-the-green-solids

slug:
the-orange-solid-on-heating-gives-a-colourless-gas-and-a-green-solid-which-can-be-reduced-to-metal-by-aluminium-powder-the-orange-and-the-green-solids-45645

question:
<div class="question">The orange solid on heating gives a colourless gas and a green solid which can be reduced to metal by aluminium powder. The orange and the green solids are, respectively</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{Na}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7}$ and $\mathrm{Cr}_{2} \mathrm{O}_{3}$</span> </div>

answer:
<div class="solution">Orange solid is ammonium dichromate.
<br/>$\left(\mathrm{NH}_{4}\right)_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} \stackrel{\Delta}{\longrightarrow} \mathrm{N}_{2} \uparrow+\mathrm{Cr}_{2} \mathrm{O}_{3}+4 \mathrm{H}_{2} \mathrm{O}$
<br/>$$
<br/>\mathrm{Cr}_{2} \mathrm{O}_{3}+2 \mathrm{Al} \longrightarrow 2 \mathrm{Cr}+\mathrm{Al}_{2} \mathrm{O}_{3}
<br/>$$</div>


title: the-order-of-decreasing-ease-of-abstraction-of-hydrogen-atoms-in-the-following-molecule
slug: the-order-of-decreasing-ease-of-abstraction-of-hydrogen-atoms-in-the-following-molecule-53521
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The order of decreasing ease of abstraction of hydrogen atoms in the following molecule <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q49.png"/></div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{H}_{a}&gt;\mathrm{H}_{c}&gt;\mathrm{H}_{\mathrm{b}}$</span> </div>

answer:
<div class="solution">Abstraction of $\left(\mathrm{H}_{a}\right)$ hydrogen atom results in the formation of $3^{\circ}$ allylic carbocation wheres abstraction of $\mathrm{H}_{c}$ gives $2^{\circ}$ allylic carbocation (which is less stable as compared to former one). However, abstraction of $\mathrm{H}_{b}$ gives only a $2^{\circ}$ carbocation which is less stable as compared to former two. More the stability of generated carbocation, higher is the tendency of abstraction of hydrogen. Thus, the order of decreasing ease of abstraction of hydrogen atoms from the given molecule is
<br/>$$
<br/>\mathrm{H}_{a}&gt;\mathrm{H}_{c}&gt;\mathrm{H}_{b}
<br/>$$</div>


title: the-order-of-equivalent-conductances-at-infinite-dilution-for-licl-nacl-and-kcl-is
slug: the-order-of-equivalent-conductances-at-infinite-dilution-for-licl-nacl-and-kcl-is-91396
tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The order of equivalent conductances at infinite dilution for LiCl, $\mathrm{NaCl}$ and $\mathrm{KCl}$ is</div>

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

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

answer:
<div class="solution">KCl is more ionic than NaCl while NaCl is more ionic than LiCl
<br/>Also, more the ionic nature, more is the conductivity thus, order of equivalent conductance at infinite dilution is $\mathrm{KCl}&gt;\mathrm{NaCl}&gt;\mathrm{LiCl}$</div>


title: the-order-of-the-differential-equation-d-x-2-d-2-y-1-d-x-d-y-2-is
slug: the-order-of-the-differential-equation-d-x-2-d-2-y-1-d-x-d-y-2-is-68402
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
answer: <div class="solution">No Solution Available</div>
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The order of the differential equation $\frac{d^2 y}{d x^2}=\sqrt{1-\left(\frac{d y}{d x}\right)^2}$ is</div>

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


title: the-order-of-the-differential-equation-of-all-parabolas-whose-axis-of-symmetry-along-x-axis-is
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P3.db
slug:
the-order-of-the-differential-equation-of-all-parabolas-whose-axis-of-symmetry-along-x-axis-is-49335

question:
<div class="question">The order of the differential equation of all parabolas whose axis of symmetry along $X$ -axis is</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">3</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">None of these</span> </li> </ul>

answer:
<div class="solution">The general equation of the parabola is $x=a y^{2}+b$.
<br/>where a and $b$ are arbitrary constants.
<br/>So, the differential equation is of order 2.</div>


title: the-ore-chromite-is
slug: the-ore-chromite-is-24943
question: <div class="question">The ore chromite is</div>
tags: ['Chemistry', 'General Principles and Processes of Isolation of Metals', 'WBJEE', 'WBJEE 2013']
answer: <div class="solution">Ore chomite is $\mathrm{FeCr}_{2} \mathrm{O}_{4}$</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{FeCr}_{2} \mathrm{O}_{4}$</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">$\mathrm{CoCr}_{2} \mathrm{O}_{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{CrFe}_{2} \mathrm{O}_{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{FeCr}_{2} \mathrm{O}_{3}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{FeCr}_{2} \mathrm{O}_{4}$</span> </div>


title: the-output-y-of-the-logic-circuit-given-below-is
slug: the-output-y-of-the-logic-circuit-given-below-is-27918
tags: ['Physics', 'Semiconductors', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The output Y of the logic circuit given below is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_q5.png"/></div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_phy_a5.png"/> <br/> <br/>By absorption laws
<br/>$$
<br/>Y=\bar{A}+\bar{A} \cdot B
<br/>$$
<br/>or $\quad Y=(\bar{A} \cdot B)+\bar{A}=\bar{A} \cdot(B+1)=\bar{A} \cdot 1=\bar{A}$</div>


title: the-oxidant-which-is-used-as-an-antiseptic-is
slug: the-oxidant-which-is-used-as-an-antiseptic-is-39890
question: <div class="question">The oxidant which is used as an antiseptic is :</div>
tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2009']
answer: <div class="solution">No Solution Available</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{KBrO}_3$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{KMnO}_4$</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">$\mathrm{CrO}_3$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">Merrifield resin</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{KMnO}_4$</span> </div>


title: the-ozone-layer-forms-naturally-by
slug: the-ozone-layer-forms-naturally-by-81186
question: <div class="question">The ozone layer forms naturally by</div>
tags: ['Chemistry', 'Environmental Chemistry', 'WBJEE', 'WBJEE 2011']
answer: <div class="solution">No Solution Available</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">the interaction of CFC with oxygen</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">the interaction of UV radiation with oxygen</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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 interaction of IR radiation with oxygen</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">the interaction of oxygen and water vapour.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the interaction of UV radiation with oxygen</span> </div>


title: the-paramagnetic-behaviour-of-b-2-is-due-to-the-presence-of
slug: the-paramagnetic-behaviour-of-b-2-is-due-to-the-presence-of-49596
tags: ['Chemistry', 'Chemical Bonding and Molecular Structure', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The paramagnetic behaviour of $\mathrm{B}_{2}$ is due to the presence of</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">two unpaired electrons in $\pi^{*}$ MO</span> </div>

answer:
<div class="solution">$B_{2}=5+5=10 e^{-}$
<br/>$$
<br/>=\sigma 1 s^{2} \dot{\sigma} 1 s^{2}, \sigma 2 s^{2} \dot{\sigma} 2 s^{2}, \pi 2 p_{x}^{1}, \pi 2 p_{y}^{\prime}
<br/>$$
<br/>Due to the presence of 2 unpaired electrons, in $\pi$ bonding orbitals, $B_{2}$ shows paramagnetic behaviour.</div>


title: the-perfect-gas-equation-for-4-g-of-hydrogen-gas-is
slug: the-perfect-gas-equation-for-4-g-of-hydrogen-gas-is-85011
question: <div class="question">The perfect gas equation for $4 \mathrm{g}$ of hydrogen gas is</div>
tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p V=R T$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$p V=2 R T$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$p V=\frac{1}{2} R T$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$p V=4 R T$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$p V=2 R T$</span> </div>

answer:
<div class="solution">Mass of gas $=4 \mathrm{g} \quad$ (given)
<br/>From pertect gas equation
<br/>$$
<br/>p V=n R T
<br/>$$
<br/>where $n=$ number of moles here, $\left(n=\frac{4}{2}=2\right)$
<br/>$\therefore \quad p V=2 R T$
<br/>It will be the perfect gas equation for $4 \mathrm{g}$ of hydrogen.</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P3.db
title:
the-period-of-oscillation-of-a-simple-pendulum-of-length-l-suspended-from-the-roof-of-a-vehicle-which-moves-without-friction-down-an-inclined-plane-of-1

slug:
the-period-of-oscillation-of-a-simple-pendulum-of-length-l-suspended-from-the-roof-of-a-vehicle-which-moves-without-friction-down-an-inclined-plane-of-1-17228

question:
<div class="question">The period of oscillation of a simple pendulum of length \(l\) suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination \(\alpha\), is given by</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(2 \pi \sqrt{\frac{1}{g \cos \alpha}}\)</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/MXvNY4ii3w0_0o73_UYY7N7aDcSAlya3M-8_xrU2KrA.original.fullsize.png"/><br/><br/>\(\begin{aligned}<br/>&amp; \mathrm{g}_{\text {eff }}=\mathrm{g} \cos \alpha \\<br/>&amp; \mathrm{T}=2 \pi \sqrt{\frac{\ell}{\mathrm{g}_{\text {eff }}}}<br/>\end{aligned}\)</div>


title: the-period-of-the-function-fxcos-4-xtan-3-x-is
slug: the-period-of-the-function-fxcos-4-xtan-3-x-is-68325
question: <div class="question">The period of the function \(f(x)=\cos 4 x+\tan 3 x\) is</div>
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2011']
answer: <div class="solution">Hints: \(\operatorname{LCM}\left(\frac{2 \pi}{4}, \pi / 3\right)=\pi\)</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\pi\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\frac{\pi}{2}\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(\frac{\pi}{3}\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\frac{\pi}{4}\)</span> </li> </ul>

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


title: the-ph-of-1-0-4-m-koh-solution-will-be
slug: the-ph-of-1-0-4-m-koh-solution-will-be-92724
question: <div class="question">The pH of $10^{-4} \mathrm{M}$ KOH solution will be</div>
tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10</span> </div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">11</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">10.5</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">10</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$10^{-4} \mathrm{M} \mathrm{KOH}=10^{-4} \mathrm{M}\left[\mathrm{OH}^{-}\right]$
<br/>We know that
<br/>$$
<br/>\left.\left[\mathrm{H}^{+}\right] \mathrm{IOH}^{-1}\right]=1 \times 10^{-14}
<br/>$$
<br/>$$
<br/>\begin{array}{c}
<br/>{\left[\mathrm{H}^{+}\right] \times 10^{-4}=1 \times 10^{-14}} \\
<br/>{\left[\mathrm{H}^{+}\right]=\frac{1 \times 10^{-14}}{1 \times 10^{-4}}=1 \times 10^{-10} \mathrm{M}} \\
<br/>\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]=-\log \left(1 \times 10^{-10}\right) \\
<br/>=10
<br/>\end{array}
<br/>$$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-plane-x-my-0-is-rotated-about-its-line-of-intersection-with-the-plane-z-0-through-an-angle-the-equation-changes-to

slug:
the-plane-x-my-0-is-rotated-about-its-line-of-intersection-with-the-plane-z-0-through-an-angle-the-equation-changes-to-40021

question:
<div class="question">The plane $\ell \mathrm{x}+\mathrm{my}=0$ is rotated about its line of intersection with the plane $\mathrm{z}=0$ through an angle $\alpha$. The equation changes to</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\ell \mathrm{x}+\mathrm{my} \pm \mathrm{z} \tan \alpha \sqrt{\ell^{2}+\mathrm{m}^{2}}=0$</span> </div>

answer:
<div class="solution">$P_{1}: \ell x+m y=0, \quad P_{2}=z=0$
<br/>Plane through common line of $P_{1}$ and $P_{2}$ 
<br/>$\mathrm{P}_{3}: \ell \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$
<br/>angle between $P_{1}$ and $P_{3}=\alpha$ 
<br/>$\therefore \cos \alpha=\hat{n}_{1}, \hat{n}_{2}$
<br/>$=\frac{\ell^{2}+\mathrm{m}^{2}}{\sqrt{\ell^{2}+\mathrm{m}^{2}} \sqrt{\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}}}=\sqrt{\frac{\ell^{2}+\mathrm{m}^{2}}{\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}}}$
<br/>$\Rightarrow \cos ^{2} \alpha=\frac{\ell^{2}+\mathrm{m}^{2}}{\ell^{2}+\mathrm{m}^{2}+\mathrm{n}^{2}}$
<br/>$\Rightarrow \mathrm{n}^{2}=\left(\ell^{2}+\mathrm{m}^{2}\right) \tan ^{2} \alpha$
<br/>$\Rightarrow \mathrm{n}=\pm \sqrt{\ell^{2}+\mathrm{m}^{2}} \tan \alpha$
<br/>$\mathrm{P}_{3}: \ell \mathrm{x}+\mathrm{my} \pm \sqrt{\ell^{2}+\mathrm{m}^{2}}$ tan $\alpha=0$</div>


tags: ['Mathematics', 'Straight Lines', 'JEE Main']
source_db: db\MarksBatch2_P3.db
title:
the-point-4-5-is-the-vertex-of-a-square-and-one-of-its-diagonals-is-7-x-y-8-0-the-equation-of-the-other-diagonal-is-1

slug:
the-point-4-5-is-the-vertex-of-a-square-and-one-of-its-diagonals-is-7-x-y-8-0-the-equation-of-the-other-diagonal-is-1-10278

question:
<div class="question">The point $(-4,5)$ is the vertex of a square and one of its diagonals is $7 x-y+8=0$. The equation of the other diagonal is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$7 x-y+23=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$7 y+x=30$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$7 y+x=31$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad x-7 y=30$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$7 y+x=31$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>\text { Hints } &amp; : x+7 y=k \\<br/>&amp; -4+35=k \\<br/>&amp; 31=k \\<br/>&amp; x+7 y-31=0<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lWqgVSss-TbmlOEPtFVLGEgvJENv07Xt5YmwB-OA_c4.original.fullsize.png"/><br/></div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
title:
the-point-4-5-is-the-vertex-of-a-square-and-one-of-its-diagonals-is-7-x-y-8-0-the-equation-of-the-other-diagonal-is

slug:
the-point-4-5-is-the-vertex-of-a-square-and-one-of-its-diagonals-is-7-x-y-8-0-the-equation-of-the-other-diagonal-is-96080

question:
<div class="question">The point $(-4,5)$ is the vertex of a square and one of its diagonals is $7 x-y+8=0$. The equation of the other diagonal is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$7 x-y+23=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$7 y+x=30$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$7 y+x=31$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\quad x-7 y=30$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$7 y+x=31$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>\text { Hints } &amp; : x+7 y=k \\<br/>&amp; -4+35=k \\<br/>&amp; 31=k \\<br/>&amp; x+7 y-31=0<br/>\end{aligned}<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/lWqgVSss-TbmlOEPtFVLGEgvJENv07Xt5YmwB-OA_c4.original.fullsize.png"/><br/></div>


title: the-point-in-the-interval-0-2-where-f-x-e-x-sin-x-has-maximum-slope-is
slug: the-point-in-the-interval-0-2-where-f-x-e-x-sin-x-has-maximum-slope-is-36707
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The point in the interval $[0,2 \pi]$, where $f(x)=e^x \sin x$ has maximum slope, is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{\pi}{4}$</span> </li><li class="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">Hints: $f^{\prime}(x)=e^x(\sin x+\cos x)$<br/>$$<br/>\begin{aligned}<br/>&amp; f^{\prime \prime}(x)=e^x(\sin x+\cos x+\cos x-\sin x) \Rightarrow f^{\prime \prime}(x)=e^x \cos x=0 \\<br/>&amp; \Rightarrow x=\frac{\pi}{2}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
title:
the-point-of-contact-of-the-tangent-to-the-parabola-y-2-9-x-which-passes-through-the-point-4-10-and-makes-an-angle-with-the-positive-side-of-the-axis-

slug:
the-point-of-contact-of-the-tangent-to-the-parabola-y-2-9-x-which-passes-through-the-point-4-10-and-makes-an-angle-with-the-positive-side-of-the-axis-52382

question:
<div class="question">The point of contact of the tangent to the parabola $y^2=9 x$ which passes through the point $(4,10)$ and makes an angle $\theta$ with the positive side of the axis of the parabola where $\tan \theta&gt;2$, is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{4}{9}, 2\right)$</span> </div>

answer:
<div class="solution">$$<br/>\tan \theta=\frac{1}{t}&gt;2<br/>$$<br/>Equation of tangent at $\mathrm{t}$<br/>$$<br/>\begin{aligned}<br/>&amp;\left.\mathrm{yt}=\mathrm{x}+\frac{9}{4} \mathrm{t}^2 \text { (passes through }(4,10)\right) \Rightarrow 10 \mathrm{t}=4+\frac{9}{4} \mathrm{t}^2 \Rightarrow 9 \mathrm{t}^2-40 \mathrm{t}+16=0 \Rightarrow \mathrm{t}=4, \frac{4}{9}\left(\frac{1}{\mathrm{t}}&gt;2\right) \Rightarrow \mathrm{t}=\frac{4}{9} \\<br/>&amp;\therefore \text { point }\left(\mathrm{at}^2, 2 \mathrm{at}\right) \Rightarrow\left(\frac{4}{9}, 2\right)<br/>\end{aligned}<br/>$$</div>


title: the-point-on-the-parabola-y-2-64-x-which-is-nearest-to-the-line-4-x-3-y-35-0-has-coordinates
slug: the-point-on-the-parabola-y-2-64-x-which-is-nearest-to-the-line-4-x-3-y-35-0-has-coordinates-27283
tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The point on the parabola $y^{2}=64 x$ which is nearest to the line $4 x+3 y+35=0$ has coordinates</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">(9,-24)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">(1,81)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(4,-16)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">(-9,-24)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(9,-24)</span> </div>

answer:
<div class="solution">Given equation of parabola is
<br/>$$
<br/>y^{2}=64 x
<br/>$$
<br/>The point at which the tangent to the curve is parallel to the line is the nearest point on the curve.
<br/>On differentiating both sides of Eq. (i), we get $2 y \frac{d y}{d x}=64$
<br/>$\Rightarrow \quad \frac{d y}{d x}=\frac{32}{y}$
<br/>Also, slope of the given line is $-\frac{4}{3}$.
<br/>$\therefore$
<br/>$$
<br/>-\frac{4}{3}=\frac{32}{y} \Rightarrow y=-24
<br/>$$
<br/>From Eq. (i), $(-24)^{2}=64 x \Rightarrow x=9$
<br/>$\therefore$ Hence, the required point is (9,-24)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2017']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">(0,0)</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-point-p-3-6-is-first-reflected-on-the-line-y-x-and-then-the-image-point-q-is-again-reflected-on-the-line-y-x-to-get-the-image-point-q-then-the-cir

slug:
the-point-p-3-6-is-first-reflected-on-the-line-y-x-and-then-the-image-point-q-is-again-reflected-on-the-line-y-x-to-get-the-image-point-q-then-the-cir-82793

question:
<div class="question">The point $P(3,6)$ is first reflected on the line $y=x$ and then the image point $Q$ is again reflected on the line $y=-x$ to get the image point $Q^{\prime} .$ Then, the circumcentre of the $\Delta P Q Q^{\prime}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(6,3)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">(6,-3)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">(3,-6)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">(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> </ul>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_mat_a17.png"/> <br/> <br/>The coordinates of $Q$ and $Q^{\prime}$ will be (6,3) 
<br/>and $(-3,-6),$ respectively. Now, slope of $P Q=\frac{6-3}{3-6}=-1$ and Slope of $Q Q^{\prime}=\frac{-6-3}{-3-6}=1$
<br/>$\therefore$ Slope of $P Q \times$ Slope of $Q Q^{\prime}=-1 \times 1=-1$
<br/>$\therefore \Delta P Q Q^{\prime}$ is right angled triangle at $Q .$
<br/>$\therefore$ Circumcentre will be the mid-point of hypotenuse $P Q^{\prime}=\left(\frac{-3+3}{2}, \frac{-6+6}{2}\right)$
<br/>$$
<br/>=(0,0)
<br/>$$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
title:
the-point-q-is-the-image-of-the-point-p-1-5-about-the-line-y-x-and-r-is-the-image-of-the-point-q-about-the-line-y-x-the-circumcentre-of-the-pqr-is

slug:
the-point-q-is-the-image-of-the-point-p-1-5-about-the-line-y-x-and-r-is-the-image-of-the-point-q-about-the-line-y-x-the-circumcentre-of-the-pqr-is-30087

question:
<div class="question">The point $Q$ is the image of the point $P(1,5)$ about the line $y=x$ and $R$ is the image of the point $Q$ about the line $y=-x .$ The circumcentre of the $\Delta P Q R$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">(5,1)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">(-5,1)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(1,-5)$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$(0,0)$</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> </ul>

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

answer:
<div class="solution">Given, point P(1,5) image of the point $P(1,5)$ about the line $y=x$ is $Q(5,1)$ and image of the point $Q$ on line $y=-x$ is $R(-1,-5)$
<br/>$\therefore$ Required circumcentre $=$ Mid-point of $P$ and $R$ $=\left(\frac{1-1}{2}, \frac{5-5}{2}\right)=(0,0$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_mat_a41.png"/></div>


title: the-points-a-b-a-b-0-0-and-a-2-ab-a-0-b-0-are-always
slug: the-points-a-b-a-b-0-0-and-a-2-ab-a-0-b-0-are-always-68322
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The points $(-a,-b),(a, b),(0,0)$ and $\left(a^{2}, a b\right), a \neq 0, b \neq 0$ are always</div>

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

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

answer:
<div class="solution">Let the four points be $A(-a,-b), B(a, b), C(0,0)$ and $D\left(a^{2}, a b\right)$
<br/>If $A,B$ and $C$ are collinear.
<br/>Then, $\left|\begin{array}{ccc}-a &amp; -b &amp; 1 \\ a &amp; b &amp; 1 \\ 0 &amp; 0 &amp; 1\end{array}\right|=0$
<br/>$\Rightarrow \quad-a(b-0)+b(a-0)+1(0)=0$
<br/>$-a b+a b=0$
<br/>Hence, $A, B$ and $C$ ae collinear.
<br/>Again, if $B, C$ and $D$ are collinear.
<br/>Then, $\left|\begin{array}{ccc}a &amp; b &amp; 1 \\ 0 &amp; 0 &amp; 1 \\ a^{2} &amp; a b &amp; 1\end{array}\right|=0$
<br/>$\Rightarrow \quad a(0-a b)-b\left(0-a^{2}\right)+1(0) ;=0$
<br/>$-a^{2} b+a^{2} b=0$
<br/>$\Rightarrow$
<br/>Hence, $B, C$ and $D$ are collinear. So, the points $A, B, C$ and $D$ are akways collinear.</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-points-of-intersection-of-two-ellipses-x-2-2-y-2-6-x-12-y-20-0-and-2-x-2-y-2-10-x-6-y-15-0-lie-on-a-circle-the-centre-of-the-circle-is

slug:
the-points-of-intersection-of-two-ellipses-x-2-2-y-2-6-x-12-y-20-0-and-2-x-2-y-2-10-x-6-y-15-0-lie-on-a-circle-the-centre-of-the-circle-is-11910

question:
<div class="question">The points of intersection of two ellipses $x^{2}+2 y^{2}-6 x-12 y+20=0$ and $2 x^{2}+y^{2}-10 x-6 y+15=0$ lie on a circle. The centre of the circle is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{8}{3}, 3\right)$</span> </div>

answer:
<div class="solution">$x^{2}+2 y^{2}-6 x-12 y+20+\lambda\left(2 x^{2}+y^{2}-10 x-6 y+15\right)=0$
<br/>$\begin{array}{l}
<br/>\Rightarrow(1+2 \lambda) \mathrm{x}^{2}+(2+\lambda) \mathrm{y}^{2}-(10 \lambda+6) \mathrm{x}-(6 \lambda+12) \mathrm{y}+15 \lambda+20=0 \\
<br/>2+\lambda=1+2 \lambda \\
<br/>\Rightarrow \lambda=1
<br/>\end{array}$
<br/>$\therefore$ Centre $:\left(\frac{8}{3}, 3\right)$</div>


tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-points-of-the-ellipse-16-x-2-9-y-2-400-at-which-the-ordinate-decreases-at-the-same-rate-at-which-the-abscissa-increases-isare-given-by

slug:
the-points-of-the-ellipse-16-x-2-9-y-2-400-at-which-the-ordinate-decreases-at-the-same-rate-at-which-the-abscissa-increases-isare-given-by-21661

question:
<div class="question">The points of the ellipse $16 x^{2}+9 y^{2}=400$ at which the ordinate decreases at the same rate at which the abscissa increases is/are given by</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left(3, \frac{16}{3}\right)$ and $\left(-3,-\frac{16}{3}\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\left(3,-\frac{16}{3}\right)$ and $\left(-3, \frac{16}{3}\right)$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{1}{16}, \frac{1}{9}\right)$ and $\left(-\frac{1}{16},-\frac{1}{9}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\frac{1}{16},-\frac{1}{9}\right)$ and $\left(-\frac{1}{16}, \frac{1}{9}\right)$</span> </li> </ul>

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

answer:
<div class="solution">Given that, the ordinate decreases at the same rate at which the abscissa increases, therefore
<br/>$$
<br/>\frac{d y}{d t}=-\frac{d x}{d t}
<br/>$$
<br/>Also, equation of ellipse
<br/>$$
<br/>16 x^{2}+9 y^{2}=400
<br/>$$
<br/>On differentiating wrt. $t$, we get 
<br/>$16 \times 2 x \frac{d x}{d t}+9 \times 2 y \frac{d y}{d t}=0$
<br/>$\Rightarrow \quad 16 x \frac{d x}{d t}+9 y \frac{d y}{d t}=0$
<br/>$\Rightarrow \quad 9 y \frac{d y}{d t}=-16 x \frac{d x}{d t}$
<br/>$\Rightarrow \quad 9 y \frac{d y}{d t}=-16 x\left(-\frac{d y}{d t}\right) \quad$ [using $\left.\mathrm{Eq.}(i)\right]$
<br/>$\Rightarrow \quad 9 y=16 x \Rightarrow y=\frac{16}{9} x$
<br/>$16 x^{2}+9\left(\frac{16}{9} x\right)^{2}=400$
<br/>$\Rightarrow \quad 16 x^{2}+\frac{(16)^{2}}{9} x^{2}=400$
<br/>$\Rightarrow \quad 16 x\left[x+\frac{16}{9} x\right]=400$
<br/>$\Rightarrow \quad 16 x\left[\frac{25}{9} x\right]=400 \Rightarrow x^{2}=\frac{400 \times 9}{25 \times 16}$
<br/>$\Rightarrow \quad x^{2}=9 \Rightarrow x=\pm 3$
<br/>$\therefore$ Required points are $\left(3, \frac{16}{3}\right)$ and $\left(-3-\frac{16}{3}\right)$</div>


title: the-points-representing-the-complex-number-z-for-which-arg-z-2-z-2-3-lie-on
slug: the-points-representing-the-complex-number-z-for-which-arg-z-2-z-2-3-lie-on-70000
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The points representing the complex number z for which arg $\left(\frac{z-2}{z+2}\right)=\frac{\pi}{3}$ lie on</div>

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

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

answer:
<div class="solution">Let $z=x+i y$ $\therefore \quad \frac{z-2}{z+2}=\frac{x+iy-2}{x+i y+2}$
<br/>$=\frac{(x-2)+i y}{(x+2)+6 y} \times \frac{(x+2)-i y}{(x+2)-i y}$
<br/>$=\frac{(x-2)(x+2)+i y(x+2)-i y(x-2)-i^{2} y^{2}}{(x+2)^{2}-(i y)^{2}}$
<br/>$=\frac{x^{2}-4+i x y+2 i y-i x y+2 i y+y^{2}}{(x+2)^{2}+y^{2}}$
<br/>$x^{2}+y^{2}-4-\frac{4}{\sqrt{3}} y=0,$ which represents a
<br/>circle.</div>


tags: ['Physics', 'Mechanical Properties of Solids', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$4 \%$</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-poissons-ratio-of-a-material-is-05-if-a-force-is-applied-to-a-wire-of-this-material-there-is-a-decrease-in-the-crosssectional-area-by-4-the-percen

slug:
the-poissons-ratio-of-a-material-is-05-if-a-force-is-applied-to-a-wire-of-this-material-there-is-a-decrease-in-the-crosssectional-area-by-4-the-percen-79208

question:
<div class="question">The poisson's ratio of a material is 0.5 . If a force is applied to a wire of this material, there is a decrease in the cross-sectional area by $4 \%$. The percentage increase in the length is :</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$1 \%$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$2 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$2.5 \%$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4 \%$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Poisson ratio $=0.5$<br/>Therefore density is constant hence change in volume is zero we have $\mathrm{V}=\mathrm{A} \times \ell=$ constant<br/>$\log \mathrm{V}=\log \mathrm{A}+\log \ell$ or $\frac{d A}{A}+\frac{d \ell}{\ell}=0 \Rightarrow \frac{d \ell}{\ell}=-\frac{d A}{A}$<br/>That is $4 \%$</div>


tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
title:
the-polar-coordinate-of-a-point-p-is-2-4-the-polar-coordinate-of-the-point-q-which-is-such-that-line-joining-pq-is-bisected-perpendicularly-by-the-ini

slug:
the-polar-coordinate-of-a-point-p-is-2-4-the-polar-coordinate-of-the-point-q-which-is-such-that-line-joining-pq-is-bisected-perpendicularly-by-the-ini-57974

question:
<div class="question">The polar coordinate of a point $P$ is $\left(2,-\frac{\pi} {4}\right)$ The polar coordinate of the point $Q$ which is such that line joining $P Q$ is bisected perpendicularly by the initial line, is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(2, \frac{\pi}{4}\right)$</span> </div>

answer:
<div class="solution">Since, $Q$ is the point for which $P Q$ is bisected perpendicularly by the initial line $(X$ -axis). Therefore, $Q$ will be the image of $P$ in $X$ -axis
<br/>$\therefore$ Coordinates of $Q$ is $(2, \pi / 4)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2019_mat_a60.png"/></div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P3.db
title:
the-portion-of-the-tangent-to-the-curve-xfrac23yfrac23afrac23-a-0-at-any-point-of-it-intercepted-between-the-axes

slug:
the-portion-of-the-tangent-to-the-curve-xfrac23yfrac23afrac23-a-0-at-any-point-of-it-intercepted-between-the-axes-38059

question:
<div class="question">The portion of the tangent to the curve \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}, a &gt; 0\) at any point of it, intercepted between the axes</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">varies as abscissa</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">varies as ordinate</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">is 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">varies as the product of abscissa and ordinate</span> </li> </ul>

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

answer:
<div class="solution">Hint : for the given curve \(x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}, a &gt; 0\)<br/>Parametric Coordinates are \(x=a \cos ^3 \theta\)<br/>\(y=a \sin ^3 \theta\)<br/>\(\begin{gathered}<br/>\frac{d y}{d x}=\text { slope of tangent at any }=\frac{d y / d \theta}{d x / d \theta} \\<br/>\text { point }(x, y)<br/>\end{gathered}\)<br/>\(=\frac{3 a \sin ^2 \theta \cdot \cos \theta}{3 a \cos ^2 \theta(-\sin \theta)}=-\tan \theta\)<br/>Equation of tangent at \(\left(a \cos ^3 \theta, a \sin ^3 \theta\right)\)<br/>\(y-a \sin ^3 \theta=-\tan \theta\left(x-a \cos^3 \theta\right)\)<br/>its \(x\)-intercept \(=a \cos \theta\)<br/>its \(y\)-intercept \(=a \sin \theta\)<br/>So, the tangent cuts the axes at \(A(a \cos \theta, 0)\) and \(B(0, a \sin \theta)\) respectively.<br/>\(A B=\sqrt{(a \cos \theta)^2+(a \sin \theta)^2}=a\), which is constant</div>


tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-4$</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-position-vectors-of-the-points-a-b-c-and-d-are-3-i-2-j-k-2-i-3-j-2-k-i-j-2-k-and-4-i-j-k-respectively-if-the-points-a-b-c-and-d-lie-on-a-plane-the

slug:
the-position-vectors-of-the-points-a-b-c-and-d-are-3-i-2-j-k-2-i-3-j-2-k-i-j-2-k-and-4-i-j-k-respectively-if-the-points-a-b-c-and-d-lie-on-a-plane-the-39623

question:
<div class="question">The position vectors of the points $A, B, C$ and $D$ are $3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \mathbf{k}, \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $4 \hat{\mathbf{i}}-\hat{\mathbf{j}}-\lambda \hat{\mathbf{k}},$ respectively. If the points $A, B, C$ and $D$ lie on a plane, the value of $\lambda$ is</div>

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

answer:
<div class="solution">The position vectors of the points $A, B, C$ and $D$ are $3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, 2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, 5 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and
<br/>$4 \hat{\mathbf{i}}-\hat{\mathbf{j}}-\lambda \hat{\mathbf{k}}$ respectively
<br/>$\begin{aligned}
<br/>\mathbf{B A} &amp;=(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\mathbf{k})-(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+2 \mathbf{k})=\hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \mathbf{k} \\
<br/>\mathbf{C A} &amp;=(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}})-(5 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \mathbf{k})=-2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-3 \hat{\mathbf{k}} \\
<br/>\mathbf{D A} &amp;=(3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}})-(4 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}) \\
<br/>&amp;=-\hat{\mathbf{i}}-\hat{\mathbf{j}}-(1+\lambda) \mathbf{k}
<br/>\end{aligned}$
<br/>These are coplanar, so
<br/>$\left|\begin{array}{ccc}1 &amp; 1 &amp; -3 \\ -2 &amp; -1 &amp; -3 \\ -1 &amp; -1 &amp; -(1+\lambda)\end{array}\right|=0$
<br/>$=\left|\begin{array}{ccc}-1 &amp; -1 &amp; 3 \\ 2 &amp; 1 &amp; 3 \\ 1 &amp; 1 &amp; 1+\lambda\end{array}\right|=0$
<br/>$-1[1+\lambda-3]+1[2+2 \lambda-3]+3[2-1]=0$
<br/>$\Rightarrow \quad-1[\lambda-2]+1[2 \lambda-1]+3=0$
<br/>$\Rightarrow -\lambda+2+2 \lambda-1+3=0$
<br/>$\Rightarrow \lambda+4=0$
<br/>$\Rightarrow \lambda=-4$</div>


title: the-possible-products-to-be-obtained-from-the-reaction-of-cyclobutyl-amine-with-hno-2-isare
slug: the-possible-products-to-be-obtained-from-the-reaction-of-cyclobutyl-amine-with-hno-2-isare-59078
tags: ['Chemistry', 'Amines', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The possible product(s) to be obtained from the reaction of cyclobutyl amine with HNO $_{2}$ is/are</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q76_opta.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q76_optb.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q76_optc.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q76_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q76_opta.png"/>, <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_q76_optc.png"/></span> </div>

answer:
<div class="solution">The possible product $(s)$ to be obtained from the reaction of cyclobutyl amine with $\mathrm{HNO}_{2}$, is/are
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2018_che_a76.png"/></div>


title: the-possible-values-of-x-which-satisfy-the-trigonometric-equation-tan-1-x-2-x-1-tan-1-x-2-x-1-4-are
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
slug:
the-possible-values-of-x-which-satisfy-the-trigonometric-equation-tan-1-x-2-x-1-tan-1-x-2-x-1-4-are-19847

question:
<div class="question">The possible values of $x$, which satisfy the trigonometric equation
<br/>$\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\tan ^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}$ are</div>

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

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

answer:
<div class="solution">We have, $\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\tan ^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}$
<br/>$\Rightarrow \quad \tan ^{-1}\left[\frac{\frac{x-1}{x-2}+\frac{x+1}{x+2}}{1-\frac{x-1}{x-2} \cdot \frac{x+1}{x+2}}\right]=\frac{\pi}{4}$
<br/>$\Rightarrow \frac{(x-1)(x+2)+(x-2)(x+1)}{(x-2)(x+2)-(x-1)(x+1)}=\tan \frac{\pi}{4}$
<br/>$\Rightarrow \quad \frac{x^{2}+x-2+x^{2}-x-2}{x^{2}-4-x^{2}+1}=1$
<br/>$\Rightarrow$ $\frac{2 x^{2}-4}{-3}=1$
<br/>$\Rightarrow$ $2 x^{2}-4=3$
<br/>$\Rightarrow$ $2 x^{2}=1$
<br/>$x^{2}=\frac{1}{2}$
<br/>$x=\pm \frac{1}{\sqrt{2}}$</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P3.db
title:
the-potential-difference-v-required-for-accelerating-an-electron-to-have-the-debroglie-wavelength-of-1-a-is

slug:
the-potential-difference-v-required-for-accelerating-an-electron-to-have-the-debroglie-wavelength-of-1-a-is-75417

question:
<div class="question">The potential difference $V$ required for accelerating an electron to have the de-Broglie wavelength of $1 Ã$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$100 \mathrm{V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$125 \mathrm{V}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$150 \mathrm{V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$200 \mathrm{V}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$150 \mathrm{V}$</span> </div>

answer:
<div class="solution">The de-Broglie wavelength associated with electron is given by.
<br/>$$
<br/>\lambda=\frac{h}{\sqrt{2 m e V}}
<br/>$$
<br/>where, $h=$ Planck's constant
<br/>$$
<br/>\begin{aligned}
<br/>m &amp;=\text { mass of electron } \\
<br/>\mathrm{e} &amp;=\text { charge on electron } \\
<br/>\mathrm{V} &amp;=\text { potential difference } \\
<br/>\lambda &amp;=1 \dot{\mathrm{A}}=10^{-10} \mathrm{m} \\
<br/>\lambda^{2} &amp;=\frac{h^{2}}{2 \mathrm{meV}} \Rightarrow \mathrm{V}=\frac{h^{2}}{2 \mathrm{me} \lambda^{2}} \\
<br/>&amp;=\frac{\left(6.6 \times 10^{-34}\right)^{2}}{2 \times 9.1 \times 10^{-31} \times 1.6 \times 10^{-19} \times\left(1 \times 10^{-10}\right)^{2}} \\
<br/>&amp;=\frac{6.6 \times 10^{-34} \times 6.6 \times 10^{-34}}{2 \times 9.1 \times 10^{-31} \times 1.6 \times 10^{-19} \times 10^{-20}} \\
<br/>&amp;=\frac{6.6 \times 6.6 \times 10^{2}}{2 \times 9.1 \times 1.6}=\frac{4356 \times 10^{2}}{2912}=150 \mathrm{V}
<br/>\end{aligned}
<br/>$$</div>


tags: ['Physics', 'Oscillations', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-potential-energy-of-a-particle-of-mass-002-kg-moving-along-x-axis-is-given-by-v-a-x-x-4-j-where-x-is-in-metres-and-a-is-a-constant-which-of-the-fo

slug:
the-potential-energy-of-a-particle-of-mass-002-kg-moving-along-x-axis-is-given-by-v-a-x-x-4-j-where-x-is-in-metres-and-a-is-a-constant-which-of-the-fo-77309

question:
<div class="question">The potential energy of a particle of mass $0.02 \mathrm{~kg}$ moving along $x$-axis is given by $V=A x(x-4) J$ where $x$ is in metres and $\mathrm{A}$ is a constant. Which of the following is/are correct statement(s)?</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">The particle executes simple harmonic motion, The speed of the particle is maximum at $x=2 m$</span> </div>

answer:
<div class="solution">$V=A x(x-4) J m=0.02=A x^{2}-4 A x$
<br/>$$
<br/>\begin{array}{l}
<br/>\mathrm{F}=-\frac{\mathrm{d} \mathrm{V}}{\mathrm{dx}}=-2 \mathrm{Ax}+4 \mathrm{~A} \text { (not constant) } \\
<br/>\mathrm{F}=-2 \mathrm{~A}(\mathrm{x}-2)
<br/>\end{array}
<br/>$$
<br/>Hence position execute S.H.M mean position is when force $=0$ i.e. $x=2$ and speed is maximum at mean position
<br/>$$
<br/>\omega^{2}=\frac{2 \mathrm{~A}}{\mathrm{~m}} \mathrm{~T}=\frac{2 \pi}{\omega}=2 \pi \sqrt{\frac{\mathrm{m}}{2 \mathrm{~A}}}=2 \pi \sqrt{\frac{1}{100 \mathrm{~A}}}=\frac{\pi}{5} \sqrt{\frac{1}{\mathrm{~A}}}
<br/>$$</div>


title: the-potential-of-a-hydrogen-electrode-at-ph-10-is
slug: the-potential-of-a-hydrogen-electrode-at-ph-10-is-88451
question: <div class="question">The potential of a hydrogen electrode at $\mathrm{pH}=10$ is</div>
tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.59 \mathrm{~V}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.00 \mathrm{~V}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-0.59 \mathrm{~V}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-0.059</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$-0.59 \mathrm{~V}$</span> </div>

answer:
<div class="solution">Hints : $\mathrm{H}^{+}(\mathrm{pH}=10)\left|\mathrm{H}_2(1 \mathrm{~atm})\right| \operatorname{Pt}(\mathrm{s})$<br/>Reaction: $2 \mathrm{H}^{+}\left(\mathrm{p}^{\mathrm{H}}=10\right)+2 \mathrm{e} \rightarrow \mathrm{H}_2(1 \mathrm{~atm})$<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{E}=\mathrm{E}^0-\frac{0.0591}{2} \log \left(\frac{\mathrm{P}_{\mathrm{H}_2}}{\left[\mathrm{H}^{+}\right]^2}\right) \\<br/>&amp; =0-\frac{0.0591}{2} \log \frac{1}{\left(10^{-10}\right)^2}=-\frac{0.0591}{2} \times 2 \log \frac{1}{10^{-10}}=-0.0591 \times 10=-0.591 \\<br/>&amp; \text { i.e. } E=-0.591 \mathrm{~V} \\<br/>&amp;<br/>\end{aligned}<br/>$$</div>


tags: ['Physics', 'Thermodynamics', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
title:
the-pressure-p-volume-v-and-temperature-t-for-a-certain-gas-are-related-by-p-v-a-t-b-t-2-where-a-and-b-are-constants-the-work-done-by-the-gas-when-the

slug:
the-pressure-p-volume-v-and-temperature-t-for-a-certain-gas-are-related-by-p-v-a-t-b-t-2-where-a-and-b-are-constants-the-work-done-by-the-gas-when-the-70864

question:
<div class="question">The pressure $p$, volume $V$ and temperature $T$ for a certain gas are related by $p=\frac{A T-B T^{2}}{V}$ where $A$ and $B$ are constants. The work done by the gas when the temperature changes from $T_{1}$ to $T_{2}$ while the pressure remains constant, is given by</div>

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

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

answer:
<div class="solution">Given $P=\frac{A T-B T^{2}}{V}$
<br/>$\begin{array}{lc}\Rightarrow &amp; P V=A T-B T^{2} \\ \Rightarrow &amp; P \Delta V=A \Delta T-B T \Delta T\end{array}$
<br/>On integrating, we get Work $=\int P d V=A \int_{T_{1}}^{T_{2}} d T-B \int_{T_{1}}^{T_{2}} T d T$
<br/>$=A\left(T_{2}-T_{1}\right)-\frac{B}{2}\left[\left(T_{2}\right)^{2}-\left(T_{1}\right)^{2}\right]$</div>


title: the-probability-that-a-nonleap-year-selected-at-random-will-have-53-sunday-is
slug: the-probability-that-a-nonleap-year-selected-at-random-will-have-53-sunday-is-59397
tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The probability that a non-leap year selected at random will have 53 Sunday is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 / 7$</span> </div>

answer:
<div class="solution">In a non-leap year,total number of days is 365 . Out of them, there are 52 weeks and 1 day extra. Thus, a non-leap year always has 52 Sunday. The remaining 1 day can be Sunday, Monday, Tuenday, Wednesday, Thursday, Friday and Sarurday. Out of these 7 cases, we have Sunday in one
<br/>case
<br/>$\therefore$ Total number of outcomes = 7 
<br/>Number of favourable outcomes = 1
<br/>Hence, required probability $=\frac{1}{7}$</div>


tags: ['Mathematics', 'Probability', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1.1</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-probability-that-at-least-one-of-a-and-b-occurs-is-06-if-a-and-b-occur-simultaneously-with-probability-03-then-p-a-p-b-is

slug:
the-probability-that-at-least-one-of-a-and-b-occurs-is-06-if-a-and-b-occur-simultaneously-with-probability-03-then-p-a-p-b-is-73172

question:
<div class="question">The probability that at least one of $A$ and $B$ occurs is 0.6 . If A and $B$ occur simultaneously with probability 0.3 , then $P^{\prime}\left(A^{\prime}\right)+P_{(}\left(B^{\prime}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.9</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.15</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">1.1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">1.2</span> </li> </ul>

answer:
<div class="solution">$$<br/>\begin{array}{cl}<br/>\text { Hints: } \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.6 &amp; \mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})=\mathrm{P}(\mathrm{A} \cup \mathrm{B})+\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.9 \\<br/>\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.3 &amp; \mathrm{P}\left(\mathrm{A}^{\prime}\right)+\mathrm{P}\left(\mathrm{B}^{\prime}\right)=2-0.9=1.1<br/>\end{array}<br/>$$</div>


title: the-product-of-the-above-reaction-is-1
slug: the-product-of-the-above-reaction-is-1-97092
tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q61.png"/> <br/> <br/>The product of the above reaction is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q61_opta.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q61_optb.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q61_optc.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q61_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q61_optc.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_a61.png"/> <br/> <br/>It is an example of intramolecular Cannizzaro reaction.</div>


title: the-product-of-the-following-hydrogenation-reaction-is
slug: the-product-of-the-following-hydrogenation-reaction-is-48284
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The product of the following hydrogenation reaction is:<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/XrVFUvb-GOohBHNdPeYKgvDLL-N-fCOPUHzcHmN37ko.original.fullsize.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/A3Cly-Wl0DOmACu-PKL85pNkvYHCz5V-OCkNcDeG3V4.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/RhCmb39qJFdOeDw0FKh4welUkAVeehMrw49Vdayw_Y4.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/uldBEOvsvmi1zDtOkR9xWmUdePPfRfCvcwDEi9UTaWs.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/A3LjKajyJ8EbL4qJtsJqcWkii0udzbCNZc0gJoAS23Q.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/uldBEOvsvmi1zDtOkR9xWmUdePPfRfCvcwDEi9UTaWs.original.fullsize.png"/></span> </div>

answer:
<div class="solution">The double bonds will continue to get reduced in one molecule one after another. This process will continue till all the hydrogens are exhausted.</div>


title: the-product-of-the-following-reaction-is-1
slug: the-product-of-the-following-reaction-is-1-36873
tags: ['Chemistry', 'Aldehydes and Ketones', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The product of the following reaction is :<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/jFadQXolb_ZTz07B1yjSw0PK8EeTyzL-NuZ9u6SjTWs.original.fullsize.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/TIhHEgt93j85GLhaFD4URUBCMNTzFrxDySf47_bhDIY.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/QhTGus0WfwUGyYpVP2yREpaYe4CU3lpckPDwUXORQQw.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2jySvGp83Ao0L6xDpApq4iwahnEzuEh3-pUdrNM_34g.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/DoJVODM3dfHJ3rbuguVI3q9_gFnKS6-iKryuZW0CohE.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/TIhHEgt93j85GLhaFD4URUBCMNTzFrxDySf47_bhDIY.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/smNm9B5aIyWfuAi3srHyU0naClpRIbcZzb-sVzfbYKQ.original.fullsize.png"/></div>


title: the-product-of-the-reaction-is-1
slug: the-product-of-the-reaction-is-1-43031
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The product of the reaction is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q45.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q45_opta.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q45_optb.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q45_optc.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q45_optd.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_q45_optc.png"/></span> </div>

answer:
<div class="solution">It is a substitution reaction, so the product
<br/>formed is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_a45.png"/></div>


title: the-product-s-from-the-above-reaction-will-be
slug: the-product-s-from-the-above-reaction-will-be-74646
tags: ['Chemistry', 'Hydrocarbons', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
question:
<div class="question"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/3__E2l7qjA_jlr4r5fUWoLouF47o7Ozd_rr5zXR79t4.original.fullsize.png"/><br/> <br/>The product (s) from the above reaction will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/mOw28hmzVM_FIDn0W_KzFqfN1Pyh3i2bWVvy7Mgomhc.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/EQ5FnMvDvi4r0KyKc2UnE4mjAVNt_LTxMa1HIoGEdjc.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/29UicIGeGkgeFizRogDrW4f_0dDzB6xSyWDbMr8Qfbs.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/_uMM1DS7VFCmZsFLymRvOUKh1Cf7ZjJ3dvXU5W7YfGw.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/mOw28hmzVM_FIDn0W_KzFqfN1Pyh3i2bWVvy7Mgomhc.original.fullsize.png"/>, <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/29UicIGeGkgeFizRogDrW4f_0dDzB6xSyWDbMr8Qfbs.original.fullsize.png"/></span> </div>

answer:
<div class="solution">Hint:
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/ftLn9WuQFECUKG-riuzEqYzN11nIKHApLKMLPDpp_Wo.original.fullsize.png"/></div>


tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-product-s-in-the-following-sequence-of-reactions-will-be-me-c-c-me-2-dil-alkaline-kmno-4-1-na-nh-3-liq-ethanol-3-3-c-product-s

slug:
the-product-s-in-the-following-sequence-of-reactions-will-be-me-c-c-me-2-dil-alkaline-kmno-4-1-na-nh-3-liq-ethanol-3-3-c-product-s-57936

question:
<div class="question">The product (s) in the following sequence of reactions will be
<br/>$\mathrm{Me}-\mathrm{C} \equiv \mathrm{C}-\mathrm{Me} \frac{\begin{array}{l}\text { 1. } \mathrm{Na} / \mathrm{NH}_{3} \text { (liq.) } \\ \text { ethanol, }-33^{\circ} \mathrm{C}\end{array}}{\text { 2. dil. alkaline } \mathrm{KMnO}_{4}}$ Product (s)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/pIKHewJIlt6PME78LMSPZZL1XCxMW2TXXaTfbl4gxH8.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/JYqzkLrZKScfbblPUdkTsl5POXo41hOknjxS_9RrFPA.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 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"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/cK_wLEoG53fRmJ71lO0hrq7Kh04U562uWHho2VxWQhM.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Zn4M4J-3VqxRA6ArWT2194vccME1tM-Hltf-_Y_68Tg.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/JYqzkLrZKScfbblPUdkTsl5POXo41hOknjxS_9RrFPA.original.fullsize.png"/>, <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/cK_wLEoG53fRmJ71lO0hrq7Kh04U562uWHho2VxWQhM.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/7Nz9MD3tCJJolpaDuk_Gwv1Tln_8pPDWuK6PfFZtHAs.original.fullsize.png"/></div>


tags: ['Chemistry', 'Alcohols Phenols and Ethers', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P3.db
title:
the-products-x-and-y-which-are-formed-in-the-following-sequence-of-reactions-are-respectively-phenol-dil-hno-3-x-2-ch-3-co-2-o-1-equiv-1-zn-hcl-y

slug:
the-products-x-and-y-which-are-formed-in-the-following-sequence-of-reactions-are-respectively-phenol-dil-hno-3-x-2-ch-3-co-2-o-1-equiv-1-zn-hcl-y-69811

question:
<div class="question">The products $\mathrm{X}$ and $\mathrm{Y}$ which are formed in the following sequence of reactions are respectively $\mathrm{Phenol} \stackrel{\text { dil } \mathrm{HNO}_{3}}{\longrightarrow} \mathrm{X} \frac{1 . \mathrm{Zn} / \mathrm{HCl}, \Delta}{2 \cdot\left(\mathrm{CH}_{3} \mathrm{CO}\right)_{2} \mathrm{O}(1 \text { equiv })} \longrightarrow \mathrm{Y}$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/DVlU8gNd7-eDbuOeRJknG7EHc7ZN6iNwgWh19SswUK8.original.fullsize.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Fi9ulsbZtepSzVMcOyHcZEAEvzb8dEW5hvXfc1HxNCY.original.fullsize.png"/></span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/XAro6siksBub3vB1dlaSMmvqqzl6g6Er42amzwAz5vc.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/48u9cT9qNeLP3LWW_0vcgcrd4OjhEhVgNcxvOpbw3Hk.original.fullsize.png"/></span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/XAro6siksBub3vB1dlaSMmvqqzl6g6Er42amzwAz5vc.original.fullsize.png"/></span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/z5cRiZEoyo9DmEtuGhu_OVKzmquVGyg16XQrXVoxXkQ.original.fullsize.png"/></div>


title: the-quadratic-equation-2-x-2-a-3-8-a-1-x-a-2-4-a-0-has-roots-of-opposite-sign-then
slug: the-quadratic-equation-2-x-2-a-3-8-a-1-x-a-2-4-a-0-has-roots-of-opposite-sign-then-30391
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The quadratic equation 
<br/>$2 x^{2}-\left(a^{3}+8 a-1\right) x+a^{2}-4 a=0$ has roots of opposite sign. Then,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$a \leq 0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0     &lt; a &lt; 4$</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">$4 \leq a     &lt; 8$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a \geq 8$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0     &lt; a &lt; 4$</span> </div>

answer:
<div class="solution">Since, roots are opposite sign.
<br/>Product of the roots $\leq 0$ $\begin{array}{ll}\Rightarrow &amp; a^{2}-4 a     &lt; 0 \\ \Rightarrow &amp; a(a-4) &lt; 0 \\ \Rightarrow &amp; 0 &lt; a &lt; 4\end{array}$</div>


title: the-quadratic-equation-whose-roots-are-three-times-the-roots-of-3-a-x-2-3-b-x-c-0-is
slug: the-quadratic-equation-whose-roots-are-three-times-the-roots-of-3-a-x-2-3-b-x-c-0-is-92817
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The quadratic equation whose roots are three times the roots of $3 a x^2+3 b x+c=0$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a x^2+3 b x+3 c=0$</span> </div>

answer:
<div class="solution">Hints: $3 a \alpha^2+3 b \alpha+c=0$<br/>$$<br/>x=3 \alpha \Rightarrow \alpha=\frac{x}{3}<br/>$$<br/>$$<br/>\begin{aligned}<br/>&amp; 3 a \frac{x^2}{9}+3 b \cdot \frac{x}{3}+c=0 \\<br/>&amp; a x^2+3 b x+3 c=0<br/>\end{aligned}<br/>$$</div>


title: the-quadratic-equation-x-2-15-x-14-0-has
slug: the-quadratic-equation-x-2-15-x-14-0-has-33658
question: <div class="question">The quadratic equation $x^2+15|x|+14=0$ has</div>
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2010']
answer: <div class="solution">Hints: $x^2+15|x|+14&gt;0 \forall x$<br/>Hence no solution</div>
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">only positive solutions</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">only negative solutions</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">no solution</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">both positive and negative solution</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">no solution</span> </div>


title: the-quadratic-expression-2-x-1-2-p-x-q-0-for-any-real-x-if
slug: the-quadratic-expression-2-x-1-2-p-x-q-0-for-any-real-x-if-56778
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The quadratic expression $(2 x+1)^{2}-p x+q \neq 0$ for any real $x,$ if</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$p^{2}-16 p-8 q    &lt; 0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$p^{2}-8 p+16 q    &lt; 0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$p^{2}-8 p-16 q    &lt; 0$</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">$p^{2}-16 p+8 q    &lt; 0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$p^{2}-8 p-16 q    &lt; 0$</span> </div>

answer:
<div class="solution">Given equation is $(2 x+1)^{2}-p x+q \neq 0$
<br/>$$
<br/>\begin{array}{lr}
<br/>\Rightarrow &amp; 4 x^{2}+4 x+1-p x+q \neq 0 \\
<br/>\Rightarrow &amp; 4 x^{2}+(4-p) x+(1+q) \neq 0
<br/>\end{array}
<br/>$$
<br/>Now. D $    &lt; $ 0
<br/>$\Rightarrow \quad(4-p)^{2}-4(4)(1+q) &lt; 0$
<br/>$\Rightarrow \quad 16-8 p+p^{2}-16-16 q &lt; 0$
<br/>$$
<br/>\Rightarrow \quad p^{2}-8 p-16 q &lt; 0
<br/>$$</div>


title: the-quantity-h-v-k-b-corresponds-to
slug: the-quantity-h-v-k-b-corresponds-to-12721
question: <div class="question">The quantity $h v / K_{\mathrm{B}}$ corresponds to</div>
tags: ['Chemistry', 'States of Matter', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">wavelength</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">velocity</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">temperature</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">angular momentum</span> </li> </ul>

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

answer:
<div class="solution">The quantity $h v / K_{\mathrm{B}}$ corresponds to temperature. $h v=3 / 2 K_{\mathrm{B}} T$
<br/>where, $T=$ temperature $K_{\mathrm{B}}=$ Boltzmann constant $$
<br/>\begin{array}{l}
<br/>\frac{h v}{K_{B}}=\frac{3}{2} T
<br/>\end{array}
<br/>$$
<br/>since, $\frac{3}{2}$ is a constant, the value of $\frac{h v}{K_{\mathrm{B}}}$ corresponds
<br/>to temperature. Therefore option
<br/>(c) is correct.</div>


tags: ['Chemistry', 'Electrochemistry', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
title:
the-quantity-of-electricity-needed-to-separately-electrolyze-1-m-solution-of-znso-4-alcl-3-and-agno-3-completely-is-in-the-ratio-of

slug:
the-quantity-of-electricity-needed-to-separately-electrolyze-1-m-solution-of-znso-4-alcl-3-and-agno-3-completely-is-in-the-ratio-of-73377

question:
<div class="question">The quantity of electricity needed to separately electrolyze 1 M solution of $\mathrm{ZnSO}_{4}$, $\mathrm{AlCl}_{3}$ and $\mathrm{AgNO}_{3}$ completely is in the ratio of</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2: 3: 1$</span> </div>

answer:
<div class="solution">From the Faradiay's IInd law, we know that
<br/>$$
<br/>W \propto \text { Zit, } \quad W \propto \frac{E}{96500} i t
<br/>$$
<br/>or $\quad$ it $\propto \frac{1}{E}$ or electricity $\propto \frac{1}{E}$ 
<br/>Thus, the amount of electricity required for $\mathrm{ZnSO}_{4}, \mathrm{AlCl}_{3}$ and $\mathrm{AgNO}_{3}$ is in the ratio of
<br/>2: 3: 1 . (Here, 2,3 and 1 are the valencies of metal atom in the given compounds). 
<br/>Alternatively 
<br/>In $Z n S O_{4}$,
<br/>$Z n^{2+}+2 e^{-}$ $\rightarrow Z \mathrm{n}$
<br/>$1mol$  $\text{2 mol or 2 Faraday}$
<br/>In $\mathrm{AlCl}_{3}$,
<br/>$\mathrm{Al}^{3+}+3 e^{-} \longrightarrow \mathrm{Al}$
<br/>$1mol$  $\text{3F}$
<br/>In $\mathrm{AgNO}_{3}$,
<br/>$\mathrm{Ag}^{+}+e^{-} \longrightarrow \mathrm{Ag}$
<br/>$1mol$  $\text{1F}$
<br/>Thus, the ratio of amounts of electricity required is 2: 3: 1</div>


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P3.db
title:
the-radius-of-the-first-bohr-orbit-of-a-hydrogen-atom-is-053-1-0-8-cm-the-velocity-of-the-electron-in-the-first-bohr-orbit-is

slug:
the-radius-of-the-first-bohr-orbit-of-a-hydrogen-atom-is-053-1-0-8-cm-the-velocity-of-the-electron-in-the-first-bohr-orbit-is-74603

question:
<div class="question">The radius of the first Bohr orbit of a hydrogen atom is $0.53 \times 10^{-8} \mathrm{~cm}$. The velocity of the electron in the first Bohr orbit is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$2.188 \times 10^{9} \mathrm{~cm} \mathrm{~s}^{-1}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4.376 \times 10^{8} \mathrm{~cm} \mathrm{~s}^{-1}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$1.094 \times 10^{8} \mathrm{~cm} \mathrm{~s}^{-1}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2.188 \times 10^{9} \mathrm{~cm} \mathrm{~s}^{-1}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$2.188 \times 10^{9} \mathrm{~cm} \mathrm{~s}^{-1}$</span> </div>

answer:
<div class="solution">Hint:
<br/>$r=0.53 \times 10^{-9} \mathrm{~cm}=0.53 \times 10^{-9} n^{2}$
<br/>$\therefore \mathrm{n}=1$
<br/>$\mathrm{~V}=2.18 \times 10^{9} \mathrm{cm} / \mathrm{s}$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
title:
the-radius-of-the-light-circle-observed-by-a-fish-at-a-depth-of-12-meter-is-refractive-index-of-water-43

slug:
the-radius-of-the-light-circle-observed-by-a-fish-at-a-depth-of-12-meter-is-refractive-index-of-water-43-72374

question:
<div class="question">The radius of the light circle observed by a fish at a depth of 12 meter is (refractive index of water $=4 / 3$ )</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$36 \sqrt{7}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{36}{\sqrt{7}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$36 \sqrt{5}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$4 \sqrt{5}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{36}{\sqrt{7}}$</span> </div>

answer:
<div class="solution">$$<br/>\text { Hints : } r=\frac{h}{\sqrt{\mu^2-1}}=\frac{12}{\sqrt{\frac{16}{9}-1}}=\frac{12 \times 3}{\sqrt{7}}=\frac{36}{\sqrt{7}}<br/>$$</div>


title: the-range-of-the-function-y-3-sin-16-2-x-2-is
slug: the-range-of-the-function-y-3-sin-16-2-x-2-is-38915
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The range of the function $y=3 \sin \left(\sqrt{\frac{\pi^{2}}{16}-x^{2}}\right)$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$[0, \sqrt{3 / 2}]$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">[0,1]</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$[0,3 / \sqrt{2}]$</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">$[0, \infty)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$[0,3 / \sqrt{2}]$</span> </div>

answer:
<div class="solution">Given function is $y=3 \sin \left(\sqrt{\frac{\pi^{2}}{16}-x^{2}}\right)$
<br/>Here, domain is $\left[0, \frac{\pi}{4}\right]$
<br/>$$
<br/>\begin{array}{l}
<br/>\begin{aligned}
<br/>x=&amp; 0 \\
<br/>y &amp;=3 \sin \left(\sqrt{\frac{\pi^{2}}{16}-0}\right) \\
<br/>&amp;=3 \sin \left(\frac{\pi}{4}\right)=3 \times \frac{1}{\sqrt{2}}=\frac{3}{\sqrt{2}} \\
<br/>\text { When } x=\frac{\pi}{4}, y &amp;=3 \sin \left(\sqrt{\frac{\pi^{2}}{16}-\frac{\pi^{2}}{16}}\right) \\
<br/>&amp;=3 \sin (0)=0
<br/>\end{aligned}
<br/>\end{array}
<br/>$$
<br/>Hence, range of given function is $\left[0, \frac{3}{\sqrt{2}}\right]$</div>


tags: ['Chemistry', 'Chemical Kinetics', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P3.db
title:
the-rate-of-a-certain-reaction-is-given-by-rate-k-h-n-the-rate-increases-100-times-when-the-ph-changes-from-3-to-1-the-order-n-of-the-reaction-is

slug:
the-rate-of-a-certain-reaction-is-given-by-rate-k-h-n-the-rate-increases-100-times-when-the-ph-changes-from-3-to-1-the-order-n-of-the-reaction-is-53368

question:
<div class="question">The rate of a certain reaction is given by, rate $=k\left[\mathrm{H}^{+}\right]^{n}$. The rate increases 100 times when the pH changes from 3 to 1 . The order $(n)$ of the reaction is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">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">1.5</span> </li> </ul>

answer:
<div class="solution">Given, rate $=k\left[\mathrm{H}^{*}\right]^{n}$ 
<br/>Initial $\mathrm{pH}=3 \mathrm{so}\left[\mathrm{H}^{+}\right]=1 \times 10^{-3}$, initial rate $=\mathrm{r}_{\mathrm{1}}$
<br/>Final $\mathrm{pH}=1$ so $\left[\mathrm{H}^{+}\right]=1 \times 10^{-1}$
<br/>Final rate $=r_{2}=100 r_{1}$
<br/>On substituting values, we get
<br/>$$
<br/>\begin{array}{l}
<br/>r_{1}=k\left[1 \times 10^{-3}\right]^{n} \\
<br/>r_{2}=100 r_{1}=k\left[10^{-1}\right]^{n}
<br/>\end{array}
<br/>$$
<br/>Dividing eqs (i) by (ii).
<br/>$$
<br/>\frac{r_{1}}{100 r_{1}}=\left[\frac{1 \times 10^{-3}}{1 \times 10^{-1}}\right]^{n}
<br/>$$
<br/>$\therefore$
<br/>$$
<br/>\begin{array}{c}
<br/>\frac{1}{100}=\left[\frac{10^{-2}}{1}\right]^{n} \Rightarrow\left[\frac{1}{100}\right]^{1}=\left[\frac{1}{100}\right]^{n} \\
<br/>n=1
<br/>\end{array}
<br/>$$
<br/>Thus, the reaction is of first order.</div>


tags: ['Physics', 'Gravitation', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P3.db
title:
the-ratio-of-accelerations-due-to-gravity-g-1-g-2-on-the-surfaces-of-two-planets-is-5-2-and-the-ratio-of-their-respective-average-densities-1-2-is-2-1

slug:
the-ratio-of-accelerations-due-to-gravity-g-1-g-2-on-the-surfaces-of-two-planets-is-5-2-and-the-ratio-of-their-respective-average-densities-1-2-is-2-1-35862

question:
<div class="question">The ratio of accelerations due to gravity $g_{1}:g_{2}$ on the surfaces of two planets is 5: 2 and the ratio of their respective average densities $\rho_{1}: \rho_{2}$ is 2: 1 . What is the ratio of respective escape velocities $v_{1}: v_{2}$ from the surface of the planets?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">5: 2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\sqrt{5}: \sqrt{2}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$5: 2 \sqrt{2}$</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">25.4</span> </li> </ul>

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

answer:
<div class="solution">Escape velocity from a planet. $v_{0}=\sqrt{\frac{2 G M}{R}}=\sqrt{\frac{2 G M}{R^{2}}} R$
<br/>$$
<br/>=\sqrt{2 g R}
<br/>$$
<br/>According due to gravity $g=\frac{G M}{R^{2}}=\frac{G\frac{4}{3}\pi R^{3} \cdot \rho}{R^{2}}=\frac{4}{3} G\pi R_{p}$
<br/>$\therefore \quad$ Radius $R=\frac{3 g}{4 \pi G \rho}$
<br/>From Eqs. (i) and (ii), we get $v_{e}=\sqrt{2 g \cdot \frac{3 g}{4 \pi G \rho}}=\sqrt{\frac{3}{2} \frac{g^{2}}{\pi G \rho}}$
<br/>Thus, $\quad v_{e} \propto \frac{g}{\sqrt{\rho}}$
<br/>$\therefore \quad \frac{v_{n}}{v_{r_{2}}}=\frac{g_{1}}{\sqrt{\rho_{1}}} \times \frac{\sqrt{\rho_{2}}}{g_{2}}=\frac{5}{2} \times \frac{1}{\sqrt{2}}$
<br/>(given, $\frac{g_{1}}{g_{2}}=\frac{5}{2}$ and $\frac{Q}{\rho_{2}}=2: 1$ )
<br/>$=\frac{5}{2 \sqrt{2}}$</div>


tags: ['Physics', 'Magnetic Effects of Current', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P3.db
title:
the-ratio-of-magnetic-field-and-magnetic-moment-at-the-centre-of-a-current-carrying-circular-loop-is-x-when-both-the-current-and-radius-is-doubled-the

slug:
the-ratio-of-magnetic-field-and-magnetic-moment-at-the-centre-of-a-current-carrying-circular-loop-is-x-when-both-the-current-and-radius-is-doubled-the-61887

question:
<div class="question">The ratio of magnetic field and magnetic moment at the centre of a current carrying circular loop is $x$. When both the current and radius is doubled the ratio will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$x / 8$</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">$x / 4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x / 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$2 x$</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } \mathrm{B}=\frac{\mu_0 \mathrm{I}}{2 a} \quad \quad \mathrm{M}=\mathrm{I}\left(\pi a^2\right) \\<br/>&amp; \frac{\mathrm{B}}{\mathrm{M}}=\frac{\mu_0 \mathrm{I}}{2 a} \times \frac{1}{\mathrm{I} \pi a^2}=\frac{\mu_0}{2 \pi a^3}=x<br/>\end{aligned}<br/>$$<br/>Again, Ratio $=\frac{\mu_0}{2 \pi(2 a)^3}=\frac{1}{8}\left(\frac{\mu_0}{2 \pi a^3}\right)=\frac{x}{8}$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P3.db
title:
the-ratio-of-the-diameter-of-the-sun-to-the-distance-between-the-earth-and-the-sun-is-approximately-0009-the-approximate-diameter-of-the-image-of-the-

slug:
the-ratio-of-the-diameter-of-the-sun-to-the-distance-between-the-earth-and-the-sun-is-approximately-0009-the-approximate-diameter-of-the-image-of-the-79554

question:
<div class="question">The ratio of the diameter of the sun to the distance between the earth and the sun is approximately $0.009 .$ The approximate diameter of the image of the sun formed by a concave spherical mirror of radius of curvature $0.4 \mathrm{m}$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$4.5 \times 10^{-6} \mathrm{m}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$4.0 \times 10^{-6} \mathrm{m}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$3.6 \times 10^{-3} \mathrm{m}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$1.8 \times 10^{-3} \mathrm{m}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1.8 \times 10^{-3} \mathrm{m}$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_phy_a18.png"/> <br/> <br/>The angle subtended by sun on mirror
<br/>;
<br/>$$
<br/>\begin{array}{l}
<br/>Q=\frac{D}{d_{S E}}=\frac{d}{f} \\
<br/>d=\frac{D}{d_{s E}} \times f
<br/>\end{array}
<br/>$$
<br/>According to the question,
<br/>$$
<br/>\begin{aligned}
<br/>\frac{D}{d_{s k}} &amp;=0 \cdot 009 \\
<br/>f &amp;=\frac{r}{2}=\frac{0.4}{2}=0 \cdot 2 \mathrm{m}
<br/>\end{aligned}
<br/>$$
<br/>$\therefore \quad d=0 \cdot 009 \times 0 \cdot 2 \mathrm{m}$
<br/>$$
<br/>\begin{array}{l}
<br/>=9 \times 2 \times 10^{-4} \mathrm{m} \\
<br/>=18 \times 10^{-4} \mathrm{m} \\
<br/>=1.8 \times 10^{-3} \mathrm{m}
<br/>\end{array}
<br/>$$</div>


tags: ['Chemistry', 'Ionic Equilibrium', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
title:
the-ratio-of-volumes-of-ch-3-cooh-01-n-to-ch-coona-01-n-required-to-prepare-a-buffer-solution-of-ph-574-is-given-p-k-a-of-c-h-3-coo-h-is-474

slug:
the-ratio-of-volumes-of-ch-3-cooh-01-n-to-ch-coona-01-n-required-to-prepare-a-buffer-solution-of-ph-574-is-given-p-k-a-of-c-h-3-coo-h-is-474-67904

question:
<div class="question">The ratio of volumes of $\mathrm{CH}_{3} \mathrm{COOH } \text{ 0.1}$ (N) to CH COONa 0.1 (N) required to prepare a buffer solution of $\mathrm{pH} 5.74$ is  (Given, $p K_{a}$ of $C H_{3} C O O H$ is $\left.4.74\right)$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$1 : 10$</span> </div>

answer:
<div class="solution">Given, that $\mathrm{pH}=5.74, \mathrm{pK}_{a}=4.74$
<br/>Suppose that volume of acid solution $=x \mathrm{L}$ Volume of salt solution $=y \mathrm{L}$
<br/>From Henderson equation,
<br/>$$
<br/>\mathrm{pH}=\mathrm{p} K_{\mathrm{a}}+\log \frac{[\mathrm{Salt}]}{[\mathrm{Acid}]}
<br/>$$
<br/>or, $\mathrm{pH}-\mathrm{p} K_{\mathrm{a}}=\log \frac{\left[\mathrm{CH}_{3} \mathrm{COONa}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]}$
<br/>or, $5.74-4.74=1=\log \frac{\left[\mathrm{CH}_{3} \mathrm{COONa}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]}$
<br/>or $\frac{\left.\mathrm{CH}_{3} \mathrm{COONa}\right]}{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]}=10$
<br/>or $\frac{\left[\mathrm{CH}_{3} \mathrm{COOH}\right]}{\left[\mathrm{CH}_{3} \mathrm{COONa}\right]}=\frac{1}{10}=\frac{\frac{x+y}{0.1 \mathrm{y}}}{\mathrm{x}+\mathrm{y}}$
<br/>Thus, $\frac{x}{y}=\frac{1}{10}$</div>


tags: ['Chemistry', 'd and f Block Elements', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">Iron</span> </div>
answer: <div class="solution">Hint: <br/>Thermite Process</div>
source_db: db\MarksBatch2_P3.db
title:
the-reaction-for-obtaining-the-metal-m-from-its-oxide-m-2-o-3-ore-is-given-by-m-2-o-3-s-2-al-heat-al-2-o-3-2-m-s-s-solid-liquid-in-that-case-m-is

slug:
the-reaction-for-obtaining-the-metal-m-from-its-oxide-m-2-o-3-ore-is-given-by-m-2-o-3-s-2-al-heat-al-2-o-3-2-m-s-s-solid-liquid-in-that-case-m-is-14315

question:
<div class="question">The reaction for obtaining the metal (M) from its oxide $\left(\mathrm{M}_{2} \mathrm{O}_{3}\right)$ ore is given by
<br/>$\mathrm{M}_{2} \mathrm{O}_{3}(\mathrm{~s})+2 \mathrm{Al}(\ell) \stackrel{\text { Heat }}{\rightarrow} \mathrm{Al}_{2} \mathrm{O}_{3}(\ell)+2 \mathrm{M}(\mathrm{s}),(\mathrm{s}=$ solid, $\ell=$ liquid $)$
<br/>in that case, $M$ is</div>

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


title: the-reaction-of-aniline-with-chloroform-under-alkaline-conditions-leads-to-the-formation-of-1
slug: the-reaction-of-aniline-with-chloroform-under-alkaline-conditions-leads-to-the-formation-of-1-73076
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Main']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The reaction of aniline with chloroform under alkaline conditions leads to the formation of</div>

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

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

answer:
<div class="solution">Aniline is a primary amine (contains - $\mathrm{NH}_{2}$ group). When it is treated with chloroform under alkaline conditions, it results in the formation of a bad smelling compound, known by the name isonitrile or carbylamine. This reaction is known as carbylamine reaction. <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_a44.png"/></div>


title: the-reaction-of-methyltrichloroacetate-cl-3-cco-2-me-with-sodium-methoxide-n-a-om-e-generates
slug: the-reaction-of-methyltrichloroacetate-cl-3-cco-2-me-with-sodium-methoxide-n-a-om-e-generates-41215
tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The reaction of methyltrichloroacetate $\left(\mathrm{Cl}_{3} \mathrm{CCO}_{2} \mathrm{Me}\right)$ with sodium methoxide $\mathrm({NaOMe})$ generates</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_che_a67.png"/></div>


tags: ['Chemistry', 'General Organic Chemistry', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P3.db
title:
the-reaction-of-nitroprusside-anion-with-sulphide-ion-gives-purple-colouration-due-to-the-formation-of

slug:
the-reaction-of-nitroprusside-anion-with-sulphide-ion-gives-purple-colouration-due-to-the-formation-of-81320

question:
<div class="question">The reaction of nitroprusside anion with sulphide ion gives purple colouration due to the formation of</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">the tetranionic complex of iron (II) coordinating to one $NOS^{-}$ 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">B</span> <span class="option-data">the dianoinic complex of iron (II) coordinating to one $NCS^{-}$ ion</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">the trianoionic complex of iron (III) coordinating to one $NOS^{-}$ ion</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">the tetranionic complex of iron (III) coordinating to one $NCS^{-}$ ion</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">the tetranionic complex of iron (II) coordinating to one $NOS^{-}$ ion</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2013_che_a21.png"/></div>


title: the-reaction-sequence-given-below-gives-product-r-the-structure-of-the-product-r-is-1
slug: the-reaction-sequence-given-below-gives-product-r-the-structure-of-the-product-r-is-1-17803
tags: ['Chemistry', 'Haloalkanes and Haloarenes', 'JEE Main']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The reaction sequence given below gives product $R$
<br/>The structure of the product $R$ is
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q73.png"/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q73_opta.png"/></span> </li><li class=""> <span class="option-label">B</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q73_optb.png"/></span> </li><li class=""> <span class="option-label">C</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_q73_optc.png"/></span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Xbfs4l1DJ30bsvlTtMhiX_iQB_Zmg78cUmybEwYd7nw.original.fullsize.png"/></span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/Xbfs4l1DJ30bsvlTtMhiX_iQB_Zmg78cUmybEwYd7nw.original.fullsize.png"/></span> </div>

answer:
<div class="solution">The reaction occurs as follows :
<br/>The above reaction is an example of Borodine Hunsdiecker reaction.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2017_che_a73.png"/></div>


title: the-reaction-that-takes-place-when-cl-2-gas-is-passed-through-conc-naoh-solution-is
slug: the-reaction-that-takes-place-when-cl-2-gas-is-passed-through-conc-naoh-solution-is-60868
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The reaction that takes place when $\mathrm{Cl}_2$ gas is passed through conc. $\mathrm{NaOH}$ solution is :</div>

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/14LN3LkuyBdwCMmWPnwl3iil4osCS2_YKNtI9kadVgA.original.fullsize.png"/><br/><br/>Hence the reaction is disproportionation</div>


title: the-reactive-species-in-chlorine-bleach-is
slug: the-reactive-species-in-chlorine-bleach-is-87321
question: <div class="question">The reactive species in chlorine bleach is</div>
tags: ['Chemistry', 'p Block Elements (Group 15, 16, 17 & 18)', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P3.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{Cl}_{2} \mathrm{O}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\mathrm{OCl}^{-}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$\mathrm{Cl} \mathrm{O}_{2}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{HCl}$</span> </li> </ul>

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

answer:
<div class="solution">Chlorine bleach is $\mathrm{NaOCl}$. The hypochlorite ion in chlorine bleach dissociates to give nascent oxygen as shown below.
<br/>$$
<br/>\mathrm{OCl}^{-} \longrightarrow[\mathrm{O}]+\mathrm{Cl}^{-}
<br/>$$
<br/>Therefore, the reactive species in chlorine bleach is $\mathrm{OCl}^{-}$
<br/>So, the option (b) is correct.</div>


title: the-reading-of-the-ammeter-in-the-following-figure-will-be-1
slug: the-reading-of-the-ammeter-in-the-following-figure-will-be-1-84951
tags: ['Physics', 'Current Electricity', 'JEE Main']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">$$<br/>\text { The reading of the ammeter in the following figure will be }<br/>$$<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/sMOnbBXBeV9DbjsnOtRTKZqiOZHSwqU0AZVaO6NSsr0.original.fullsize.png"/><br/></div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.8 \mathrm{~A}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$0.6 \mathrm{~A}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$0.4 \mathrm{~A}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.2 \mathrm{~A}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.4 \mathrm{~A}$</span> </div>

answer:
<div class="solution">Hints: $\frac{1}{R}=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}=\frac{3+2+1}{6}=1 \Omega$<br/>$$<br/>\begin{aligned}<br/>&amp; \mathrm{R}_{\text {eq }}=1+4=5 \Omega \\<br/>&amp; \mathrm{I}=\frac{2}{5}=0.4 \mathrm{~A}<br/>\end{aligned}<br/>$$</div>


title: the-reagent-with-which-the-following-reaction-is-best-accomplished-is
slug: the-reagent-with-which-the-following-reaction-is-best-accomplished-is-83031
tags: ['Chemistry', 'Amines', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P3.db
question:
<div class="question">The reagent with  which the following reaction is best accomplished is <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2014_che_q42.png"/></div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\mathrm{H}_{3} \mathrm{PO}_{2}$</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">$\mathrm{H}_{3} \mathrm{PO}_{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{H}_{3} \mathrm{PO}_{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$NaHSO_{3}$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{H}_{3} \mathrm{PO}_{2}$ is the reagent that converts $-\mathrm{N}_{2}^{+} \mathrm{Cl}^{-}$ group into H. During this process $\mathrm{N}_{2}$ and $\mathrm{HCl}$ gases are released.</div>


