# questions

tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-i$</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-p-q-and-r-are-angles-of-an-isosceles-triangle-and-p-2-then-the-value-of-cos-3-p-i-sin-3-p-3-cos-q-i-sin-q-cos-r-i-sin-r-cos-p-i-sin-p-cos-q-i-sin-q

slug:
if-p-q-and-r-are-angles-of-an-isosceles-triangle-and-p-2-then-the-value-of-cos-3-p-i-sin-3-p-3-cos-q-i-sin-q-cos-r-i-sin-r-cos-p-i-sin-p-cos-q-i-sin-q-66660

question:
<div class="question">If $P, Q$ and $R$ are angles of an isosceles triangle and $\angle P=\frac{\pi}{2},$ then the value of $\left(\cos \frac{P}{3}-i \sin \frac{P}{3}\right)^{3}+(\cos Q+i \sin Q) \\ (\cos R-i \sin R)+(\cos P-i \sin P) \\ (\cos Q-i \sin Q)(\cos R-i \sin R) \text { is }$</div>

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

answer:
<div class="solution">Given that, $P, Q$ and $R$ are angles of an isosceles triangle and $\angle P=\frac{\pi}{2}$
<br/>$\therefore \quad Q=R=\frac{\pi}{4}\left(\because P+Q+R=180^{\circ}\right)$
<br/>Now, $\left(\cos \frac{P}{3}-i \sin \frac{P}{3}\right)^{3}+(\cos Q+i \sin Q)$
<br/>$(\cos R-i \sin R)$
<br/>$+(\cos P-i \sin P)(\cos Q-i \sin Q)$
<br/>$\quad(\cos R-i \sin R)$
<br/>$=\left(e^{-i \frac{P}{3}}\right)^{3}+e^{i Q} \cdot e^{-i R}+e^{-i \rho} e^{-i Q} \cdot e^{-i R}$
<br/>$\left(\because \cos \theta+i \sin \theta=e^{i \theta}\right)$
<br/>$=e^{-i P}+e^{\gamma Q-R)}+e^{-i(P+Q+R)}$
<br/>$=e^{-i \pi / 2}+e^{(0)}+e^{-i \pi} \quad\left(Q=R=\frac{\pi}{2} P=\frac{\pi}{2}\right)$
<br/>$=-i+1+(-1)=-i$</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-p-q-and-r-are-angles-of-pqr-then-the-value-of-1-cos-r-cos-q-cos-r-1-cos-p-cos-q-cos-p-1-is-equal-to

slug:
if-p-q-and-r-are-angles-of-pqr-then-the-value-of-1-cos-r-cos-q-cos-r-1-cos-p-cos-q-cos-p-1-is-equal-to-52390

question:
<div class="question">If $P, Q$ and $R$ are angles of $\Delta P Q R$, then the value of $\left|\begin{array}{ccc}-1 &amp; \cos R &amp; \cos Q \\ \cos R &amp; -1 &amp; \cos P \\ \cos Q &amp; \cos P &amp; -1\end{array}\right|$ is equal to</div>

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

answer:
<div class="solution">$$
<br/>\begin{aligned}
<br/>&amp;\text { Multiplying } C_{1} \text { by } p \text { and then doing }\\
<br/>&amp;\begin{aligned}
<br/>C_{1} &amp; \rightarrow C_{1}+q C_{2}+r C_{3} \text { , we get } \\
<br/>\Delta &amp;=\frac{1}{p}\left[\begin{array}{ccc}
<br/>-p &amp; \cos R &amp; \cos Q \\
<br/>p \cos R &amp; -1 &amp; \cos P \\
<br/>p \cos Q &amp; \cos P &amp; -1
<br/>\end{array} \mid\right.\\
<br/>&amp;=\frac{1}{p}\left|\begin{array}{ccc}
<br/>-p+q \cos R+r \cos Q &amp; \cos R &amp; \cos Q \\
<br/>p \cos R-q+r \cos P &amp; -1 &amp; \cos P \\
<br/>p \cos Q+q \cos P-r &amp; \cos P &amp; -1
<br/>\end{array}\right| \\
<br/>=\frac{1}{p}\left|\begin{array}{ccc}
<br/>0 &amp; \cos R &amp; \cos Q \\
<br/>0 &amp; -1 &amp; \cos P \\
<br/>0 &amp; \cos P &amp; -1
<br/>\end{array}\right|=0
<br/>\end{aligned}
<br/>\end{aligned}
<br/>$$
<br/>Using the prejection formula $p=q \cos R+r \cos Q$</div>


title: if-p-q-are-odd-integers-then-the-roots-of-the-equation-2-p-x-2-2-p-q-x-q-0-are
slug: if-p-q-are-odd-integers-then-the-roots-of-the-equation-2-p-x-2-2-p-q-x-q-0-are-33956
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $p, q$ are odd integers, then the roots of the equation $2 p x^{2}+(2 p+q) x+q=0$ are</div>

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

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

answer:
<div class="solution">Given equation.
<br/>$$
<br/>\begin{array}{l}
<br/>2 p x^{2}+(2 p+q) x+q=0 \\
<br/>\therefore D=(2 p+q)^{2}-4(2 p)(q) \\
<br/>\quad=4 p^{2}+q^{2}+4 p q-8 p q \\
<br/>\quad=4 p^{2}+q^{2}-4 p q \\
<br/>\quad=(2 p-q)^{2}
<br/>\end{array}
<br/>$$
<br/>$=$ a perfect square
<br/>Given equation has rational roots</div>


title: if-p-q-are-the-roots-of-the-equation-x-2-p-x-q-0-then
slug: if-p-q-are-the-roots-of-the-equation-x-2-p-x-q-0-then-10885
question: <div class="question">If $p, q$ are the roots of the equation $x^{2}+p x+q=0,$ then</div>
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$p=1, q=-2$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$p=0, q=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$p=-2, q=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$p=-2, q=1$</span> </li> </ul>

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

answer:
<div class="solution">Given, $\rho, q$ are roots of equation $x^{2}+p x+q=0$
<br/>$\therefore \quad p+q=-p \Rightarrow 2 p+q=0$
<br/>$\begin{array}{lr}\text { and } &amp; p q=q \\ \Rightarrow &amp; \rho=1\end{array}$
<br/>Put $p=1$ in Eq, we get $2 \times 1+q=0$
<br/>$\begin{array}{lr}\Rightarrow &amp; q+2=0 \\ \Rightarrow &amp; q=-2 \\ \text { Hence, } p=1, q=-2\end{array}$</div>


title: if-p-x-a-x-2-bx-c-and-q-x-ax-2-dx-c-where-ac-0-a-b-c-d-are-all-real-then-p-x-q-x-0-has
slug: if-p-x-a-x-2-bx-c-and-q-x-ax-2-dx-c-where-ac-0-a-b-c-d-are-all-real-then-p-x-q-x-0-has-64872
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\mathrm{P}(\mathrm{x})=a \mathrm{x}^{2}+\mathrm{bx}+\mathrm{c}$ and $\mathrm{Q}(\mathrm{x})=-\mathrm{ax}^{2}+\mathrm{dx}+\mathrm{c}$, where $\mathrm{ac} \neq 0 \quad[\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ are all real], then $\mathrm{P}(\mathrm{x}) \cdot \mathrm{Q}(\mathrm{x})=0$ has</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">at least two real roots</span> </div>

answer:
<div class="solution">Hint:
<br/>If $P(x)=a x^{2}+b x+c, Q(x)=-a x^{2}+d x+c$
<br/>$\mathrm{D}_{1}=\mathrm{b}^{2}-4 \mathrm{ac}$
<br/>$\mathrm{D}_{2}=\mathrm{d}^{2}+4 \mathrm{ac}$
<br/>$\Rightarrow \mathrm{D}_{1}+\mathrm{D}_{2}&gt;0$
<br/>Atleast two real roots.</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
title:
if-pq-is-a-double-ordinate-of-the-hyperbola-a-2-x-2-b-2-y-2-1-such-that-opq-is-equilateral-o-being-the-centre-then-the-eccentricity-e-satisfies

slug:
if-pq-is-a-double-ordinate-of-the-hyperbola-a-2-x-2-b-2-y-2-1-such-that-opq-is-equilateral-o-being-the-centre-then-the-eccentricity-e-satisfies-36806

question:
<div class="question">If $P Q$ is a double ordinate of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ such that $\Delta O P Q$ is equilateral. $O$ being the centre. Then, the eccentricity $e$ satisfies</div>

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

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

answer:
<div class="solution">Given equation of hyperbola is $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ and $\Delta O P Q$ is an equlateral triangle. $P Q$ is double ordinate of the hyperbola.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_mat_a39.png"/> <br/> <br/>Let the coordinates of $P$ and $Q$ be $(a \sec \theta, b$ tan $\theta)$ and $(a \sec \theta,-b$ tane $),$ respectively.
<br/>In$\Delta O P Q$,
<br/>$O P=P Q$
<br/>$\Rightarrow \quad a^{2} \sec ^{2} \theta+b^{2} \tan ^{2} \theta=(2 b \tan \theta)^{2}$
<br/>$\Rightarrow \quad a^{2} \sec ^{2} \theta=3 b^{2} \tan ^{2} \theta$
<br/>$\Rightarrow \quad \sin ^{2} \theta=\frac{a^{2}}{3 b^{2}}$
<br/>Now, $\sin ^{2} \theta &lt; 1$
<br/>$\Rightarrow \quad \frac{a^{2}}{3 b^{2}} &lt; 1$
<br/>$\frac{b^{2}}{a^{2}}&gt;\frac{1}{3}$
<br/>$\Rightarrow \quad 1+\frac{b^{2}}{a^{2}}&gt;\frac{4}{3}$
<br/>$\therefore$
<br/>$e^{2}&gt;\frac{4}{3} \Rightarrow e&gt;\frac{2}{\sqrt{3}}$</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
title:
if-pressure-of-real-gas-o-2-in-a-container-is-given-by-p-2-v-b-rt-4-b-2-a-then-the-mass-of-the-gas-in-the-container-is

slug:
if-pressure-of-real-gas-o-2-in-a-container-is-given-by-p-2-v-b-rt-4-b-2-a-then-the-mass-of-the-gas-in-the-container-is-45289

question:
<div class="question">If pressure of real gas $\mathrm{O}_{2}$ in a container is given by $\mathrm{P}=\frac{\mathrm{RT}}{2 \mathrm{~V}-\mathrm{b}}-\frac{\mathrm{a}}{4 \mathrm{~b}^{2}}$, then the mass of the gas in the container is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$32 \mathrm{gm}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$16 \mathrm{gm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$4 \mathrm{gm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$64 \mathrm{gm}$</span> </li> </ul>

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

answer:
<div class="solution">$\mathrm{P}=\frac{\mathrm{RT}}{2 \mathrm{~V}-\mathrm{b}}-\frac{\mathrm{a}}{4 \mathrm{~b}^{2}}$
<br/>$\left[\mathrm{P}+\frac{\mathrm{a}}{4 \mathrm{~b}^{2}}\right]=\frac{\mathrm{RT}}{2 \mathrm{~V}-\mathrm{b}}$
<br/>$2\left[\mathrm{P}+\frac{\mathrm{a}}{4 \mathrm{~b}^{2}}\right]\left[\mathrm{V}-\frac{\mathrm{b}}{2}\right]=\mathrm{RT}$
<br/>$\left[\mathrm{P}+\frac{\mathrm{a}}{4 \mathrm{~b}^{2}}\right]\left[\mathrm{V}-\frac{\mathrm{b}}{2}\right]=\frac{\mathrm{RT}}{2}$
<br/>$\therefore \mathrm{n}=1 / 2 \mathrm{~mole}$
<br/>$\therefore$ Mass of $\mathrm{O}_{2}=\frac{1}{2} \times 32=16 \mathrm{gm}$</div>


title: if-r-and-1-2-i-c-o-s-1-i-c-o-s-is-real-number-then-will-be-when-i-set-of-integers
slug: if-r-and-1-2-i-c-o-s-1-i-c-o-s-is-real-number-then-will-be-when-i-set-of-integers-80548
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\theta \in \mathbb{R}$ and $\frac{1-i \cos \theta}{1+2 i \cos \theta}$ is real number, then $\theta$ will be (when $I:$ Set of integers)</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$(2 n+1) \frac{\pi}{2}, n \in l$</span> </div>

answer:
<div class="solution">Let $Z=\frac{1-i \cos \theta}{1+2 i \cos \theta}$
<br/>$\therefore \quad \bar{z}=\frac{1+i \cos \theta}{1-2 i \cos \theta}$
<br/>Since, $z$ is a real number, then $z-\bar{z}=0$ $\Rightarrow \frac{1-i \cos \theta}{1+2 i \cos \theta}=\frac{1+i \cos \theta}{1-2 i \cos \theta}$
<br/>$\Rightarrow(1-i \cos \theta)(1-2 i \cos \theta)=(1+i \cos \theta)(1+2 i \cos \theta)$
<br/>$\Rightarrow 1-2 i \cos \theta-i \cos \theta-2 \cos ^{2} \theta$ $=1+2 i \cos \theta+i \cos \theta-2 \cos ^{2} \theta$
<br/>$\Rightarrow \quad 6 i \cos \theta=0$
<br/>$\Rightarrow \quad \cos \theta=0$
<br/>$\Rightarrow \quad \theta=(2 n+1) \frac{\pi}{2}, n \in I$</div>


title: if-r-and-mathrmr1-are-equivalence-relations-on-a-set-a-then-so-are-the-relation
slug: if-r-and-mathrmr1-are-equivalence-relations-on-a-set-a-then-so-are-the-relation-45676
tags: ['Mathematics', 'Sets and Relations', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(R\) and \(\mathrm{R}^1\) are equivalence relations on a set \(A\), then so are the relation</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">\(\mathrm{R}^{-1}\), \(\mathrm{R} \cap \mathrm{R}^1\)</span> </div>

answer:
<div class="solution">Hint: If \(R ~\&amp;~ R^1\) are equivalence relation<br/>\(\mathrm{R}^{-1} \Rightarrow\) Equivalence<br/>\(\mathrm{R} \cap \mathrm{R}^1 \Rightarrow\) Equivalence<br/>\(R \cup R^1\) may or may not equivalence</div>


title: if-r-be-the-set-of-all-real-numbers-and-f-r-r-is-given-by-f-x-3-x-2-1-then-the-set-f-1-1-6-is
slug: if-r-be-the-set-of-all-real-numbers-and-f-r-r-is-given-by-f-x-3-x-2-1-then-the-set-f-1-1-6-is-42748
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $R$ be the set of all real numbers and $f: R \rightarrow R$ is given by $f(x)=3 x^{2}+1$. Then, the set $f^{-1}([1,6])$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left\{-\sqrt{\frac{5}{3}}, 0, \sqrt{\frac{5}{3}}\right\}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left[-\sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right]$</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">C</span> <span class="option-data">$\left[-\sqrt{\frac{1}{3}}, \sqrt{\frac{1}{3}}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(-\sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right)$</span> </li> </ul>

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

answer:
<div class="solution">Given, $f(x)=3 x^{2}+1$
<br/>Let $\quad y=3 x^{2}+1$
<br/>$\Rightarrow \quad 3 x^{2}=y-1 \Rightarrow x^{2}=\frac{y-1}{3}$
<br/>$\Rightarrow \quad x=\pm \sqrt{\frac{y-1}{3}}$
<br/>$\therefore \quad f^{-1}(x)=\pm \sqrt{\frac{x-1}{3}}$
<br/>When $x \in[1,6]$
<br/>Then, $\quad f^{-1}(x) \in\left[-\sqrt{\frac{5}{3}}, \sqrt{\frac{5}{3}}\right]$</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
if-r-is-the-rydberg-constant-in-cm-1-then-hydrogen-atom-does-not-emit-any-radiation-of-wavelength-in-the-range-of

slug:
if-r-is-the-rydberg-constant-in-cm-1-then-hydrogen-atom-does-not-emit-any-radiation-of-wavelength-in-the-range-of-66381

question:
<div class="question">If $\mathrm{R}$ is the Rydberg Constant in $\mathrm{cm}^{-1}$, then hydrogen atom does not emit any radiation of wave-length in the range of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{1}{\mathrm{R}}$ to $\frac{4}{3 \mathrm{R}} \mathrm{cm}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{7}{5 \mathrm{R}}$ to $\frac{19}{5 \mathrm{R}} \mathrm{cm}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{4}{\mathrm{R}}$ to $\frac{36}{5 \mathrm{R}} \mathrm{cm}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{9}{\mathrm{R}}$ to $\frac{144}{7 \mathrm{R}} \mathrm{cm}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{7}{5 \mathrm{R}}$ to $\frac{19}{5 \mathrm{R}} \mathrm{cm}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$\frac{1}{\lambda}=\mathrm{R}\left[\frac{1}{\mathrm{n}_{\mathrm{f}^{2}}}-\frac{1}{\mathrm{n}_{\mathrm{i}^{2}}}\right]$
<br/>For range of wavelengths:$\mathrm{n}_{\mathrm{i}}=1,2,3, \ldots . .$ for Lyman, Balmer, Paschen, $\ldots \ldots$
<br/>$\mathrm{n}_{\mathrm{f}}=\mathrm{n}_{\mathrm{i}}+1$ and $\mathrm{n}_{\mathrm{f}}=\infty$ for upper and lower range
<br/>Thus, Lyman : $\left[\frac{1}{\mathrm{R}}\right.$ to $\left.\frac{4}{3 \mathrm{R}}\right]$, Balmer : $\left[\frac{4}{\mathrm{R}}\right.$ to $\left.\frac{36}{5 \mathrm{R}}\right]$, Paschen : $\left[\frac{9}{\mathrm{R}} \mathrm{to} \frac{144}{36 \mathrm{R}}\right]$, Bracket : $\left[\frac{16}{\mathrm{R}}\right.$ to $\left.\frac{400}{9 \mathrm{R}}\right]$, Pfund : $\left[\frac{25}{\mathrm{R}}\right.$ to $\left.\frac{900}{11 \mathrm{R}}\right]$</div>


title: if-radium-and-chlorine-combine-to-form-radium-chloride-the-compound-would-be
slug: if-radium-and-chlorine-combine-to-form-radium-chloride-the-compound-would-be-55082
tags: ['Chemistry', 's Block Elements', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If radium and chlorine combine to form radium chloride, the compound would be</div>

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

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

answer:
<div class="solution">Radium in isolated form and in its chloride (i.e. $Ra^{+}$ ) only differ in the number of orbital electron. But, radioactivity is independent of chemical environment of an ion or atom, i.e. radiouctivity is the phenomenon which does not depend on the orbital electrons but depends only on the composition of nucleus</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-s-r-2-r-6-r-2-1-4-r-3-2-n-r-x-y-z-n-n-1-n-2-2-n-3-n-3-n-1-then-the-value-of-r-1-n-s-r-is-independent-of

slug:
if-s-r-2-r-6-r-2-1-4-r-3-2-n-r-x-y-z-n-n-1-n-2-2-n-3-n-3-n-1-then-the-value-of-r-1-n-s-r-is-independent-of-81925

question:
<div class="question">If $S_{r}=\left|\begin{array}{ccc}2 r &amp; x &amp; n(n+1) \\ 6 r^{2}-1 &amp; y &amp; n^{2}(2 n+3) \\ 4 r^{3}-2 n r &amp; z &amp; n^{3}(n+1)\end{array}\right|$, then the
<br/>value of $\sum_{r=1}^{n} S_{r}$ is independent of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">only $x$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">only $y$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">only $n$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$x y, z$ and $n$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x y, z$ and $n$</span> </div>

answer:
<div class="solution">We have, $S_{r}=\left|\begin{array}{ccc}2 r &amp; x &amp; n(n+1) \\ 6 r^{2}-1 &amp; y &amp; n^{2}(2 n+3) \\ 4 r^{3}-2 n r &amp; z &amp; n^{3}(n+1)\end{array}\right|$
<br/>$\Rightarrow \quad \sum_{r=1}^{n} S_{r}=\left|\begin{array}{ccc}2 \sum_{r=1}^{n} r &amp; x &amp; n(n+1) \\ \sum_{r=1}^{n}\left(6 r^{2}-1\right) &amp; y &amp; n^{2}(2 n+3) \\ \sum_{r=1}^{n}\left(4 r^{3}-2 n r\right) &amp; z &amp; n^{3}(n+1)\end{array}\right|$
<br/>$=\left|\begin{array}{ccc}n(n+1) &amp; x &amp; n(n+1) \\ n^{2}(2 n+3) &amp; y &amp; n^{2}(2 n+3) \\ n^{3}(n+1) &amp; z &amp; n^{3}(n+1)\end{array}\right|$
<br/>$=0 \quad\left(\because C_{1}\right.$ and $C_{3}$ are identical $)$
<br/>Hence, $\sum_{r=1}^{n} S_{r}$ is independent of $x, y, z$ and $n$.</div>


title: if-sin-1-13-x-cosec-1-12-13-2-then-the-value-of-x-is
slug: if-sin-1-13-x-cosec-1-12-13-2-then-the-value-of-x-is-93820
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">5</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2},$ then the value of $x$ is</div>

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

answer:
<div class="solution">Given,<br/>$\sin ^{-1}\left(\frac{x}{13}\right)+\operatorname{cosec}^{-1}\left(\frac{13}{12}\right)=\frac{\pi}{2} ..(i)$<br/>Let $\operatorname{cosec}^{-1} \frac{13}{12}=y$<br/>Then. $\operatorname{cosec} y=\frac{13}{12} \Rightarrow \sin y=\frac{12}{13}$<br/>$\therefore \cos y=\sqrt{1-\sin ^{2} y}$<br/>$\begin{array}{l}<br/>=\sqrt{1-\left(\frac{12}{13}\right)^{2}}=\sqrt{1-\frac{144}{169}} \\<br/>=\sqrt{\frac{25}{169}}=\frac{5}{13} \Rightarrow y=\cos ^{-1} \frac{5}{13}<br/>\end{array}$<br/>Eq. (i) becomes, $\sin ^{-1}\left(\frac{x}{13}\right)+\cos ^{-1}\left(\frac{5}{13}\right)=\frac{\pi}{2}$<br/>We know that<br/>$\sin ^{-1} \theta+\cos ^{-1} \theta=\frac{\pi}{2}$<br/>$\therefore$ Both angles of Eq. (i) should be same.<br/>$\therefore x=5$</div>


title: if-sin-1-x-2-x-2-4-x-3-8-x-4-6-where-x-2-then-the-value-of-x-is
slug: if-sin-1-x-2-x-2-4-x-3-8-x-4-6-where-x-2-then-the-value-of-x-is-63693
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1}\left(x-\frac{x^{2}}{2}+\frac{x^{3}}{4}-\frac{x^{4}}{8}+\ldots\right)=\frac{\pi}{6},$ where $|x|    &lt; 2$ then the value of $x$ is</div>

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

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

answer:
<div class="solution">We have, $\sin ^{-1}\left(x-\frac{x^{2}}{2}+\frac{x^{3}}{4}-\frac{x^{4}}{8}+\ldots\right)=\frac{\pi}{6}$
<br/>$\Rightarrow \sin ^{-1}\left(\frac{x}{1-\left(\frac{-x}{2}\right)}\right)=\frac{\pi}{6} \quad\left[\because S_{\infty}=\frac{a}{1-r}\right]$
<br/>$\Rightarrow \quad \sin ^{-1}\left(\frac{2 x}{2+x}\right)=\frac{\pi}{6}$
<br/>$\Rightarrow \frac{2 x}{2+x}=\sin \frac{\pi}{6} \Rightarrow \frac{2 x}{2+x}=\frac{1}{2}$
<br/>$\Rightarrow \quad 4 x=2+x \quad \Rightarrow \quad 3 x=2 \quad \Rightarrow \quad x=\frac{2}{3}$</div>


title: if-sin-1-x-sin-1-y-sin-1-z-2-3-then-the-value-of-x-9-y-9-z-9-x-9-y-9-z-9-1-is-equal-to
slug: if-sin-1-x-sin-1-y-sin-1-z-2-3-then-the-value-of-x-9-y-9-z-9-x-9-y-9-z-9-1-is-equal-to-99161
tags: ['Mathematics', 'Inverse Trigonometric Functions', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{-1} x+\sin ^{-1} y+\sin ^{-1} z=\frac{3 \pi}{2},$ then the value of $x^{9}+y^{9}+z^{9}-\frac{1}{x^{9} y^{9} z^{9}}$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">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">We know that $\left|\sin ^{-1} x\right| \leq \frac{\pi}{2}$
<br/>Hence, from the given relation we observe that each of $\sin ^{-1} x, \sin ^{-1} y$ and $\sin ^{-1} x$ will be $\frac{\pi}{2}$
<br/>so that $x=y=x=\sin \frac{\pi}{2}=1$
<br/>$\begin{aligned} \therefore \quad &amp; x^{9}+y^{9}+z^{9}-\frac{1}{x^{9} y^{9} z^{9}} \\ &amp;=(1)^{9}+(1)^{9}+(1)^{9}-\frac{1}{(1)^{9}(1)^{9}(1)^{9}} \\ &amp;=1+1+1-\frac{1}{1 \times 1 \times 1} \\ &amp;=3-1=2 \end{aligned}$</div>


title: if-sin-2-3-cos-2-then-cos-3-sec-3-is-equal-out-to
slug: if-sin-2-3-cos-2-then-cos-3-sec-3-is-equal-out-to-56205
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">18</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\sin ^{2} \theta+3 \cos \theta=2,$ then $\cos ^{3} \theta+\sec ^{3} \theta$ is equal out to</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">4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">9</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">18</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Given, $\sin ^{2} \theta+3 \cos \theta=2$
<br/>$\begin{array}{lr}\Rightarrow &amp; 1-\cos ^{2} \theta+3 \cos \theta=2 \\ \Rightarrow &amp; \cos ^{2} \theta-3 \cos \theta+1=0\end{array}$
<br/>$\Rightarrow \quad \cos \theta=\frac{3 \pm \sqrt{9-4}}{2}$ (by quadratic formula)
<br/>$\Rightarrow \quad \cos \theta=\frac{3 \pm \sqrt{5}}{2}$
<br/>Here, $\cos \theta=\frac{3-\sqrt{5}}{2}$
<br/>$\left(\because \frac{3+\sqrt{5}}{2} &gt; 1 \text{ and } -1 \leq \cos \theta \leq 1\right)$
<br/>and $\sec \theta=\frac{2}{3-\sqrt{5}} \times \frac{3+\sqrt{5}}{3-\sqrt{5}}$
<br/>$=\frac{2(3+\sqrt{5})}{4}=\frac{3+\sqrt{5}}{2}$
<br/>Now, $\cos ^{3} \theta+\sec ^{3} \theta=(\cos \theta+\sec \theta)$
<br/>$\quad\left(\cos ^{2} \theta+\sec ^{2} \theta-\cos \theta \cdot \sec \theta\right)$
<br/>$\begin{array}{l}
<br/>=\left(\frac{3-\sqrt{5}+3+\sqrt{5}}{2}\right) \\
<br/>\quad \quad\left\{(\cos \theta+\sec \theta)^{2}-3 \cos \theta \cdot \sec \theta\right\} \\
<br/>=3 \cdot\left\{(3)^{2}-3\right\}=3(9-3)=3 \times 6=18
<br/>\end{array}$</div>


title: if-sin-6-sin-4-sin-2-0-then-general-value-of-is
slug: if-sin-6-sin-4-sin-2-0-then-general-value-of-is-50755
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\sin 6 \theta+\sin 4 \theta+\sin 2 \theta=0$, then general
<br/>value of $\theta$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{n \pi}{4}, n \pi \pm \frac{\pi}{3}$</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">$\frac{n \pi}{4}, n \pi \pm \frac{\pi}{6}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{n \pi}{4}, 2 n \pi \pm \frac{\pi}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{n \pi}{4}, 2 n \pi \pm \frac{\pi}{6}$</span> </li> </ul>

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

answer:
<div class="solution">We have, $\sin 6 \theta+\sin 4 \theta+\sin 2 \theta=0$
<br/>$\Rightarrow \quad \sin 6 \theta+\sin 2 \theta+\sin 4 \theta=0$
<br/>$\Rightarrow \quad 2 \sin 4 \theta \cdot \cos 2 \theta+\sin 4 \theta=0$
<br/>$\Rightarrow \quad \sin 4 \theta(2 \cos 2 \theta+1)=0$
<br/>$\Rightarrow \quad \sin 4 \theta=0$ or $2 \cos 2 \theta+1=0$
<br/>$\Rightarrow \quad 4 \theta=n \pi$ or $\cos 2 \theta=\frac{-1}{2}=\cos \frac{2 \pi}{3}$
<br/>$\Rightarrow \quad \theta=\frac{n \pi}{4}$ or $2 \theta=2 n \pi \pm \frac{2 \pi}{3}$
<br/>$\Rightarrow \quad \theta=\frac{n \pi}{4}$ or $\theta=n \pi \pm \frac{\pi}{3},$ where $n \in z$</div>


title: if-sin-6-sin-4-sin-2-0-then-the-general-value-of-is
slug: if-sin-6-sin-4-sin-2-0-then-the-general-value-of-is-35566
tags: ['Mathematics', 'Trigonometric Equations', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\sin 6 \theta+\sin 4 \theta+\sin 2 \theta=0$ then the general value of $\theta$ is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{\mathrm{n} \pi}{4}, \mathrm{n} \pi \pm \frac{\pi}{3}$</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">$\frac{\mathrm{n} \pi}{4}, \mathrm{n} \pi \pm \frac{\pi}{6}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{\mathrm{n} \pi}{4}, 2 \mathrm{n} \pi \pm \frac{\pi}{3}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{\mathrm{n} \pi}{4}, 2 \mathrm{n} \pi \pm \frac{\pi}{6}$</span> </li> </ul>

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

answer:
<div class="solution">Hints : $2 \sin 4 \theta \cos 2 \theta+\sin 4 \theta=0$<br/>$\sin 4 \theta=0$<br/>$2 \cos 2 \theta=-1$<br/>$4 \theta=n \pi$<br/>$\cos 2 \theta=-\frac{1}{2}=\cos \frac{2 \pi}{3}$<br/>$\theta=\frac{n \pi}{4}$<br/>$2 \theta=2 \mathrm{n} \pi \pm \frac{2 \pi}{3}, \Rightarrow \theta=\mathrm{n} \pi \pm \frac{\pi}{3}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
title:
if-sin-cos-be-the-roots-of-the-equation-x-2-b-x-c-0-then-which-of-the-following-statements-isare-correct

slug:
if-sin-cos-be-the-roots-of-the-equation-x-2-b-x-c-0-then-which-of-the-following-statements-isare-correct-15253

question:
<div class="question">If $\sin \alpha, \cos \alpha$ be the roots of the equation $x^{2}-b x+c=0 .$ Then, which of the following statements is/are correct?</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$c \leq \frac{1}{2}$, $b \leq \sqrt{2}$</span> </div>

answer:
<div class="solution">Given equation is $x^{2}-b x+c=0$
<br/>and roots are $\sin \alpha$ and $\cos \alpha$
<br/>Also $\sin \alpha+\cos \alpha=b$
<br/>and $\sin \alpha \cdot \cos \alpha=c$
<br/>$$
<br/>\begin{array}{r}
<br/>\because \sin ^{2} \alpha+\cos ^{2} \alpha=(\sin \alpha+\cos \alpha)^{2} \\
<br/>-2 \sin \alpha \cdot \cos \alpha
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad 1=b^{2}-2 c$
<br/>[from Eqs. (ii) and (iii)]
<br/>$-\sqrt{1+1} \leq(\sin \alpha+\cos \alpha) \leq \sqrt{1+1}$
<br/>$\Rightarrow \quad-\sqrt{2} \leq b \leq \sqrt{2}$
<br/>and $2 \sin \alpha ; \cos \alpha=x 2 c$
<br/>[from Eq. (iii)] $\begin{array}{ll}\Rightarrow &amp; \sin 2 \alpha=2 c(\because-1 \leq \sin 2 \alpha \leq 1) \\ \Rightarrow &amp; 2 c \leq 1 \Rightarrow c \leq \frac{1}{2}\end{array}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-sin-theta-and-cos-theta-are-the-roots-of-the-equation-a-x2b-xc0-then-mathrma-mathrmb-and-mathrmc-satisfy-the-relation

slug:
if-sin-theta-and-cos-theta-are-the-roots-of-the-equation-a-x2b-xc0-then-mathrma-mathrmb-and-mathrmc-satisfy-the-relation-17336

question:
<div class="question">If \(\sin \theta\) and \(\cos \theta\) are the roots of the equation \(a x^2-b x+c=0\), then \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) satisfy the relation</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{a}^2-\mathrm{b}^2+2 \mathrm{ac}=0\)</span> </div>

answer:
<div class="solution">Hints: \(\sin \theta+\cos \theta=\frac{b}{a}\)<br/>\(\begin{aligned}<br/>&amp; \sin \theta \cdot \cos \theta=\frac{c}{a} \\<br/>&amp; \left(\frac{b}{a}\right)^2=1+\frac{2 c}{a} \\<br/>&amp; b^2=a^2+2 a c \\<br/>&amp; a^2-b^2+2 a c=0<br/>\end{aligned}\)</div>


title: if-sin-thetacos-theta0-and-0-theta-pi-then-theta
slug: if-sin-thetacos-theta0-and-0-theta-pi-then-theta-30827
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(\sin \theta+\cos \theta=0\) and \(0  &lt; \theta &lt; \pi\), then \(\theta\)</div>

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

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

answer:
<div class="solution">Hints: \(\sin \theta+\cos \theta=0\)<br/>\(\Rightarrow \tan \theta=-1 \quad \theta=\frac{3 \pi}{4}\)</div>


title: if-sin-thetafrac2-t1t2-and-theta-lies-in-the-second-quadrant-then-cos-theta-is-equal-to
slug: if-sin-thetafrac2-t1t2-and-theta-lies-in-the-second-quadrant-then-cos-theta-is-equal-to-60289
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(\sin \theta=\frac{2 t}{1+t^2}\) and \(\theta\) lies in the second quadrant, then \(\cos \theta\) is equal to</div>

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

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

answer:
<div class="solution">Hints: \(\theta\) in 2nd quad \(\cos \theta  &lt; 0\)<br/>\(\begin{aligned}<br/>&amp; |\cos \theta|=\left|\frac{1-t^2}{1+t^2}\right|=\frac{\left|1-t^2\right|}{1+t^2} \\<br/>&amp; \cos \theta=-\frac{\left|1-t^2\right|}{1+t^2}<br/>\end{aligned}\)</div>


tags: ['Physics', 'Magnetic Properties of Matter', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
if-stands-for-the-magnetic-susceptibility-of-a-substance-for-its-magnetic-permeability-and-0-for-the-permeability-of-free-space-then

slug:
if-stands-for-the-magnetic-susceptibility-of-a-substance-for-its-magnetic-permeability-and-0-for-the-permeability-of-free-space-then-14910

question:
<div class="question">If $\chi$ stands for the magnetic susceptibility of a substance, $\mu$ for its magnetic permeability and $\mu_{0}$ for the permeability of free space, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">for a paramagnetic substance: $\chi    &lt; 0$, $\mu&gt;0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">for a paramagnetic substance: $\chi&gt;0$, $\mu&gt;\mu_{0}$</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">for a diamagnetic substance: $\chi&gt;0, \mu    &lt; 0$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">for a ferromagnetic substance: $\chi&gt;1$, $\mu&gt;\mu_{0}$</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 answers are:
<span class="option-value">for a paramagnetic substance: $\chi&gt;0$, $\mu&gt;\mu_{0}$, for a ferromagnetic substance: $\chi&gt;1$, $\mu&gt;\mu_{0}$</span> </div>

answer:
<div class="solution">We know that magnetic susceptibility<br/>$\chi=\mu_{r}-1$<br/>and  \mu_r=\frac{\mu}{\mu_{0}}$<br/>=Relative permeability <br/>For paramagnetic substance,<br/>$\begin{array}{ll} <br/>&amp; \chi&gt;0 \\<br/>\therefore &amp; \mu_{r}&gt;1 \\<br/>\therefore \mu&gt;\mu_{0}<br/>\end{array}$<br/>For diamagnetic substance,<br/>$\chi&lt; 0$<br/>$\therefore \mu_r &lt; 1$<br/>$\therefore \mu &lt; \mu_{0}$<br/>For ferromagnetic substance,<br/>$\begin{array}{l}<br/>\chi&gt;&gt;1 \\<br/>\mu&gt;&gt;M_{0}<br/>\end{array}$<br/>If<br/>$0 &lt; \mu &lt; \mu_{0}$<br/>Then substance will not be paramagnetic. Hence option (a) is incorrect and option (b) and (d) are correct.</div>


title: if-sum-of-an-infinite-geometric-series-is-5-4-and-its-1-st-term-is-4-3-then-its-common-ratio-is
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
slug:
if-sum-of-an-infinite-geometric-series-is-5-4-and-its-1-st-term-is-4-3-then-its-common-ratio-is-43417

question:
<div class="question">If sum of an infinite geometric series is $\frac{4}{5}$ and its 1 st term is $\frac{3}{4}$, then its common ratio is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\frac{7}{16}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{9}{16}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{9}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{7}{9}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{7}{16}$</span> </div>

answer:
<div class="solution">Hints: $\frac{\mathrm{a}}{1-\mathrm{r}}=\frac{4}{3} \quad$ Then $\frac{\frac{3}{4}}{1-\mathrm{r}}=\frac{4}{3} \Rightarrow \mathrm{r}=1-\frac{9}{16}=\frac{7}{16}$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2010']
answer: <div class="solution">Hints : $\mathrm{t}_1 \mathrm{t}_2=-1$ Fact</div>
source_db: db\MarksBatch2_P1.db
title:
if-t-1-and-t-2-be-the-parameters-of-the-end-points-of-a-focal-chord-for-the-parabola-y-2-4-a-x-then-which-one-is-true

slug:
if-t-1-and-t-2-be-the-parameters-of-the-end-points-of-a-focal-chord-for-the-parabola-y-2-4-a-x-then-which-one-is-true-73474

question:
<div class="question">If $t_1$ and $t_2$ be the parameters of the end points of a focal chord for the parabola $y^2=4 a x$, then which one is true?</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\mathrm{t}_1 \mathrm{t}_2=1$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{t_1}{t_2}=1$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\mathrm{t}_1 \mathrm{t}_2=-1$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{t}_1+\mathrm{t}_2=-1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\mathrm{t}_1 \mathrm{t}_2=-1$</span> </div>


title: if-t-1-for-0-t-1-0-otherwise-then-3000-3000-r-2014-2016-t-r-t-2016-d-t-is
slug: if-t-1-for-0-t-1-0-otherwise-then-3000-3000-r-2014-2016-t-r-t-2016-d-t-is-97544
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\phi(t)=\left\{\begin{array}{l}1, \text { for } 0 \leq t    &lt; 1 \\ 0, \text { otherwise }\end{array},\right.$ then $\int_{-3000}^{3000}\left(\sum_{r^{\prime}=2014}^{2016} \phi(t-r') \phi(t-2016)\right) d t$ is</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">a real number, 1</span> </div>

answer:
<div class="solution">$\int_{-3000}^{3000}\left(\sum_{r'=2014}^{2016} \phi\left(t-r^{\prime}\right) \phi(t-2016)\right) d t$
<br/>$=\int_{-3000}^{3000}[\phi(t-2014) \phi(t-2016)+\phi(t-2015) \phi(t-2016) +\phi(t-2016) \phi(t-2016)] \mathrm{dt}$
<br/>$=\int_{-3000}^{3000} \phi(t-2016)[\phi(t-2014)+\phi(t-2015)$ $+\phi(t-2016)] d t$
<br/>$=\int_{-3000}^{2016} 0 d t+\int_{2016}^{2017} 1 \cdot(0+0+1) dt+\int_{2017}^{3000} 0 d t$
<br/>$=0+1+0=1$ which is a real number.</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2022']
answer: <div class="solution">$\left(\frac{2+0+1}{3}, \frac{0+2+1}{3}\right)$ i.e. $(1,1)$.</div>
source_db: db\MarksBatch2_P1.db
title:
if-the-algebraic-sum-of-the-distances-from-the-points-2-0-0-2-and-1-1-to-a-variable-straight-line-be-zero-then-the-line-passes-through-the-fixed-point

slug:
if-the-algebraic-sum-of-the-distances-from-the-points-2-0-0-2-and-1-1-to-a-variable-straight-line-be-zero-then-the-line-passes-through-the-fixed-point-65825

question:
<div class="question">If the algebraic sum of the distances from the points $(2,0),(0,2)$ and $(1,1)$ to a variable straight line be zero, then the line passes through the fixed point</div>

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

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


tags: ['Physics', 'Capacitance', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
title:
if-the-charge-on-a-capacitor-is-increased-by-2-coulomb-the-energy-stored-in-it-increases-by-21-the-original-charge-on-the-capacitor-is

slug:
if-the-charge-on-a-capacitor-is-increased-by-2-coulomb-the-energy-stored-in-it-increases-by-21-the-original-charge-on-the-capacitor-is-98273

question:
<div class="question">If the charge on a capacitor is increased by 2 coulomb, the energy stored in it increases by $21 \%$. The original charge on the capacitor is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10 \mathrm{C}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$20 \mathrm{C}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$30 \mathrm{C}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$40 \mathrm{C}$</span> </li> </ul>

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

answer:
<div class="solution">$\frac{\frac{q_f^2}{2 C}-\frac{q_i^2}{2 C}}{\frac{q_i^2}{2 C}} \times 100=21$ and $q_f-q_i=2$<br/>solving we get $q_i=20$ coulomb</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
if-the-circle-x-2-y-2-2-gx-2-f-y-c-0-cuts-the-three-circles-x-2-y-2-5-0-x-2-y-2-8-x-6-y-10-0-and-x-2-y-2-4-x-2-y-2-0-at-the-extremities-of-their-diame

slug:
if-the-circle-x-2-y-2-2-gx-2-f-y-c-0-cuts-the-three-circles-x-2-y-2-5-0-x-2-y-2-8-x-6-y-10-0-and-x-2-y-2-4-x-2-y-2-0-at-the-extremities-of-their-diame-35550

question:
<div class="question">If the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ cuts the three circles $x^{2}+y^{2}-5=0$ $x^{2}+y^{2}-8 x-6 y+10=0$ and $x^{2}+y^{2}-4 x+2 y-2=0$ at the extremities of their diameters, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$c=-5$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$f g=147 / 25$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$g+2 t=c+2$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$4 f=3 g$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$4 f=3 g$</span> </div>

answer:
<div class="solution">Centres and constant terms in the circles $x^{2}+y^{2}-5=0, x^{2}+y^{2}-8 x-6 y+10=0$ and
<br/>$x^{2}+y^{2}-4 x+2 y-2=0$ are $C_{1}^{\prime}(0,0), c_{1}=-5$
<br/>$C_{2}^{\prime}(4,3), c_{2}=10$ and $C_{3}^{\prime}(2,-1), c_{3}=-2$
<br/>Also, centre and constant of circle $x^{2}+y^{2}+2 g x+2 f y+c=0$
<br/>is $\quad C_{4}^{\prime}(-g,-f)$ and $c_{4}=c$
<br/>since, the first circle intersect all the three at the
<br/>extremities of diameter, therefore they are orthogonal to each other.
<br/>$\therefore \quad 2\left(g_{1} g_{2}+f_{1} f_{2}\right)=c_{1}+c$
<br/>$\therefore \quad 2[-g(0)+(-f)(0)]=c-5$
<br/>$\Rightarrow \quad c=5$
<br/>$$
<br/>2[-g(4)+(-f)(3)]=c+10
<br/>$$
<br/>$\Rightarrow \quad-2(4 g+3 f)=5+10$
<br/>$\Rightarrow \quad 4 g+3 f=-\frac{15}{2}$
<br/>and $2[-g(2)+(-f)(-1)]=c-2$
<br/>$\Rightarrow \quad 2[-2 g+f]=5-2$
<br/>$\Rightarrow \quad-2 g+f=\frac{3}{2}$
<br/>On solving Eqs. (i) and (ii), we get $f=-\frac{9}{10}$ and $g=-\frac{12}{10}$
<br/>$\therefore \quad f g=\frac{-9}{10} \times-\frac{12}{10}=\frac{27}{25}$
<br/>and $\quad 4 f=4 \times \frac{-9}{10}$
<br/>$\Rightarrow \quad 4 f=\frac{-36}{10}$
<br/>$\Rightarrow \quad 4 f=3 g$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
title:
if-the-circles-x-2-y-2-2-x-2-k-y-6-0-and-x-2-y-2-2-k-y-k-0-intersect-orthogonally-then-k-is-equal-to

slug:
if-the-circles-x-2-y-2-2-x-2-k-y-6-0-and-x-2-y-2-2-k-y-k-0-intersect-orthogonally-then-k-is-equal-to-26193

question:
<div class="question">If the circles $x^{2}+y^{2}+2 x+2 k y+6=0$ and $x^{2}+y^{2}+2 k y+k=0$ intersect orthogonally. then $k$ is equal to</div>

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

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

answer:
<div class="solution">Two circles are orthogonally if and only if $\quad 2\left(g_{1} g_{2}+f_{1} f_{2}\right)=c_{1}+c_{2}$
<br/>$\Rightarrow \quad 2[(1 \times 0+(k) k]=6+k$
<br/>$\Rightarrow \quad 2 k^{2}=6+k$
<br/>$\Rightarrow \quad 2 k^{2}-k-6=0$
<br/>$\Rightarrow \quad 2 k^{2}-4 k+3 k-6=0$
<br/>$\Rightarrow \quad \quad 2 k(k-2)+3(k-2)=0$
<br/>$\Rightarrow \quad(k-2)(2 k+3)=0$
<br/>$\Rightarrow \quad k=2-\frac{3}{2}$</div>


tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
if-the-coefficient-of-x-8-in-a-x-2-b-x-1-13-is-equal-to-the-coefficient-of-x-8-in-a-x-b-x-2-1-13-then-a-and-b-will-satisfy-the-relation

slug:
if-the-coefficient-of-x-8-in-a-x-2-b-x-1-13-is-equal-to-the-coefficient-of-x-8-in-a-x-b-x-2-1-13-then-a-and-b-will-satisfy-the-relation-39649

question:
<div class="question">If the coefficient of $x^{8}$ in $\left(a x^{2}+\frac{1}{b x}\right)^{13}$ is equal to the coefficient of $x^{-8}$ in $\left(a x-\frac{1}{b x^{2}}\right)^{13},$ then $a$ and $b$ will satisfy the relation</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$a b+1=0$</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">$a b=1$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$a=1-b$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$a+b=-1$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a b+1=0$</span> </div>

answer:
<div class="solution">The general term in $\left(a x^{2}+\frac{1}{b x}\right)^{13}$ is
<br/>$T_{r+1}={ }^{13} C_{r}\left(a x^{2}\right)^{13-r}\left(\frac{1}{b x}\right)^{r}$
<br/>$$
<br/>={ }^{13} C_{r} a^{13-r} \times b^{-r}(x)^{26-3 r}
<br/>$$
<br/>For coefficient of $x^{8}$. put $26-3 r=8$
<br/>$\Rightarrow \quad 3 r=18 \Rightarrow r=6$
<br/>$\therefore$
<br/>$T_{7}={ }^{13} C_{6} a^{13-6} b^{-6} x^{8}$
<br/>$={ }^{13} C_{6} a^{7} b^{-6} x^{8}$
<br/>Now, the general term in $\left(a x-\frac{1}{b x^{2}}\right)^{13}$ is
<br/>$$
<br/>\begin{aligned}
<br/>T_{r+1}^{'} &amp;={ }^{13} C_{r}(a x)^{13-r}\left(-\frac{1}{b x^{2}}\right)^{r} \\
<br/>&amp;={ }^{13} C_{r}^{*} a^{13-r} \times b^{-r} \times(x)^{13-3 r}(-1)^{\prime}
<br/>\end{aligned}
<br/>$$
<br/>For coefficient of $x^{-8}$, put $13-3 r=-8$
<br/>According to the given condition, Coefficient of $x^{8}$ in $\left(a x^{2}+\frac{1}{b x}\right)^{13}$
<br/>$$
<br/>\begin{array}{l}
<br/>=\text { Coefficient of } x^{-8} \text {in }\left(a x-\frac{1}{b x^{2}}\right)^{13} \\
<br/>\therefore{ }^{13} C_{6} a^{7} b^{-6}=-{ }^{13} C_{7} a^{5} b^{-7} \\
<br/>\Rightarrow{ }^{13} C_{7} \frac{a^{7}}{b^{6}}=-{ }^{13} C_{7} \frac{a^{6}}{b^{7}} \\
<br/>\Rightarrow \quad \frac{a^{7}}{a^{6}}=-\frac{b^{6}}{b^{7}} \Rightarrow \quad a=-\frac{1}{b} \Rightarrow a b+1=0
<br/>\end{array}
<br/>$$</div>


title: if-the-coefficients-of-x-2-and-x-3-in-the-expansion-of-3-a-x-9-be-same-then-the-value-of-a-is
slug: if-the-coefficients-of-x-2-and-x-3-in-the-expansion-of-3-a-x-9-be-same-then-the-value-of-a-is-32004
tags: ['Mathematics', 'Binomial Theorem', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the co-efficients of $x^2$ and $x^3$ in the expansion of $(3+a x)^9$ be same, then the value of ' $a$ ' is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{3}{7}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$\frac{7}{3}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{7}{9}$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">$\frac{9}{7}$</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> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{9}{7}$</span> </div>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints : }(3+a x)^9={ }^9 \mathrm{C}_0 3^9+{ }^9 \mathrm{C}_1 3^8(a x)+{ }^9 \mathrm{C}_2 37(a x)^2+{ }^9 \mathrm{C}_3 3{ }^6(a x)^3 \\<br/>&amp; { }^9 \mathrm{C}_2 3^7 a^2={ }^9 \mathrm{C}_3 3{ }^6 \mathrm{a}^3 \\<br/>&amp; \frac{9}{7}=a<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-the-coordinates-of-one-end-of-a-diameter-of-the-circle-x2y24-x8-y50-is-21-the-coordinates-of-the-other-end-is

slug:
if-the-coordinates-of-one-end-of-a-diameter-of-the-circle-x2y24-x8-y50-is-21-the-coordinates-of-the-other-end-is-61866

question:
<div class="question">If the coordinates of one end of a diameter of the circle \(x^2+y^2+4 x-8 y+5=0\), is \((2,1)\), the coordinates of the other end is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\((-6,-7)\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\((6,7)\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\((-6,7)\)</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">D</span> <span class="option-data">\((7,-6)\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((-6,7)\)</span> </div>

answer:
<div class="solution">Hints: \(x^2+y^2+9 x-8 y+5=0\)<br/>Centre circle \((-2,4)\)<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/IrPVNzbE5UZz2YIP_XO_M5ZN0aKLk3KvBeQhE-Vx7Yw.original.fullsize.png"/><br/><br/>\(\begin{aligned}<br/>&amp; \frac{\mathrm{h}+2}{2}=-2 \\<br/>&amp; \mathrm{~h}=-4-2=-6 \\<br/>&amp; \frac{\mathrm{k}+1}{2}=4 \Rightarrow \mathrm{k}=7 \\<br/>&amp; (\mathrm{~h}, \mathrm{k}) \rightarrow(-6,7)<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
title:
if-the-displacement-velocity-and-acceleration-of-a-particle-at-time-t-be-x-v-and-f-respectively-then-which-one-is-true

slug:
if-the-displacement-velocity-and-acceleration-of-a-particle-at-time-t-be-x-v-and-f-respectively-then-which-one-is-true-98319

question:
<div class="question">If the displacement, velocity and acceleration of a particle at time, $\mathrm{t}$ be $\mathrm{x}, \mathrm{v}$ and $\mathrm{f}$ respectively, then which one is true?</div>

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

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } \frac{d^2 t}{d x^2}=\frac{d\left(\frac{d t}{d x}\right)}{d x}=\frac{d\left(\frac{1}{v}\right)}{d x}=-\frac{1}{v^2} \frac{d v}{d t} \times \frac{1}{v} \\<br/>&amp; \Rightarrow f=-v^3 f \frac{d^2 t}{d x^2}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2013']
source_db: db\MarksBatch2_P1.db
title:
if-the-distance-between-the-foci-of-an-ellipse-is-equal-to-the-length-of-the-latusrectum-then-its-eccentricity-is

slug:
if-the-distance-between-the-foci-of-an-ellipse-is-equal-to-the-length-of-the-latusrectum-then-its-eccentricity-is-39871

question:
<div class="question">If the distance between the foci of an ellipse is equal to the length of the latusrectum, then its eccentricity is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{2}(\sqrt{5}-1)$</span> </div>

answer:
<div class="solution">Given,
<br/>In ellipse the distance between the foci
<br/>= Length of the latusrectum 
<br/>$\begin{array}{ll}\Rightarrow \quad &amp; \text { 2ae }=\frac{2 b^{2}}{a} \\ \Rightarrow \quad &amp; \quad a^{2} e=b^{2} \Rightarrow e=\frac{b^{2}}{a^{2}}\end{array}$
<br/>$\because \quad e^{2}=1-\frac{b^{2}}{a^{2}}$
<br/>$\Rightarrow \quad e^{2}=1-e \quad$ [from Eq.
<br/>$\Rightarrow e^{2}+e-1=0$
<br/>$\Rightarrow \quad e=\frac{-1 \pm \sqrt{1+4}}{2}$
<br/>(by quadratic formula) $\Rightarrow \quad e=\frac{-1 \pm \sqrt{5}}{2}$
<br/>$\Rightarrow$
<br/>$$
<br/>e=\frac{\sqrt{5}-1}{2} \quad(\because 1&gt;e&gt;0)
<br/>$$</div>


tags: ['Mathematics', 'Three Dimensional Geometry', 'WBJEE', 'WBJEE 2023']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">6</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-the-distance-between-the-plane-a-x2-yzk-and-the-plane-containing-the-lines-fracx12fracy23fracz34-and-fracx23fracy34fracz45-is-sqrt6-then-k-is

slug:
if-the-distance-between-the-plane-a-x2-yzk-and-the-plane-containing-the-lines-fracx12fracy23fracz34-and-fracx23fracy34fracz45-is-sqrt6-then-k-is-77409

question:
<div class="question">If the distance between the plane \(a x-2 y+z=k\) and the plane containing the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}\) and \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) is \(\sqrt{6}\), then \(|k|\) is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">36</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">12</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">6</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(2 \sqrt{3}\)</span> </li> </ul>

answer:
<div class="solution">Hint: Equation of plane<br/>\(\begin{aligned}<br/>&amp; \left|\begin{array}{ccc}<br/>x-1 &amp; y-2 &amp; z-3 \\<br/>2 &amp; 3 &amp; 4 \\<br/>3 &amp; 4 &amp; 5<br/>\end{array}\right|=0 \\<br/>&amp; \Rightarrow x-2 y+z=0 \quad ---(1)<br/>\end{aligned}\)<br/>&amp; given \(\alpha x-2 y+z=k\)  ---(2)<br/>(1) // (2) \(\Rightarrow \alpha=1, k \neq 0\)<br/>\(\therefore \frac{|k|}{\sqrt{6}}=\sqrt{6} \Rightarrow|k|=6\)</div>


tags: ['Physics', 'Rotational Motion', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-the-earth-were-to-suddenly-contract-to-frac1n-th-of-its-present-radius-without-any-change-in-its-mass-the-duration-of-the-new-day-will-be-nearly

slug:
if-the-earth-were-to-suddenly-contract-to-frac1n-th-of-its-present-radius-without-any-change-in-its-mass-the-duration-of-the-new-day-will-be-nearly-58880

question:
<div class="question">If the Earth were to suddenly contract to \(\frac{1}{n}\) th of its present radius without any change in its mass, the duration of the new day will be nearly</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(24 / \mathrm{n}^2 \mathrm{hr}\).</span> </div>

answer:
<div class="solution">Hints: \(\mathrm{I}_1 \omega_1=\mathrm{I}_2 \omega_2\)<br/>\(\begin{aligned}<br/>&amp; \frac{2}{5} \mathrm{MR}^2\left(\frac{2 \pi}{\mathrm{T}_1}\right)=\frac{2}{5} \mathrm{M} \cdot \frac{\mathrm{R}^2}{\mathrm{n}^2}\left(\frac{2 \pi}{\mathrm{T}_2}\right) \\<br/>&amp; \mathrm{T}_2=\frac{\mathrm{T}_1}{\mathrm{n}^2}=\frac{24}{\mathrm{n}^2}<br/>\end{aligned}\)</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
title:
if-the-electron-in-a-hydrogen-atom-jumps-from-an-orbit-with-level-n-1-2-to-an-orbit-with-level-n-2-1-the-emitted-radiation-has-a-wavelength-given-by

slug:
if-the-electron-in-a-hydrogen-atom-jumps-from-an-orbit-with-level-n-1-2-to-an-orbit-with-level-n-2-1-the-emitted-radiation-has-a-wavelength-given-by-39142

question:
<div class="question">If the electron in a hydrogen atom jumps from an orbit with level $\mathrm{n}_1=2$ to an orbit with level $\mathrm{n}_2=1$ the emitted radiation has a wavelength given by</div>

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

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

answer:
<div class="solution">$\frac{1}{\lambda}=R\left(\frac{1}{n_1^2}-\frac{1}{n_2^2}\right)=R\left(\frac{1}{1^2}-\frac{1}{2^2}\right)=\frac{3 R}{4}$<br/>$\Rightarrow \lambda=\frac{4}{3 R}$</div>


tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
if-the-equation-of-one-tangent-to-the-circle-with-centre-at-2-1-from-the-origin-is-3-x-y-0-then-the-equation-of-the-other-tangent-through-the-origin-i

slug:
if-the-equation-of-one-tangent-to-the-circle-with-centre-at-2-1-from-the-origin-is-3-x-y-0-then-the-equation-of-the-other-tangent-through-the-origin-i-78951

question:
<div class="question">If the equation of one tangent to the circle with centre at $(2,-1)$ from the origin is $3 x+y=0$, then the equation of the other tangent through the origin is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$3 x-y=0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x+3 y=0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$x-3 y=0$</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">$x+2 y=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$x-3 y=0$</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/6cGx9rcIKCPJNcWGJyjwrvPnj5KRPnrLXNciOXJ6VrM.original.fullsize.png"/><br/><br/>$$<br/>\begin{aligned}<br/>&amp;\because \mathrm{CA}=\mathrm{CB} \\<br/>&amp;\Rightarrow \frac{5}{\sqrt{10}}=\left|\frac{2 \mathrm{~m}+1}{\sqrt{1+\mathrm{m}^2}}\right|<br/>\end{aligned}<br/>$$<br/>squaring<br/>$$<br/>\frac{25}{10}=\frac{4 m^2+4 m+1}{1+m^2}<br/>$$<br/>$$<br/>\Rightarrow 5+5 m^2=8 m^2+8 m+2 \Rightarrow 3 m^2+8 m-3=0 \Rightarrow m=\frac{1}{3},-3<br/>$$<br/>$\therefore$ Equation of tangent $\mathrm{OB}$ is $y=\frac{x}{3}$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-the-equation-x-2-c-a-d-0-has-roots-equal-to-the-fourth-powers-of-the-roots-of-x-2-a-x-b-0-where-a-2-4-b-then-the-roots-of-x-2-4-b-x-2-b-2-c-0-will-

slug:
if-the-equation-x-2-c-a-d-0-has-roots-equal-to-the-fourth-powers-of-the-roots-of-x-2-a-x-b-0-where-a-2-4-b-then-the-roots-of-x-2-4-b-x-2-b-2-c-0-will-99660

question:
<div class="question">If the equation $x^{2}-ca+d=0$ has roots equal to the fourth powers of the roots of $x^{2}+a x+b=0,$ where $a^{2}&gt;4 b,$ then the roots of $x^{2}-4 b x+2 b^{2}-c=0$ will be</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">both real, one positive and one negative</span> </div>

answer:
<div class="solution">Let $\alpha$ and $\beta$ are the roots of $x^{2}+a x+b=0$ and the roots of $x^{2}-cx+d=0$ are $\alpha^{4}$ and $\beta^{4}$
<br/>Now, $\alpha+\beta=-a, \alpha \beta=b$ ...(i)
<br/>and $\alpha^{4}+\beta^{4}-c, \alpha^{4} \beta^{4}=d$ ...(ii)
<br/>From Eqs. (i) and (ii),
<br/>$b^{4}=d$ and $\alpha^{4}+\beta^{4}=c$
<br/>$\left(\alpha^{2}+\beta^{2}\right)^{2}-2(\alpha \beta)^{2}=c$
<br/>$\left.\Rightarrow \quad\left[(\alpha+\beta)^{2}-2 \alpha \beta\right]^{2}-2 (\alpha\beta\right)^{2}=c$
<br/>$\Rightarrow \quad\left(a^{2}-2 b\right)^{2}-2 b^{2}=c$
<br/>$\Rightarrow \quad 2 b^{2}+c=\left(a^{2}-2 b\right)^{2}$
<br/>$\Rightarrow \quad 2 b^{2}+c=a^{4}+4 b^{2}-4 a^{2} b$
<br/>$\Rightarrow \quad 2 b^{2}-c=4 a^{2} b-a^{4}$
<br/>$\Rightarrow \quad 2 b^{2}-c=a^{2}\left(4 b-a^{2}\right)$
<br/>and for equation $x^{2}-4 b x+2 b^{2}-c=0$
<br/>$$
<br/>D=(4 b)^{2}-4(1)\left(2 b^{2}-c\right)
<br/>$$
<br/>$=16 b^{2}-8 b^{2}+4 c=8 b^{2}+4 c$
<br/>$=4\left(2 b^{2}+c\right)=4\left(a^{2}-2 b\right)^{2}&gt;0=$ real root
<br/>Now, $f(0)=2 b^{2}-c=a^{2}\left(4 b-a^{2}\right)    &lt; 0 \quad\left(\because a^{2}&gt;4 b\right)$
<br/>= roots are opposite in sign</div>


title: if-the-equation-x-2-y-2-10-x-21-0-has-real-roots-x-and-y-then
slug: if-the-equation-x-2-y-2-10-x-21-0-has-real-roots-x-and-y-then-51152
tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the equation $x^{2}+y^{2}-10 x+21=0$ has real roots $x=\alpha$ and $y=\beta,$ then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$3 \leq x \leq 7$</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">B</span> <span class="option-data">$3 \leq y \leq 7$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$-2 \leq y \leq 2$</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">$-2 \leq x \leq 2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$3 \leq x \leq 7$, $-2 \leq y \leq 2$</span> </div>

answer:
<div class="solution">Given, $\quad x^{2}+y^{2}-10 x+21=0$
<br/>$\Rightarrow$
<br/>$x^{2}-10 x+y^{2}+21=0$
<br/>Now, for real roots of $x, D \geq 0$
<br/>$\Rightarrow$
<br/>$$
<br/>\begin{array}{r}
<br/>100-4\left(y^{2}+21\right) \geq 0 \\
<br/>y^{2} \leq 4 \Rightarrow-2 \leq y \leq 2
<br/>\end{array}
<br/>$$
<br/>Also.
<br/>$$
<br/>y^{2}=-x^{2}+10 x-21
<br/>$$
<br/>Now, for real roots of $y$.
<br/>$$
<br/>\begin{array}{r}
<br/>-4\left(x^{2}-10 x+21\right) \geq 0 \\
<br/>-x^{2}+10 x-21 \geq 0
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad(x-7)(x-3) \leq 0 \therefore 3 \leq x \leq 7$</div>


tags: ['Chemistry', 'Chemical Equilibrium', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-the-equilibrium-constants-of-the-following-equilibria-mathrmso2frac12-mathrmo2-rightleftharpoons-mathrmso3-text-and-2-mathrmso3-rightleftharpoons-2

slug:
if-the-equilibrium-constants-of-the-following-equilibria-mathrmso2frac12-mathrmo2-rightleftharpoons-mathrmso3-text-and-2-mathrmso3-rightleftharpoons-2-22278

question:
<div class="question">If the equilibrium constants of the following equilibria<br/>\(\mathrm{SO}_2+\frac{1}{2} \mathrm{O}_2 \rightleftharpoons \mathrm{SO}_3 \text { and } 2 \mathrm{SO}_3 \rightleftharpoons 2 \mathrm{SO}_2+\mathrm{O}_2\)<br/>are given by \(\mathrm{K}_1\) and \(\mathrm{K}_2\) respectively, which of the following relations is correct</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(\mathrm{K}_2=\left(\frac{1}{\mathrm{~K}_1}\right)^2\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(K_1=\left(\frac{1}{K_2}\right)^3\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(K_2=\left(\frac{1}{K_1}\right)\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(K_2=\left(K_1\right)^2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{K}_2=\left(\frac{1}{\mathrm{~K}_1}\right)^2\)</span> </div>

answer:
<div class="solution">Hints: \(\mathrm{K}_1=\frac{\left[\mathrm{SO}_3\right]}{\left[\mathrm{SO}_2\right]\left[\mathrm{O}_2\right]^{1 / 2}}\)<br/>\(\mathrm{K}_2=\frac{\left[\mathrm{SO}_2\right]^2\left[\mathrm{O}_2\right]}{\left[\mathrm{SO}_3\right]^2}\)<br/>Thus \(\mathrm{K}_2=\left(\frac{1}{\mathrm{~K}_1}\right)^2\)</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
title:
if-the-first-and-2-n-1-th-terms-of-an-ap-gp-and-hp-are-equal-and-their-n-th-terms-are-respectively-a-b-c-then-always

slug:
if-the-first-and-2-n-1-th-terms-of-an-ap-gp-and-hp-are-equal-and-their-n-th-terms-are-respectively-a-b-c-then-always-40685

question:
<div class="question">If the first and $(2 n-1)$ th terms of an $\mathrm{AP}$, GP and $\mathrm{HP}$ are equal and their $n$ th terms are respectively $a, b, c,$ then always</div>

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

answer:
<div class="solution">$\because a, b$ and $c$ are respectively the $A M,$ GM and HM of first and $(2 n-1)$ th terms. and also, we know $A M \geq G M \geq H M$ and $A M \cdot H M=G M^{2}$
<br/>$\therefore \quad a \geq b \geq c$ and $a \cdot c=b^{2}$</div>


title: if-the-first-ionization-energy-of-h-atom-is-136-ev-then-the-second-ionization-energy-of-he-atom-is
tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2012']
source_db: db\MarksBatch2_P1.db
slug:
if-the-first-ionization-energy-of-h-atom-is-136-ev-then-the-second-ionization-energy-of-he-atom-is-98147

question:
<div class="question">If the first ionization energy of $\mathrm{H}$ atom is $13.6 \mathrm{eV},$ then the second ionization energy of He atom is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$27.2 \mathrm{eV}$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$40.8 \mathrm{eV}$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$54.4 \mathrm{eV}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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">$108.8 \mathrm{eV}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$54.4 \mathrm{eV}$</span> </div>

answer:
<div class="solution">Second ionisation energy $=13.6 \times \frac{2^{2}}{1^{2}}$
<br/>$$
<br/>=54.4 \mathrm{eV}
<br/>$$</div>


title: if-the-focal-length-of-the-eye-piece-of-a-telescope-is-doubled-its-magnifying-power-mathrmm-will-be
tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
slug:
if-the-focal-length-of-the-eye-piece-of-a-telescope-is-doubled-its-magnifying-power-mathrmm-will-be-69323

question:
<div class="question">If the focal length of the eye piece of a telescope is doubled, its magnifying power \((\mathrm{m})\) will be</div>

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

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

answer:
<div class="solution">Hints: \(\mathrm{m}=\frac{-\mathrm{f}_0}{\mathrm{f}_{\mathrm{e}}}\)<br/>\(\mathrm{m}^{\prime}=\frac{\mathrm{m}}{2}\)</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-the-following-three-linear-equations-have-a-nontrivial-solution-then-x-4-a-y-a-z-0-x-3-b-y-b-z-0-x-2-cy-cz-0

slug:
if-the-following-three-linear-equations-have-a-nontrivial-solution-then-x-4-a-y-a-z-0-x-3-b-y-b-z-0-x-2-cy-cz-0-42732

question:
<div class="question">If the following three linear equations have a non-trivial solution, then
<br/>$$
<br/>\begin{array}{l}
<br/>x+4 a y+a z=0 \\
<br/>x+3 b y+b z=0 \\
<br/>x+2 c y+c z=0
<br/>\end{array}
<br/>$$</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a, b, c$ are in $\mathrm{HP}$</span> </div>

answer:
<div class="solution">For non-trivial solution, We have, $\left|\begin{array}{ccc}1 &amp; 4 a &amp; a \\ 1 &amp; 3 b &amp; b \\ 1 &amp; 2 c &amp; c\end{array}\right|=0$
<br/>$\Rightarrow \quad 1(3 b c-2 b c)-(4 a c-2 a c)+1(4 a b-3 a b)=0$
<br/>$\Rightarrow b c-2 a c+a b=0$
<br/>$\Rightarrow b c+a b=2 a c$
<br/>$\Rightarrow b(a+c)=2 a c$
<br/>$\Rightarrow b=\frac{2 a c}{a+c}$
<br/>$\Rightarrow a, b, c$ are in $\mathrm{HP}$</div>


title: if-the-four-points-with-position-vectors-2-i-j-k-i-j-k-j-k-and-j-k-are-coplanar-then-is-equal-to
tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P1.db
slug:
if-the-four-points-with-position-vectors-2-i-j-k-i-j-k-j-k-and-j-k-are-coplanar-then-is-equal-to-22629

question:
<div class="question">If the four points with position vectors $-2\hat i+\hat j + \hat k, \hat i+\hat j+\hat k,\hat j- \hat k$ and $\lambda \hat j + \hat k$ are coplanar, then $\lambda$ is equal to</div>

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

answer:
<div class="solution">If four points $\left(x_{1}, y_{1}, z_{1}\right),\left(x_{2}, y_{2}, z_{2}\right),\left(x_{3}, y_{3}, z_{3}\right)$ and
<br/>$\left(x_{4}, y_{4}, z_{4}\right)$ are coplanar, then $\left|\begin{array}{lll}x_{2}-x_{1} &amp; y_{2}-y_{1} &amp; z_{2}-z_{1} \\ x_{3}-x_{1} &amp; y_{3}-y_{1} &amp; z_{3}-z_{1} \\ x_{4}-x_{1} &amp; y_{4}-y_{1} &amp; z_{4}-z_{1}\end{array}\right|=0$
<br/>Now, $\quad\left|\begin{array}{ccc}3 &amp; 0 &amp; 0 \\ 2 &amp; 0 &amp; -2 \\ 2 &amp; \lambda-1 &amp; 0\end{array}\right|=0$
<br/>$\Rightarrow \quad 3(0+2 \lambda-2)=0$
<br/>$\lambda=1$</div>


title: if-the-function-f-r-r-is-defined-by-f-x-x-2-1-35-x-r-then-f-is
slug: if-the-function-f-r-r-is-defined-by-f-x-x-2-1-35-x-r-then-f-is-21839
tags: ['Mathematics', 'Functions', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the function $f: R \rightarrow R$ is defined by $f(x)=\left(x^{2}+1\right)^{35}, \forall x \in R,$ then $f$ is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">one-one but not onto</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">onto but not one-one</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">neither one-one nor onto</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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 one-one and onto</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">neither one-one nor onto</span> </div>

answer:
<div class="solution">Given, $f(x)=\left(x^{2}+1\right)^{35}, \forall x \in R$
<br/>Since, $f(x)$ is even function. Hence, the function is not one-one. And $f(x)&gt;0, y \times \epsilon R$
<br/>Hence, the function is not onto.
<br/>Given, the function is neither one-one nor onto.</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-the-function-f-x-2-x-3-9-a-x-2-12-a-2-x-1-a-0-attains-its-maximum-and-minimum-at-p-and-q-respectively-such-that-p-2-q-then-a-is-equal-to

slug:
if-the-function-f-x-2-x-3-9-a-x-2-12-a-2-x-1-a-0-attains-its-maximum-and-minimum-at-p-and-q-respectively-such-that-p-2-q-then-a-is-equal-to-72356

question:
<div class="question">If the function $f(x)=2 x^{3}-9 a x^{2}+12 a^{2} x+1[a&gt;0]$ attains its maximum and minimum at $p$ and $q$ respectively such that $\mathrm{p}^{2}=\mathrm{q}$, then $\mathrm{a}$ is equal to</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">$\frac{1}{2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{1}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">3</span> </li> </ul>

answer:
<div class="solution">Hint:
<br/>$f^{\prime}(x)=6 x^{2}-18 a x+12 a^{2} \Rightarrow f^{\prime \prime}(x)=12 x-18 a \Rightarrow f^{\prime}(x)=0 \Rightarrow x=a, 2 a$
<br/>$f^{\prime}(a)    &lt; 0 ; p=a$ (maximum)
<br/>$f^{\prime}(2 a)&gt;0 ; q=2 a$ (minimum)
<br/>$a^{2}=2 a ; a(a-2)=0, \quad a=2$</div>


tags: ['Mathematics', 'Continuity and Differentiability', 'WBJEE', 'WBJEE 2011']
answer: <div class="solution">Hints: \(\frac{4-(A+2) 2+A}{0}=\frac{-A}{0}\)<br/>Put \(\mathrm{A}=0\).</div>
source_db: db\MarksBatch2_P1.db
title:
if-the-function-fx-begincasesfracx2a2-xax2-text-for-x-neq-2-2-text-for-x2endcases-is-continuous-at-x2-then

slug:
if-the-function-fx-begincasesfracx2a2-xax2-text-for-x-neq-2-2-text-for-x2endcases-is-continuous-at-x2-then-44828

question:
<div class="question">If the function<br/>\(f(x)= \begin{cases}\frac{x^2-(A+2) x+A}{x-2} &amp; \text { for } x \neq 2 \\ 2 &amp; \text { for } x=2\end{cases}\)<br/>is continuous at \(x=2\), then</div>

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

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


tags: ['Chemistry', 'Structure of Atom', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
if-the-given-four-electronic-configurations-i-n-4-l-1-ii-n-4-l-0-iii-n-3-l-2-iv-n-3-l-1-are-arranged-in-order-of-increasing-energy-then-the-order-will

slug:
if-the-given-four-electronic-configurations-i-n-4-l-1-ii-n-4-l-0-iii-n-3-l-2-iv-n-3-l-1-are-arranged-in-order-of-increasing-energy-then-the-order-will-95030

question:
<div class="question">If the given four electronic configurations.
<br/>(i) $n=4, l=1$
<br/>(ii) $n=4, l=0$
<br/>(iii) $n=3, l=2$
<br/>(iv) $n=3, l=1$
<br/>are arranged in order of increasing energy, then the order will be</div>

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

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

answer:
<div class="solution">According to $(n+l)$ rule;
<br/>(i) More be the sum of $(n+l)$ value, more be the energy.
<br/>(ii) For same value of sum, more be the value of $n$, more be the energy.
<br/>For
<br/>(i) $n+l \rightarrow 4+1=5$
<br/>(ii) $n+l \rightarrow 4+0=4$
<br/>(iii) $n+l \rightarrow 3+2=5$
<br/>(iv) $n+l \rightarrow 3+1=4$
<br/>Hence, order of energy will be
<br/>(iv) $    &lt; $ (ii) $ &lt; $ (iii) $ &lt; $ (i)</div>


tags: ['Physics', 'Nuclear Physics', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.13</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-the-halflife-of-a-radioactive-nucleus-is-3-days-nearly-what-fraction-of-the-initial-number-of-nuclei-will-decay-on-the-third-day-given-3-025-063

slug:
if-the-halflife-of-a-radioactive-nucleus-is-3-days-nearly-what-fraction-of-the-initial-number-of-nuclei-will-decay-on-the-third-day-given-3-025-063-94573

question:
<div class="question">If the half-life of a radioactive nucleus is 3 days, nearly what fraction of the initial number of nuclei will decay on the third day? (Given, $\sqrt[3]{0.25} \approx 0.63)$</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">0.63</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">0.5</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">0.37</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">0.13</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">Given, $t_{1 / 2}=3$ days<br/>Number of active nuclei remaining after time $t$,<br/>$N=N_{0}\left(\frac{1}{2}\right)^{\frac{t}{t_{1 / 2}}}$<br/>After $t=2$ days,<br/>$N_{1}=N_{0}\left(\frac{1}{2}\right)^{2 / 3}=\frac{N_{0}}{2^{2 / 3}}=\frac{N_{0}}{4^{1 / 3}}=0.63 \mathrm{N}_{0}$<br/>$\therefore \quad N_{1}=0.63 \mathrm{N}_{0}$<br/>After $t=3$ days,<br/>$N_{2}=\frac{N_{0}}{2}=0.5 \mathrm{N}_{0}$<br/>Number of nuclei that will decay on the 3rd day. $N_{3}=N_{2}-N_{1}=0.63 \mathrm{N}_{0}-0.5 \mathrm{N}_{0}=013 \mathrm{N}_{0}$<br/>In the term, fraction is 0.13.</div>


tags: ['Physics', 'Dual Nature of Matter', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
if-the-kinetic-energies-of-an-electron-an-alpha-particle-and-a-proton-having-same-debroglie-wavelength-are-1-2-and-3-respectively-then

slug:
if-the-kinetic-energies-of-an-electron-an-alpha-particle-and-a-proton-having-same-debroglie-wavelength-are-1-2-and-3-respectively-then-18449

question:
<div class="question">If the kinetic energies of an electron, an alpha particle and a proton having same de-Broglie wavelength are $\varepsilon_1, \varepsilon_2$ and $\varepsilon_3$ respectively, then</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\varepsilon_1&gt;\varepsilon_3&gt;\varepsilon_2$</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">$\varepsilon_1=\varepsilon_2=\varepsilon_2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\varepsilon_1 &lt; \varepsilon_3 &lt; \varepsilon_2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\varepsilon_1&gt;\varepsilon_2&gt;\varepsilon_3$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\varepsilon_1&gt;\varepsilon_3&gt;\varepsilon_2$</span> </div>

answer:
<div class="solution">$\lambda=\frac{\mathrm{h}}{\sqrt{2 \mathrm{mk}}}, \sqrt{\mathrm{mk}}=\mathrm{const}$<br/>$\mathrm{k} \propto \frac{1}{\mathrm{~m}}, \varepsilon_1&gt;\varepsilon_3&gt;\varepsilon_2$</div>


title: if-the-kinetic-energy-of-a-body-changes-by-20-then-its-momentum-would-change-by
slug: if-the-kinetic-energy-of-a-body-changes-by-20-then-its-momentum-would-change-by-47976
tags: ['Physics', 'Center of Mass Momentum and Collision', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the kinetic energy of a body changes by $20 \%$ then its momentum would change by -</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$20 \%$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$24 \%$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$40 \%$</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">None of the above</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 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">$\frac{\frac{p_f^2}{2 m}-\frac{p_i^2}{2 m}}{\frac{p_i^2}{2 m}} \times 100=20$<br/>$\Rightarrow \frac{p_f}{p_i}=\sqrt{1.2}=1.095 \Rightarrow \frac{p_f-p_i}{p_i}=0.095$<br/>Therefore $\%$ increase $=9.5 \%$</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
if-the-letters-of-the-word-probability-are-written-down-at-random-in-a-row-then-probability-that-two-bs-are-together-is

slug:
if-the-letters-of-the-word-probability-are-written-down-at-random-in-a-row-then-probability-that-two-bs-are-together-is-81496

question:
<div class="question">If the letters of the word 'PROBABILITY' are written down at random in a row, then probability that two B's are together, is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{2}{11}$</span> </div>

answer:
<div class="solution">Total number of ways in which the letters of the word 'PROBABILITY can be amanged in a row $=\frac{11!}{2!2 !}$
<br/>Number of ways in which two B's are togother $=\frac{10!}{2!}$
<br/>$\therefore$ Required probability
<br/>$=\frac{\frac{10}{2 !}}{\frac{11!}{2!2!}}=\frac{10 ! \times 2!}{11!}=\frac{2}{11}$</div>


title: if-the-line-a-x-b-y-c-0-ab-0-is-a-tangent-to-the-curve-x-y-1-2-x-then
slug: if-the-line-a-x-b-y-c-0-ab-0-is-a-tangent-to-the-curve-x-y-1-2-x-then-66626
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the line $a x+b y+c=0, a b \neq 0,$ is a tangent
<br/>to the curve $x y=1-2 x,$ then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$a&gt;0, b&gt;0$, $a    &lt; 0, b &lt; 0$</span> </div>

answer:
<div class="solution">We have, $xy=1-2 x$
<br/>$\Rightarrow y=\frac{1-2 x}{x}$
<br/>$\Rightarrow \frac{d y}{d x}=\frac{-2 x-(1-2 x) \cdot 1}{x^{2}}$
<br/>$\quad=\frac{-2 x-1+2 x}{x^{2}}=\frac{-1}{x^{2}}    &lt; 0$
<br/>since, $a x+b y+c=0$ is tangent to the curve $xy=1-2 x$
<br/>$\therefore \quad \frac{-a}{b} &lt; 0 \Rightarrow \frac{a}{b}&gt;0$
<br/>$\Rightarrow$ Either $a&gt;b&gt;0$ or $a &lt; 0, b &lt; 0$</div>


title: if-the-line-a-x-b-y-c-0-is-a-tangent-to-the-curve-x-y-4-then
slug: if-the-line-a-x-b-y-c-0-is-a-tangent-to-the-curve-x-y-4-then-25692
question: <div class="question">If the line $a x+b y+c=0$ is a tangent to the curve $x y=4$, then</div>
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">a $ &lt; 0$, b $&gt;0$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">a $\leq 0$, b $&gt;0$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">a $ &lt; 0$, b $ &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">a $\leq 0$, b $ &lt; 0$</span> </li> </ul>

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

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: Slope of line }=-\frac{a}{b} \\<br/>&amp; y=\frac{4}{x}=1, \frac{d y}{d x}=-\frac{4}{x^2},-\frac{a}{b}=-\frac{4}{x^2} \Rightarrow \frac{a}{b}=\frac{4}{x^2}&gt;0 \\<br/>&amp; a &lt; 0, b &lt; 0<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
if-the-line-y-x-is-a-tangent-to-the-parabola-y-ax-2-bx-c-at-the-point-1-1-and-the-curve-passes-through-1-0-then

slug:
if-the-line-y-x-is-a-tangent-to-the-parabola-y-ax-2-bx-c-at-the-point-1-1-and-the-curve-passes-through-1-0-then-13896

question:
<div class="question">If the line $\mathrm{y}=\mathrm{x}$ is a tangent to the parabola $\mathrm{y}=\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}$ at the point $(1,1)$ and the curve passes through $(-1,0)$, then</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$a=c=\frac{1}{4}, b=\frac{1}{2}$</span> </div>

answer:
<div class="solution">Hint: 
<br/>$\frac{d y}{d x}=2 a x+b$
<br/>at $(1,1) \frac{d y}{d x}=2 a+b=1$  ...(1)
<br/>Now $(1,1)$ and $(-1,0)$ satisfies the curve
<br/>$a-b+c=0$ ...(2)
<br/>$a+b+c=1$ ...(3)
<br/>$\mathrm{b}=\frac{1}{2} ; \mathrm{a}=\frac{1}{4} ; \mathrm{c}=\frac{1}{4}$</div>


tags: ['Mathematics', 'Ellipse', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
if-the-lines-joining-the-focii-of-the-ellipse-fracx2a2fracy2b21-where-a-b-and-an-extremity-of-its-minor-axis-is-inclined-at-an-angle-60circ-then-the-e

slug:
if-the-lines-joining-the-focii-of-the-ellipse-fracx2a2fracy2b21-where-a-b-and-an-extremity-of-its-minor-axis-is-inclined-at-an-angle-60circ-then-the-e-91847

question:
<div class="question">If the lines joining the focii of the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) where \(a &gt; b\), and an extremity of its minor axis is inclined at an angle \(60^{\circ}\), then the eccentricity of the ellipse is</div>

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

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

answer:
<div class="solution">Hint: \(\frac{b}{a e}=\tan 60^{\circ} \Rightarrow \frac{b}{a}=e \sqrt{3}\)<br/>\(\begin{aligned}<br/>&amp; \because e^2=1-\frac{b^2}{a^2}=1-3 e^2 \\<br/>&amp; \Rightarrow e=+\frac{1}{2} \quad\left(e \neq-\frac{1}{2}\right)<br/>\end{aligned}\)</div>


title: if-the-matrices-a-2-4-1-1-3-0-and-b-1-0-5-1-2-0-then-ab-will-be
slug: if-the-matrices-a-2-4-1-1-3-0-and-b-1-0-5-1-2-0-then-ab-will-be-15667
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the matrices $A=\left[\begin{array}{lll}2 &amp; 1 &amp; 3 \\ 4 &amp; 1 &amp; 0\end{array}\right]$ and $B=\left[\begin{array}{rr}1 &amp; -1 \\ 0 &amp; 2 \\ 5 &amp; 0\end{array}\right]$, then $\mathrm{AB}$ will be</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\left[\begin{array}{cr}17 &amp; 0 \\ 4 &amp; -2\end{array}\right]$</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">$\left[\begin{array}{ll}4 &amp; 0 \\ 0 &amp; 4\end{array}\right]$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left[\begin{array}{cr}17 &amp; 4 \\ 0 &amp; -2\end{array}\right]$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left[\begin{array}{ll}0 &amp; 0 \\ 0 &amp; 0\end{array}\right]$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left[\begin{array}{cr}17 &amp; 0 \\ 4 &amp; -2\end{array}\right]$</span> </div>

answer:
<div class="solution">$$<br/>\text { Hints : } \mathrm{AB}=\left[\begin{array}{lll}<br/>2 &amp; 1 &amp; 3 \\<br/>4 &amp; 1 &amp; 0<br/>\end{array}\right]\left[\begin{array}{rr}<br/>1 &amp; -1 \\<br/>0 &amp; 2 \\<br/>5 &amp; 0<br/>\end{array}\right]=\left[\begin{array}{lr}<br/>17 &amp; 0 \\<br/>4 &amp; -2<br/>\end{array}\right]<br/>$$</div>


title: if-the-matrix-a-2-0-2-0-2-0-0-0-2-then-a-n-a-0-b-0-a-0-0-0-a-n-n-where
slug: if-the-matrix-a-2-0-2-0-2-0-0-0-2-then-a-n-a-0-b-0-a-0-0-0-a-n-n-where-43149
tags: ['Mathematics', 'Matrices', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the matrix $A=\left[\begin{array}{lll}2 &amp; 0 &amp; 0 \\ 0 &amp; 2 &amp; 0 \\ 2 &amp; 0 &amp; 2\end{array}\right],$ then
<br/>$A^{n}=\left[\begin{array}{lll}a &amp; 0 &amp; 0 \\ 0 &amp; a &amp; 0 \\ b &amp; 0 &amp; a\end{array}\right], n \in N,$ where</div>

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

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

answer:
<div class="solution">We have. $A=\left[\begin{array}{lll}2 &amp; 0 &amp; 0 \\ 0 &amp; 2 &amp; 0 \\ 2 &amp; 0 &amp; 2\end{array}\right]$
<br/>$\therefore \quad A^{2}=\left[\begin{array}{lll}2 &amp; 0 &amp; 0 \\ 0 &amp; 2 &amp; 0 \\ 2 &amp; 0 &amp; 2\end{array}\right]\left[\begin{array}{lll}2 &amp; 0 &amp; 0 \\ 0 &amp; 2 &amp; 0 \\ 2 &amp; 0 &amp; 2\end{array}\right]=\left[\begin{array}{lll}4 &amp; 0 &amp; 0 \\ 0 &amp; 4 &amp; 0 \\ 8 &amp; 0 &amp; 4\end{array}\right]$
<br/>$=\left[\begin{array}{ccc}2^{2} &amp; 0 &amp; 0 \\ 0 &amp; 2^{2} &amp; 0 \\ 2 \cdot 2^{2} &amp; 0 &amp; 2^{2}\end{array}\right]$
<br/>Now, $\quad A^{n}=\left[\begin{array}{ccc}2^{n} &amp; 0 &amp; 0 \\ 0 &amp; 2^{n} &amp; 0 \\ n-2^{n} &amp; 0 &amp; 2^{n}\end{array}\right]$
<br/>$A^{n}=\left[\begin{array}{lll}a &amp; 0 &amp; 0 \\ 0 &amp; a &amp; 0 \\ b &amp; 0 &amp; a\end{array}\right]$
<br/>$\therefore \quad\left[\begin{array}{lll}a &amp; 0 &amp; 0 \\ 0 &amp; a &amp; 0 \\ b &amp; 0 &amp; a\end{array}\right]=\left[\begin{array}{ccc}2^{n} &amp; 0 &amp; 0 \\ 0 &amp; 2^{n} &amp; 0 \\ n \cdot 2^{n} &amp; 0 &amp; 2^{n}\end{array}\right]$
<br/>Hence, $\quad a=2^{n}$
<br/>and $\quad b=n 2^{n}$</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
if-the-matrix-mr-is-given-by-mrleftbeginarrayccr-r1-r1-rendarrayright-for-r123-ldots-then-operatornamedetleftmathrmm1rightoperatornamedetleftmathrmm2r

slug:
if-the-matrix-mr-is-given-by-mrleftbeginarrayccr-r1-r1-rendarrayright-for-r123-ldots-then-operatornamedetleftmathrmm1rightoperatornamedetleftmathrmm2r-42510

question:
<div class="question">If the matrix \(M_r\) is given by \(M_r=\left(\begin{array}{cc}r &amp; r-1 \\ r-1 &amp; r\end{array}\right)\) for \(r=1,2,3, \ldots\) then \(\operatorname{det}\left(\mathrm{M}_1\right)+\operatorname{det}\left(\mathrm{M}_2\right)+\ldots . .+\operatorname{det}\left(\mathrm{M}_{2008}\right)=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2007</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">2008</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\((2008)^2\)</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">\((2007)^2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\((2008)^2\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hint : } \operatorname{det}\left(M_r\right)=r^2-(r-1)^2=r^2-\left(r^2-2 r+1\right)=2 r-1 \\<br/>&amp; \begin{aligned}<br/>\therefore \sum_{r=1}^{2008} \operatorname{det}\left(M_r\right)=\sum_{r=1}^{2008}(2 r-1)= &amp; +3+5+\cdots+4015 \\<br/>= &amp; (2008)^2<br/>\end{aligned}<br/>\end{aligned}\)</div>


tags: ['Chemistry', 'Redox Reactions', 'JEE Main']
source_db: db\MarksBatch2_P1.db
title:
if-the-molecular-wt-of-mathrmna2-mathrms2-mathrmo3-and-mathrmi2-are-mathrmm1-and-mathrmm2-respectively-then-what-will-be-the-equivalent-wt-of-mathrmna-1

slug:
if-the-molecular-wt-of-mathrmna2-mathrms2-mathrmo3-and-mathrmi2-are-mathrmm1-and-mathrmm2-respectively-then-what-will-be-the-equivalent-wt-of-mathrmna-1-27686

question:
<div class="question">If the molecular wt. of \(\mathrm{Na}_2 \mathrm{~S}_2 \mathrm{O}_3\) and \(\mathrm{I}_2\) are \(\mathrm{M}_1\) and \(\mathrm{M}_2\) respectively, then what will be the equivalent wt. of \(\mathrm{Na}_2 \mathrm{~S}_2 \mathrm{O}_3\) and \(\mathrm{I}_2\) in the following reaction?<br/>\(2 \mathrm{~S}_2 \mathrm{O}_3^{2-}+\mathrm{I}_2 \longrightarrow \mathrm{S}_4 \mathrm{O}_6^{2-}+2 \mathrm{I}^{-}\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{M}_1, \mathrm{M}_2\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{M}_1, \mathrm{M}_2 / 2\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(2 \mathrm{M}_1, \mathrm{M}_2\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{M}_1, 2 \mathrm{M}_2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{M}_1, \mathrm{M}_2 / 2\)</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/_oP7eEyNc13nMegNF-tTyp1pydATUZOwXF2W2VDcZZo.original.fullsize.png"/><br/><br/>\(\begin{aligned}<br/>&amp; \text { n.f. }\left(\mathrm{S}_2 \mathrm{O}_3^{2-}\right)=1, \text { Equivalent mass }=\frac{\mathrm{M}_1}{1}=\mathrm{M}_1 \\<br/>&amp; \mathrm{n} \mathrm{f} .\left(\mathrm{I}_2\right)=2, \quad \text { Equivalent mass }=\frac{\mathrm{M}_2}{2}<br/>\end{aligned}\)</div>


tags: ['Chemistry', 'Redox Reactions', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-the-molecular-wt-of-mathrmna2-mathrms2-mathrmo3-and-mathrmi2-are-mathrmm1-and-mathrmm2-respectively-then-what-will-be-the-equivalent-wt-of-mathrmna

slug:
if-the-molecular-wt-of-mathrmna2-mathrms2-mathrmo3-and-mathrmi2-are-mathrmm1-and-mathrmm2-respectively-then-what-will-be-the-equivalent-wt-of-mathrmna-30406

question:
<div class="question">If the molecular wt. of \(\mathrm{Na}_2 \mathrm{~S}_2 \mathrm{O}_3\) and \(\mathrm{I}_2\) are \(\mathrm{M}_1\) and \(\mathrm{M}_2\) respectively, then what will be the equivalent wt. of \(\mathrm{Na}_2 \mathrm{~S}_2 \mathrm{O}_3\) and \(\mathrm{I}_2\) in the following reaction?<br/>\(2 \mathrm{~S}_2 \mathrm{O}_3^{2-}+\mathrm{I}_2 \longrightarrow \mathrm{S}_4 \mathrm{O}_6^{2-}+2 \mathrm{I}^{-}\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{M}_1, \mathrm{M}_2\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(\mathrm{M}_1, \mathrm{M}_2 / 2\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(2 \mathrm{M}_1, \mathrm{M}_2\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\mathrm{M}_1, 2 \mathrm{M}_2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\mathrm{M}_1, \mathrm{M}_2 / 2\)</span> </div>

answer:
<div class="solution">Hints: <img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/_oP7eEyNc13nMegNF-tTyp1pydATUZOwXF2W2VDcZZo.original.fullsize.png"/><br/><br/>\(\begin{aligned}<br/>&amp; \text { n.f. }\left(\mathrm{S}_2 \mathrm{O}_3^{2-}\right)=1, \text { Equivalent mass }=\frac{\mathrm{M}_1}{1}=\mathrm{M}_1 \\<br/>&amp; \mathrm{n} \mathrm{f} .\left(\mathrm{I}_2\right)=2, \quad \text { Equivalent mass }=\frac{\mathrm{M}_2}{2}<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
if-the-n-terms-a1-a2-ldots-ldots-ldots-an-are-in-ap-with-increment-r-then-the-difference-between-the-mean-of-their-squares-the-square-of-their-mean-is

slug:
if-the-n-terms-a1-a2-ldots-ldots-ldots-an-are-in-ap-with-increment-r-then-the-difference-between-the-mean-of-their-squares-the-square-of-their-mean-is-62487

question:
<div class="question">If the \(n\) terms \(a_1, a_2, \ldots \ldots \ldots . ., a_n\) are in A.P. with increment \(r\), then the difference between the mean of their squares &amp; the square of their mean is</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">(C) \(\frac{r^2\left(n^2-1\right)}{12}\)</span> </div>

answer:
<div class="solution">Hint: \(\frac{a_1^2+a_2^2+\ldots+a_n^2}{n}-\left(\frac{a_1+a_2+\ldots+a_n}{n}\right)^2\)<br/>\(=\frac{a_1^2+\left(a_1+r\right)^2+\ldots+\left\{a_1+(n-1) r\right\}^2}{n}-\left(\frac{n a_1+r \cdot \frac{n(n-1)}{2}}{n}\right)^2\)<br/>\(\begin{aligned}<br/>&amp; =\frac{n a_1^2+r^2 \cdot \frac{(n-1) \cdot n(2 n-1)}{6}+a_1 r(n-1) \cdot n}{n}-a_1^2-a_1 r(n-1)-\frac{r^2(n-1)^2}{4} \\<br/>&amp; =a_1^{2}+r^2 \frac{(n-1)(2 n-1)}{6}+a_1 r(n-1)-a_1^2-a_1 r(n-1)-\frac{r^2(n-1)^2}{4}<br/>\end{aligned}\)<br/>\(\begin{aligned} &amp; =\frac{r^2(n-1)}{2}\left(\frac{2 n-1}{3}-\frac{n-1}{2}\right) \\ &amp; =\frac{r^2(n-1)}{2} \cdot\left(\frac{4 n-2-3 n+3}{6}\right) \\ &amp; =\frac{r^2(n-1)(n+1)}{12} \\ &amp; =\frac{r^2\left(n^2-1\right)}{12}\end{aligned}\)</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_P1.db
title:
if-the-normal-to-the-curve-y-f-x-at-the-point-3-4-make-an-angle-3-4-with-the-positive-x-axis-then-f-3-is

slug:
if-the-normal-to-the-curve-y-f-x-at-the-point-3-4-make-an-angle-3-4-with-the-positive-x-axis-then-f-3-is-70684

question:
<div class="question">If the normal to the curve $y=f(x)$ at the point $(3,4)$ make an angle $3 \pi / 4$ with the positive $x$-axis, then $f(3)$ 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">-1</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$-\frac{3}{4}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{3}{4}$</span> </li> </ul>

answer:
<div class="solution">Hints: $\frac{d y}{d x}=f^{\prime}(x)$, Slope of normal $=-\frac{1}{f^{\prime}(x)},-\frac{1}{f^{\prime}(3)}=\tan \frac{3 \pi}{4}=-1$ $f^{\prime}(3)=1$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
title:
if-the-parabola-x-2-a-y-makes-an-intercept-of-length-40-units-on-the-line-y-2-x-1-then-a-is-equal-to

slug:
if-the-parabola-x-2-a-y-makes-an-intercept-of-length-40-units-on-the-line-y-2-x-1-then-a-is-equal-to-40409

question:
<div class="question">If the parabola $x^{2}=a y$ makes an intercept of length $\sqrt{40}$ units on the line $y-2 x=1$, then $a$ is equal to</div>

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

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

answer:
<div class="solution">Gven, equation of the parabola
<br/>$x^{2}=a y$
<br/>ie $y=\frac{x^{2}}{a}$
<br/>and the equation of line
<br/>$y-2 x=1$ 
<br/>$y=2 x+1$
<br/>$\frac{x^{2}}{a}=2 x+1$
<br/>$x^{2}=2 a x+a$
<br/>$x^{2}-2 a x-a=0$
<br/>$\begin{aligned} x &amp;=\frac{2 a \pm \sqrt{4 a^{2}+4 a}}{2} \\ &amp;=\frac{2\left(a \pm \sqrt{a^{2}+a}\right)}{2}=a \pm \sqrt{a^{2}+a} \end{aligned}$
<br/>Points of intersection of the parabola and the line are
<br/>$$
<br/>\left(a+\sqrt{a^{2}+a}, \left(2\left(a+\sqrt{a^{2}+a}\right)+1\right)\right)
<br/>$$
<br/>and $\quad\left(a-\sqrt{a^{2}+a}, \left(2 ( a-\sqrt{a^{2}+a})+1\right)\right)$
<br/>$\because$ Distance between the points $=\sqrt{40}$ $\therefore \quad \sqrt{40}=\sqrt{\left(2 \sqrt{a^{2}+a}\right)^{2}+\left(4 \sqrt{a^{2}+a}\right)^{2}}$
<br/>$\Rightarrow \quad 40=4\left(a^{2}+a\right)+16\left(a^{2}+a\right)$
<br/>$\Rightarrow \quad 2=a^{2}+a \Rightarrow a=1-2$</div>


tags: ['Mathematics', 'Area Under Curves', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
if-the-point-2-cos-2-sin-for-0-0-2-lies-in-the-region-between-the-lines-x-y-2-and-x-y-2-containing-the-origin-then-lies-in

slug:
if-the-point-2-cos-2-sin-for-0-0-2-lies-in-the-region-between-the-lines-x-y-2-and-x-y-2-containing-the-origin-then-lies-in-27723

question:
<div class="question">If the point $(2 \cos \theta, 2 \sin \theta)$ for $0 \in(0,2 \pi)$ lies in the region between the lines $x+y=2$ and $x-y=2$ containing the origin, then $\theta$ lies in</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\theta, \frac{\pi}{2}\right) \cup\left(\frac{3 \pi}{2}, 2 \pi\right)$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$[0, n]$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$\left(\frac{\pi}{2} \cdot \frac{3 \pi}{2}\right)$</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">$\left[\frac{\pi}{4}, \frac{\pi}{2}\right]$</span> </li> </ul>

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

answer:
<div class="solution">Given that $(2 \cos \theta, 2 \sin \theta)$ will lie on the circle $x^{2}+y^{2}=4$ (from the given figure). Since, point lies on the region containing origin.
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2015_mat_a136.png"/> <br/>So, point will be on the shaded region.
<br/>$\therefore$
<br/>$\theta \in\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$2: 1$</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-the-point-of-intersection-of-the-lines-2-a-x-4-a-y-c-0-and-7-b-x-3-b-y-d-0-lies-in-the-4-th-quadrant-and-is-equidistant-from-the-two-axes-where-a-b

slug:
if-the-point-of-intersection-of-the-lines-2-a-x-4-a-y-c-0-and-7-b-x-3-b-y-d-0-lies-in-the-4-th-quadrant-and-is-equidistant-from-the-two-axes-where-a-b-40738

question:
<div class="question">If the point of intersection of the lines $2 a x+4 a y+c=0$ and $7 b x+3 b y-d=0$ lies in the 4 th quadrant and is equidistant from the two axes, where $a, b, c$ and $d$ are non-zero numbers, then $a d: b c$ equals to</div>

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

answer:
<div class="solution">Let coordinate of the point be $(\alpha,-\alpha)$
<br/>since, $(\alpha,-\alpha)$ lie on $2 a x+4 a y+c=0$
<br/>and $\quad 7 b x+3 b y-d=0$
<br/>$\therefore \quad 2 a \alpha-4 a \alpha+c=0$
<br/>$\Rightarrow \quad-2 a \alpha+c=0$
<br/>$\Rightarrow \quad \alpha=\frac{c}{2 a}$
<br/>Also, $7 b \alpha-3 b \alpha-d=0$
<br/>$\Rightarrow \quad 4 b \alpha-d=0$
<br/>$\Rightarrow \quad \alpha=\frac{d}{4 b}$
<br/>From Eqs. (i) and (ii), 
<br/>$\begin{aligned} \frac{c}{2 a} &amp;=\frac{d}{4 b} \\ 
<br/>\Rightarrow 2 a d &amp;=4 b c \end{aligned}$
<br/>$\Rightarrow \quad \frac{a d}{b c}=\frac{4}{2}$
<br/>$\Rightarrow \quad \frac{a d}{b c}=\frac{2}{1}$
<br/>$a d: b c=2: 1$</div>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2018']
source_db: db\MarksBatch2_P1.db
title:
if-the-polynomial-f-x-1-x-a-1-2-x-b-2-x-b-1-x-a-1-1-2-x-b-1-x-a-then-the-constant-term-of-f-x-is-a-and-b-are-positive-integers

slug:
if-the-polynomial-f-x-1-x-a-1-2-x-b-2-x-b-1-x-a-1-1-2-x-b-1-x-a-then-the-constant-term-of-f-x-is-a-and-b-are-positive-integers-12214

question:
<div class="question">If the polynomial $f(x)=\left|\begin{array}{ccc}(1+x)^{a} &amp; (2+x)^{b} &amp; 1 \\ 1 &amp; (1+x)^{a} &amp; (2+x)^{b} \\ (2+x)^{b} &amp; 1 &amp; (1+x)^{a}\end{array}\right|,$ then the
<br/>constant term of $f(x)$ is $[a$ and $b$ are positive integers $]$</div>

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

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

answer:
<div class="solution">Given, $f(x)=\left|\begin{array}{ccc}(1+x)^{a} &amp; (2+x)^{b} &amp; 1 \\ 1 &amp; (1+x)^{4} &amp; (2+x)^{b} \\ (2+x)^{b} &amp; 1 &amp; (1+x)^{a}\end{array}\right|$
<br/>For constant term put $x=0$ $f(0)=\left|\begin{array}{ccc}1 &amp; 2^{b} &amp; 1 \\ 1 &amp; 1 &amp; 2 \\ 2^{b} &amp; 1 &amp; 1\end{array}\right|$
<br/>$=1\left(1-2^{b}\right)-2^{b}\left(1-2^{2 b}\right)+1\left(1-2^{b}\right)$
<br/>$=1-2^{b}-2^{b}+2^{3 b}+1-2^{b}$
<br/>$=2-3 \cdot 2^{b}+2^{3 b}$</div>


tags: ['Physics', 'Atomic Physics', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
if-the-potential-energy-of-a-hydrogen-atom-in-the-first-excited-state-is-assumed-to-be-zero-then-the-total-energy-of-n-infty-state-is

slug:
if-the-potential-energy-of-a-hydrogen-atom-in-the-first-excited-state-is-assumed-to-be-zero-then-the-total-energy-of-n-infty-state-is-58563

question:
<div class="question">If the potential energy of a hydrogen atom in the first excited state is assumed to be zero, then the total energy of \(n\) \(=\infty\) state is,</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(3.4 \mathrm{eV}\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(6.8 \mathrm{eV}\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 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</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(\infty\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(6.8 \mathrm{eV}\)</span> </div>

answer:
<div class="solution">Hint : Potential energy in 1 st emited stage \(\left(\mathrm{U}_2\right)=-6.8 \mathrm{eV}\). When \(U_2\) is assumed to be zero then potential energy in \((n=\infty)\)<br/>will be \(6.8 \mathrm{eV}\)<br/>so, total energy for \((n=\infty)=6.8 \mathrm{eV}\)</div>


tags: ['Physics', 'Waves and Sound', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
if-the-pressure-temperature-and-density-of-an-ideal-gas-are-denoted-by-p-t-and-respectively-the-velocity-of-sound-in-the-gas-is

slug:
if-the-pressure-temperature-and-density-of-an-ideal-gas-are-denoted-by-p-t-and-respectively-the-velocity-of-sound-in-the-gas-is-38862

question:
<div class="question">If the pressure, temperature and density of an ideal gas are denoted by $p, T$ and $\rho$, respectively, the velocity of sound in the gas is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">proportional to $\sqrt{p},$ when $T$ is constant.</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">proportional to $\sqrt{T}$</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="correct"> <span class="option-label">C</span> <span class="option-data">proportional to $\sqrt{p}$, when $\rho$ 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">proportional to $T$.</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">proportional to $\sqrt{T}$, proportional to $\sqrt{p}$, when $\rho$ is constant.</span> </div>

answer:
<div class="solution">$v=\sqrt{\frac{\gamma R T}{M}}=\sqrt{\frac{\gamma P}{\rho}}$
<br/>(i) Velocity of sound is proportional to $\sqrt{T}$
<br/>$\therefore$
<br/>$v \propto \sqrt{T}$
<br/>(ii) When $\rho$ is constant, the velocity of sound is proportional to root of pressure $P$
<br/>$\therefore$
<br/>$$
<br/>v \propto \sqrt{P}
<br/>$$</div>


title: if-the-radius-of-a-spherical-balloon-increases-by-01-then-its-volume-increases-approximately-by
tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2019']
source_db: db\MarksBatch2_P1.db
slug:
if-the-radius-of-a-spherical-balloon-increases-by-01-then-its-volume-increases-approximately-by-43517

question:
<div class="question">If the radius of a spherical balloon increases by $0.1 \%$, then its volume increases approximately by</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$0.2 \%$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$0.3 \%$</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">$0.4 \%$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$0.05 \%$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$0.3 \%$</span> </div>

answer:
<div class="solution">Let $V$ be the volume of spherical ballon of
<br/>radius $r$ 
<br/>Then, $V=\frac{4}{3} \pi r^{3}$
<br/>$\Rightarrow \quad \log V=\log \left(\frac{4 \pi}{3}\right)+3 \log r \Rightarrow \frac{1}{V} \frac{d V}{d r}=0+\frac{3}{r}$
<br/>$\Rightarrow \quad \frac{1}{V} \frac{d V}{d r}=\frac{3}{r} \Rightarrow \frac{1}{V} \frac{\Delta V}{\Delta r}=\frac{3}{r}$
<br/>$\Rightarrow \quad \frac{\Delta V}{V}=3 \frac{\Delta r}{r}$
<br/>$\Rightarrow \frac{\Delta V}{V} \times 100=3 \frac{\Delta r}{r} \times 100$
<br/>$\Rightarrow \frac{\Delta V}{V} \times 100=3 \times 01=0.3$
<br/>$\therefore$ Percentage increase in volume is $0.3 \%$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
title:
if-the-rate-of-increase-of-the-radius-of-a-circle-is-5-cm-sec-then-the-rate-of-increase-of-its-area-when-the-radius-is-20-cm-will-be

slug:
if-the-rate-of-increase-of-the-radius-of-a-circle-is-5-cm-sec-then-the-rate-of-increase-of-its-area-when-the-radius-is-20-cm-will-be-53744

question:
<div class="question">If the rate of increase of the radius of a circle is $5 \mathrm{~cm} / \mathrm{sec}$, then the rate of increase of its area, when the radius is $20 \mathrm{~cm}$, will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$10 \pi$</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$20 \pi$</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">$200 \pi$</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">D</span> <span class="option-data">$400 \pi$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$200 \pi$</span> </div>

answer:
<div class="solution">Hints : $\mathrm{A}=\pi \mathrm{r}^2 \quad \frac{d r}{d t}=5$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{\mathrm{dA}}{\mathrm{dt}}=2 \pi r \frac{\mathrm{dr}}{\mathrm{dt}}=2 \pi 20(5) \\<br/>&amp; =200 \pi<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Quadratic Equation', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
title:
if-the-ratio-of-the-roots-of-the-equation-mathrmpx2mathrmqxmathrmr0-is-a-b-then-fracmathrmabmathrmamathrmb2

slug:
if-the-ratio-of-the-roots-of-the-equation-mathrmpx2mathrmqxmathrmr0-is-a-b-then-fracmathrmabmathrmamathrmb2-68383

question:
<div class="question">If the ratio of the roots of the equation \(\mathrm{px}^2+\mathrm{qx}+\mathrm{r}=0\) is \(a: b\), then \(\frac{\mathrm{ab}}{(\mathrm{a}+\mathrm{b})^2}=\)</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(\frac{\mathrm{pr}}{\mathrm{q}^2}\)</span> </div>

answer:
<div class="solution">Hints: Let roots are \(a \alpha\) and \(b \alpha\)<br/>\(\Rightarrow(\mathrm{a}+\mathrm{b}) \alpha=\frac{-\mathrm{q}}{\mathrm{p}}\)<br/>\(\begin{aligned} &amp; a b \alpha^2=\frac{r}{p} \\ &amp; \frac{a b \alpha^2}{(a+b)^2 \alpha^2}=\frac{r}{p} \cdot \frac{p^2}{q^2} \\ &amp; \frac{a b}{(a+b)^2}=\frac{1 p}{q^2}\end{aligned}\)</div>


tags: ['Physics', 'Kinetic Theory of Gases', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
title:
if-the-rms-velocity-of-hydrogen-gas-at-a-certain-temperature-is-c-then-the-ras-velocity-of-oxygen-gas-at-the-same-temperature-is

slug:
if-the-rms-velocity-of-hydrogen-gas-at-a-certain-temperature-is-c-then-the-ras-velocity-of-oxygen-gas-at-the-same-temperature-is-87289

question:
<div class="question">If the rms velocity of hydrogen gas at a certain temperature is $c,$ then the ras velocity of oxygen gas at the same temperature is</div>

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

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

answer:
<div class="solution">$v_{\ln m}=\sqrt{\frac{3 R T}{M}}$
<br/>$R=$ gas constant 
<br/>$T=$ absolute temperature 
<br/>$M=$ mass of gas
<br/>$v_{\text {rms }}$ of $\mathrm{H}_{2}=\sqrt{\frac{3 R T}{M_{\mathrm{H}_{2}}}}$
<br/>$v_{\text {rms }}$ of $\mathrm{O}_{2}=\sqrt{\frac{3 R T}{M_{\mathrm{O}_{2}}}}$
<br/>$\frac{v_{\mathrm{rms}} \cdot \mathrm{H}_{2}}{v_{\mathrm{rma}} \cdot \mathrm{O}_{2}}=\sqrt{\frac{M_{O_{2}}}{M_{\mathrm{H}_{1}}}}$
<br/>Given $\quad v_{\operatorname{rms}} H_{2}=c$
<br/>$\Rightarrow \quad \frac{c}{v_{\operatorname{rms}} \mathrm{O}_{2}}=\sqrt{\frac{M_{0}}{M_{H_{1}}}}=\sqrt{\frac{32}{2}}=\sqrt{16}=4$
<br/>$\therefore \quad v_{rms}=\frac{C}{4}$</div>


title: if-the-solution-of-the-differential-equation-x-d-x-d-y-y-x-e-x-be-x-y-e-x-x-c-then-x-is-equal-to
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$x-1$</span> </div>
source_db: db\MarksBatch2_P1.db
slug:
if-the-solution-of-the-differential-equation-x-d-x-d-y-y-x-e-x-be-x-y-e-x-x-c-then-x-is-equal-to-66871

question:
<div class="question">If the solution of the differential equation $x \frac{d y}{d x}+y=x e^{x}$ be $x y=e^{x} \phi(x)+C,$ then $\phi(x)$ is equal to</div>

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

answer:
<div class="solution">Given, $\quad x \frac{d y}{d x}+y=x e^{x}$
<br/>$\Rightarrow \quad \frac{d y}{d x}+\frac{y}{x}=e^{x}$
<br/>On comparing Eq. by $\frac{d y}{d x}+P y=Q$, we get
<br/>$\therefore$
<br/>$$
<br/>\begin{array}{l}
<br/>P=\frac{1}{x} \text { and } Q=e^{x} \\
<br/>IF=e^{\int \frac{1}{x} d x}
<br/>\end{array}
<br/>$$
<br/>$$
<br/>=e^{log^{x}}=x
<br/>$$
<br/>Hence, solution of differential equation,
<br/>$$
<br/>\begin{array}{ll}
<br/>\quad y \cdot x=\int x e^{x} d x+C \\
<br/>\Rightarrow \quad x y=x e^{x}-\int e^{x} d x+C \\
<br/>\Rightarrow \quad x y=x e^{x}-e^{x}+C
<br/>\end{array}
<br/>$$
<br/>$\Rightarrow \quad x y=e^{x}(x-1)+C$
<br/>$\because xy=e^{x} \phi(x)+c$ [given]
<br/>$$
<br/>\phi(x)=(x-1)
<br/>$$</div>


tags: ['Mathematics', 'Hyperbola', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
if-the-straight-line-a-1-x-b-y-4-0-is-normal-to-the-hyperbola-x-y-1-then-which-of-the-following-does-not-hold

slug:
if-the-straight-line-a-1-x-b-y-4-0-is-normal-to-the-hyperbola-x-y-1-then-which-of-the-following-does-not-hold-88513

question:
<div class="question">If the straight line $(a-1) x-b y+4=0$ is normal to the hyperbola $x y=1$, then which of the following does not hold?</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$a&gt;1, b&gt;0$, $a    &lt; 1, b &lt; 0$</span> </div>

answer:
<div class="solution">Given equation of straight line is
<br/>$$
<br/>(a-1) x-b y+4=0
<br/>$$
<br/>Slope $=-\frac{\text { Coefficient of } x}{\text { Coefficient of } y}$ $=\frac{-(a-1)}{-b}=\frac{a-1}{b}$
<br/>Equation of hyperbola is $x y=1$
<br/>$\Rightarrow \quad y=\frac{1}{x}$
<br/>$\Rightarrow \quad \frac{d y}{d x}=-\frac{1}{x^{2}}$
<br/>Slope of normal $=x^{2}&gt;0$
<br/>$\therefore \quad \frac{a-1}{b}&gt;0$
<br/>$\Rightarrow \quad a-1&gt;0 \quad$ and $\quad b&gt;0$
<br/>or $\quad a-1    &lt; 0$ and $b &lt; 0$ $\Rightarrow \quad a&gt;1, b&gt;0$ or $a &lt; 1, b &lt; 0$</div>


title: if-the-straight-line-ym-x-lies-outside-of-the-circle-x2y220-y900-then-the-value-of-m-will-satisfy
tags: ['Mathematics', 'Circle', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
slug:
if-the-straight-line-ym-x-lies-outside-of-the-circle-x2y220-y900-then-the-value-of-m-will-satisfy-67609

question:
<div class="question">If the straight line \(y=m x\) lies outside of the circle \(x^2+y^2-20 y+90=0\), then the value of \(m\) will satisfy</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(m  &lt; 3\)</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">\(|m|  &lt; 3\)</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">\(\mathrm{m} &gt; 3\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(|\mathrm{m}| &gt; 3\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(|m|  &lt; 3\)</span> </div>

answer:
<div class="solution">Hints: \(x^2+m^2 x^2-20 m x+90\)<br/>\(\begin{aligned}<br/>&amp; x^2\left(1+m^2\right)-20 m x+90=0 \\<br/>&amp; D  &lt; 0 \\<br/>&amp; 400 m^2-4 \times 90\left(1+m^2\right) &lt; 0 \\<br/>&amp; 40 m^2 &lt; 360 \\<br/>&amp; m^2 &lt; 9 ;|m| &lt; 3<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2010']
answer: <div class="solution">Hints: $|x|+|y|=1$</div>
source_db: db\MarksBatch2_P1.db
title:
if-the-sum-of-distances-from-a-point-p-on-two-mutually-perpendicular-straight-lines-is-1-unit-then-the-locus-of-p-is

slug:
if-the-sum-of-distances-from-a-point-p-on-two-mutually-perpendicular-straight-lines-is-1-unit-then-the-locus-of-p-is-82588

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

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


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
title:
if-the-sum-of-the-distances-of-a-point-from-two-perpendicular-lines-in-a-plane-is-1-unit-then-its-locus-is

slug:
if-the-sum-of-the-distances-of-a-point-from-two-perpendicular-lines-in-a-plane-is-1-unit-then-its-locus-is-45752

question:
<div class="question">If the sum of the distances of a point from two perpendicular lines in a plane is 1 unit, then its locus is</div>

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

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

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/NzdEkE0iOQW5o5wpMlJFJLPBjUQ3bhmOn2fTA-ezXwM.original.fullsize.png"/><br/><br/>$$<br/>|x|+|y|=1<br/>$$<br/>Locus is square</div>


title: if-the-sum-of-two-unit-vectors-is-a-unit-vector-then-the-magnitude-of-their-difference-is
slug: if-the-sum-of-two-unit-vectors-is-a-unit-vector-then-the-magnitude-of-their-difference-is-19963
tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the sum of two unit vectors is a unit vector, then the magnitude of their difference is</div>

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

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

answer:
<div class="solution">$\begin{aligned} &amp;\text { Let a and } \mathbf{b} \text { are two unit vectors. }\\ &amp;\text { Then, } \quad|\mathbf{a}+\mathbf{b}|=1\\ &amp;\Rightarrow \quad|\mathbf{a}+\mathbf{b}|^{2}=1 \end{aligned}$
<br/>$\Rightarrow \quad(a+b) \cdot(a+b)=1$
<br/>$\Rightarrow|a|^{2}+|b|^{2}+2 a \cdot b=1$
<br/>$\Rightarrow \quad 1+1+2 \mathbf{a} \cdot \mathbf{b}=1$
<br/>$\Rightarrow \quad 2 \mathbf{a} \cdot \mathbf{b}=-1$
<br/>Again, $|\hat{\mathbf{a}}-\mathbf{b}|^{2}=(\mathbf{a}-\hat{\mathbf{b}}) \cdot(\mathbf{a}-\hat{\mathbf{b}})$
<br/>$=|\mathbf{a}|^{2}+|\mathbf{b}|^{2}-2 \mathbf{a} \cdot \mathbf{b}$
<br/>$=1+1-(-1)$
<br/>$=1+1+1=3$
<br/>$|\hat{a}-\hat{b}|=\sqrt{3}$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
title:
if-the-tangent-at-the-point-p-with-coordinates-h-k-on-the-curve-y-2-2-x-3-is-perpendicular-to-the-straight-line-4-x-3-y-then

slug:
if-the-tangent-at-the-point-p-with-coordinates-h-k-on-the-curve-y-2-2-x-3-is-perpendicular-to-the-straight-line-4-x-3-y-then-60385

question:
<div class="question">If the tangent at the point $P$ with co-ordinates $(h, k)$ on the curve $y^{2}=2 x^{3}$ is perpendicular to the straight line $4 x=3 y$, then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$(\mathrm{h}, \mathrm{k})=(0,0)$ only</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$(\mathrm{h}, \mathrm{k})=\left(\frac{1}{8},-\frac{1}{16}\right)$ only</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$(\mathrm{h}, \mathrm{k})=(0,0)$ or $\left(\frac{1}{8},-\frac{1}{16}\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">no such point P exists</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$(\mathrm{h}, \mathrm{k})=\left(\frac{1}{8},-\frac{1}{16}\right)$ only</span> </div>

answer:
<div class="solution">$2 y \frac{d y}{d x}=6 x^{2} \Rightarrow \frac{d y}{d x}=\frac{3 x^{2}}{y}=-\frac{3}{4} \Rightarrow y=-4 x^{2}$ and $y^{2}=2 x^{3}$
<br/>$\Rightarrow 16 x^{4}=2 x^{3} \Rightarrow x=\frac{1}{8} \quad(x \neq 0)$
<br/>$y=\frac{-1}{16} \quad(y \neq 0)$</div>


tags: ['Mathematics', 'Application of Derivatives', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
title:
if-the-tangent-to-the-curve-y-2-x-3-at-m-2-m-3-is-also-a-normal-to-the-curve-at-m-2-m-3-then-the-value-of-m-m-is

slug:
if-the-tangent-to-the-curve-y-2-x-3-at-m-2-m-3-is-also-a-normal-to-the-curve-at-m-2-m-3-then-the-value-of-m-m-is-83714

question:
<div class="question">If the tangent to the curve $y^{2}=x^{3}$ at $\left(m^{2}, m^{3}\right)$ is also a normal to the curve at $\left(M^{2}, M^{3}\right)$, then the value of $m M$ is</div>

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

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

answer:
<div class="solution">Hint: 
<br/>$2 y y_{1}=3 x^{2}$
<br/>$y_{1}=\frac{3 x^{2}}{2 y} \Rightarrow\left(y_{1}\right)_{m^{2}, m^{3}}=\frac{3 \times m^{4}}{2 \times m^{3}}=\frac{3 m}{2}$
<br/>Again; slope of normal $=-\frac{2}{3 M}, m M=-\frac{4}{9}$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
title:
if-the-tangent-to-y-2-4-a-x-at-the-point-a-t-2-2-a-t-where-t-1-is-a-normal-to-x-2-y-2-a-2-at-the-point-a-sec-a-tan-then

slug:
if-the-tangent-to-y-2-4-a-x-at-the-point-a-t-2-2-a-t-where-t-1-is-a-normal-to-x-2-y-2-a-2-at-the-point-a-sec-a-tan-then-55420

question:
<div class="question">If the tangent to $y^{2}=4 a x$ at the point $\left(a t^{2}, 2 a t\right)$ where $|t|&gt;1$ is a normal to $x^{2}-y^{2}=a^{2}$ at the point $(a \sec \theta, a \tan \theta),$ then</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$t=-\operatorname{cosec} \theta$, $t=2 \tan \theta$</span> </div>

answer:
<div class="solution">$\left(a,\right.$Equation of tangent to $y^{2}=4 a x$ at $\left(a t^{2}, 2 a t\right)$ will be
<br/>$$
<br/>x-y t=-a t^{2}
<br/>$$
<br/>Also, equation of normal to $x^{2}-y^{2}=a^{2}$ at $(a \sec \theta, a \tan \theta)$ will be
<br/>$\frac{x}{a \sec \theta}+\frac{y}{a \tan \theta}=2$
<br/>$\Rightarrow x+y \operatorname{cosec} \theta=2 a \sec \theta$
<br/>since, Eqs. (i) and (ii) are identical.
<br/>$\therefore t=-\operatorname{cosec} \theta$ or $t=2 \tan \theta$</div>


tags: ['Physics', 'Thermal Properties of Matter', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">16</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-the-temperature-of-the-sun-gets-doubled-the-rate-of-energy-received-on-the-earth-will-increase-by-a-factor-of

slug:
if-the-temperature-of-the-sun-gets-doubled-the-rate-of-energy-received-on-the-earth-will-increase-by-a-factor-of-48135

question:
<div class="question">If the temperature of the Sun gets doubled, the rate of energy received on the Earth will increase by a factor of</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">2</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">4</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">8</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">16</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">Suppose temperature of $\mathrm{Sun}=T$
<br/>When it is doubled, $T^{\prime}=2 T$
<br/>From law of radiation (Stefan's law)
<br/>$$
<br/>E \propto T^{4}=\sigma T^{4}
<br/>$$
<br/>$\sigma=$ Stefan's constant.
<br/>$$
<br/>\begin{array}{l}
<br/>E_{1}=T^{4} \\
<br/>E_{2} \approx(2 \Pi)^{4} \\
<br/>\frac{E_{1}}{E_{2}}=\frac{T^{4}}{2^{4} \cdot T^{4}} \\
<br/>E_{2}=16 E_{1}
<br/>\end{array}
<br/>$$</div>


title: if-the-three-points-3-q-0-0-3-p-and-1-1-are-collinear-then-which-one-is-true
slug: if-the-three-points-3-q-0-0-3-p-and-1-1-are-collinear-then-which-one-is-true-71055
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2010']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the three points $(3 q, 0),(0,3 p)$ and $(1,1)$ are collinear then which one is true ?</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{1}{\mathrm{p}}+\frac{1}{\mathrm{q}}=3$</span> </div>

answer:
<div class="solution">Hints: $\mathrm{A}(3 \mathrm{q}, 0) \mathrm{B}(0,3 \mathrm{p}) \mathrm{C}(11)$<br/>Slope $=1 \mathrm{AC}=5 \log \mathrm{BC}$<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{1-0}{1-3 q}=\frac{1-3 p}{1-0}=3, \frac{1}{1-3 q}=\frac{1-3 p}{1} \\<br/>&amp; 1=(1-3 p)(1-3 q), 1=1-3 q-3 p+9 p q \\<br/>&amp; \Rightarrow 3 p+3 q=9 p q, \frac{1}{q}+\frac{1}{p}=3<br/>\end{aligned}<br/>$$</div>


title: if-the-three-points-a16-b34-and-cx-y-are-collinear-then-the-equation-satisfying-by-x-and-y-is
slug: if-the-three-points-a16-b34-and-cx-y-are-collinear-then-the-equation-satisfying-by-x-and-y-is-12535
tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the three points \(A(1,6), B(3,-4)\) and \(C(x, y)\) are collinear then the equation satisfying by \(x\) and \(y\) is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">\(5 x+y-11=0\)</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">\(5 x+13 y+5=0\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(5 x-13 y+5=0\)</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">\(13 x-5 y+5=0\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(5 x+y-11=0\)</span> </div>

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text {Hints: }\left|\begin{array}{ccc}<br/>1 &amp; 1 &amp; 6 \\<br/>1 &amp; 3 &amp; -4 \\<br/>1 &amp; x &amp; y<br/>\end{array}\right|=0 \\<br/>&amp; \Rightarrow 1(3 y+4 x)-(y-6 x)+1(-4-18)=0 \Rightarrow 3 y+4 x-y+6 x-12=0 \\<br/>&amp; \Rightarrow 2 y+10 x-22=0 \\<br/>&amp; y+5 x=11<br/>\end{aligned}\)</div>


tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2020']
answer: <div class="solution">Hint: <br/>$n_{c_{m}}=k^{n-1} c_{m-1} \Rightarrow n=m k$</div>
source_db: db\MarksBatch2_P1.db
title:
if-the-total-number-of-melement-subsets-of-the-set-a-a-1-a-2-a-n-is-k-times-the-number-of-m-element-subsets-containing-a-4-then-n-is

slug:
if-the-total-number-of-melement-subsets-of-the-set-a-a-1-a-2-a-n-is-k-times-the-number-of-m-element-subsets-containing-a-4-then-n-is-82419

question:
<div class="question">If the total number of m-element subsets of the set $\mathrm{A}=\left\{\mathrm{a}{1}, \mathrm{a}{2}, \ldots . \mathrm{a}{\mathrm{n}}\right\}$ is $\mathrm{k}$ times the number of $\mathrm{m}$ element subsets containing $\mathrm{a}{4}$, then $\mathrm{n}$ is</div>

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

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


tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">4</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-the-transformation-z-lo-g-tan-2-x-reduces-the-differential-equation-d-x-2-d-2-y-cot-x-d-x-d-y-4-y-cosec-2-x-0-into-the-form-d-z-2-d-2-y-k-y-0-then-

slug:
if-the-transformation-z-lo-g-tan-2-x-reduces-the-differential-equation-d-x-2-d-2-y-cot-x-d-x-d-y-4-y-cosec-2-x-0-into-the-form-d-z-2-d-2-y-k-y-0-then-90975

question:
<div class="question">If the transformation $z=\log \tan \frac{x}{2}$ reduces the differential equation $\frac{d^2 y}{d x^2}+\cot x \frac{d y}{d x}+4 y \operatorname{cosec}{ }^2 x=0$ into the form $\frac{d^2 y}{d z^2}+k y=0$ then $k$ is equal to</div>

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

answer:
<div class="solution">$\frac{d^2 y}{d z^2}=\frac{d}{d z}\left(\frac{d y}{d z}\right)=\frac{d}{d z}\left(\frac{\frac{d y}{d x}}{\frac{d z}{d x}}\right)$<br/>$=\frac{d}{d z}\left(\frac{\frac{d y}{d x}}{\frac{1}{\sin x}}\right) \quad\left(\because \frac{d z}{d x}=\frac{1}{\sin x}\right)$<br/>$=\frac{d}{d z}\left(\sin x \cdot \frac{d y}{d x}\right)=\frac{\frac{d}{d x}\left(\sin x \cdot \frac{d y}{d x}\right)}{\frac{d z}{d x}}$<br/>$=\sin x\left[\cos x \cdot \frac{d y}{d x}+\sin x \frac{d^2 y}{d x^2}\right]$<br/>$\because \frac{d^2 y}{d z^2}+k y=0$<br/>$\therefore \sin x \cdot \cos x \cdot \frac{d y}{d x}+\sin ^2 x \frac{d^2 y}{d x^2}+k y=0$<br/>Dividing the equation by $\sin ^2 x$, we get $\frac{d^2 y}{d x^2}+\cot x \cdot \frac{d y}{d x}+k \cdot \operatorname{cosec}^2 x \cdot y=0$<br/>Comparing with the equation<br/>$\frac{d^2 y}{d x^2}+\cot x \cdot \frac{d y}{d x}+4 \operatorname{cosec}^2 x \cdot y=0$<br/>We get $k=4$</div>


tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2020']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$-1$</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-the-vectors-i-a-j-a-2-k-i-b-j-b-2-k-and-i-c-j-c-2-k-are-three-noncoplanar-vectors-and-a-b-c-a-2-b-2-c-2-1-a-3-1-b-3-1-c-3-0-then-the-value-of-abc-i

slug:
if-the-vectors-i-a-j-a-2-k-i-b-j-b-2-k-and-i-c-j-c-2-k-are-three-noncoplanar-vectors-and-a-b-c-a-2-b-2-c-2-1-a-3-1-b-3-1-c-3-0-then-the-value-of-abc-i-91446

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

answer:
<div class="solution">Hint: 
<br/>$\left|\begin{array}{lll}\mathrm{a} &amp; \mathrm{a}^{2} &amp; 1 \\ \mathrm{~b} &amp; \mathrm{~b}^{2} &amp; 1 \\ \mathrm{c} &amp; \mathrm{c}^{2} &amp; 1\end{array}\right|(1+\mathrm{abc})=0$
<br/>$a b c=-1[\because \vec{\alpha}, \vec{\beta}, \vec{\gamma}$ are non-coplanar vector $]$</div>


tags: ['Physics', 'Ray Optics', 'WBJEE', 'WBJEE 2012']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0.5</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-the-velocity-of-light-in-vacuum-is-3-1-0-8-ms-1-the-time-taken-in-nanosecond-to-travel-through-a-glass-plate-of-thickness-10-cm-and-refractive-inde

slug:
if-the-velocity-of-light-in-vacuum-is-3-1-0-8-ms-1-the-time-taken-in-nanosecond-to-travel-through-a-glass-plate-of-thickness-10-cm-and-refractive-inde-22727

question:
<div class="question">If the velocity of light in vacuum is $3 \times 10^{8} \mathrm{ms}^{-1}$, the time taken (in nanosecond) to travel through a glass plate of thickness $10 \mathrm{cm}$ and refractive index 1.5 is</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">0.5</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">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">3</span> </li> </ul>

answer:
<div class="solution">Given $v=3 \times 10^{8} \mathrm{ms}^{-1}, d=10 \mathrm{m}$ and $\mu=1.5$
<br/>Here, $t=\frac{d}{c / \mu}$
<br/>$$
<br/>t=\frac{10 \times 10^{-2}}{2 \times 10^{8}}
<br/>$$
<br/>$\left(\therefore \frac{c}{\mu}=\frac{3 \times 10^{8}}{1.5}=2 \times 10^{8} \mathrm{ms}^{-1}\right)$
<br/>$t=0.5 \times 10^{-9} \mathrm{s}$</div>


tags: ['Mathematics', 'Straight Lines', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
title:
if-the-vertex-of-the-conic-y-2-4-y-4-x-4-a-always-lies-between-the-straight-lines-x-y-3-and-2-x-2-y-1-0-then-1

slug:
if-the-vertex-of-the-conic-y-2-4-y-4-x-4-a-always-lies-between-the-straight-lines-x-y-3-and-2-x-2-y-1-0-then-1-52062

question:
<div class="question">If the vertex of the conic $y^{2}-4 y=4 x-4 a$ always lies between the straight lines ${x}+y=3$ and $2 x+2 y-1=0,$ then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2    &lt; a &lt; 4$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{2}    &lt; a &lt; 2$</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">$0    &lt; a &lt; 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{1}{2}    &lt; a &lt; \frac{3}{2}$</span> </li> </ul>

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

answer:
<div class="solution">Equation of the vertex of conic
<br/>$$
<br/>y^{2}-4 y=4 x-4 a
<br/>$$
<br/>$\Rightarrow \quad y^{2}-4 y+4=4 x-4 a+4$
<br/>$\Rightarrow \quad(y-2)^{2}=4[x-(a-1)]$
<br/>$\therefore$
<br/>Vertex $=(a-12)$
<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/2016_mat_a40.png"/> <br/> <br/>From figure,
<br/>clearly. $-\frac{3}{2} &lt; a-1 &lt; 1$
<br/>$\therefore$
<br/>$-\frac{1}{2} &lt; a &lt; 2$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2015']
source_db: db\MarksBatch2_P1.db
title:
if-the-vertex-of-the-conic-y-2-4-y-4-x-4-a-always-lies-between-the-straight-lines-x-y-3-and-2-x-2-y-1-0-then

slug:
if-the-vertex-of-the-conic-y-2-4-y-4-x-4-a-always-lies-between-the-straight-lines-x-y-3-and-2-x-2-y-1-0-then-91145

question:
<div class="question">If the vertex of the conic $y^{2}-4 y=4 x-4 a$ always lies between the straight lines $x+y=3$ and $2 x+2 y-1=0 .$ then</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$2    &lt; a &lt; 4$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$-\frac{1}{2}    &lt; a &lt; 2$</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">$0    &lt; a &lt; 2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$-\frac{1}{2}    &lt; a &lt; \frac{3}{2}$</span> </li> </ul>

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

answer:
<div class="solution">We have, $y^{2}-4 y=4 x-4 a$ $\Rightarrow \quad(y-2)^{2}-4=4 x-4 a$
<br/>$\Rightarrow \quad(y-2)^{2}=4 x-4 a+4$
<br/>$\Rightarrow \quad(y-2)^{2}=4|x-(a-1)|$
<br/>Hence, vertex is $(a-1,2)$ 
<br/>Since, vertex lies between the lines $x+y=3$ and $2 x+2 y-1=0,$ then
<br/>$(a-1+2-3)(2 a-2+4-1)    &lt; 0$
<br/>$\Rightarrow(a-2)(2 a+1) &lt; 0 \Rightarrow a \in\left(-\frac{1}{2}, 2\right)$</div>


title: if-the-vertices-of-a-square-are-z1-z2-z3-and-z4-taken-in-the-anticlockwise-order-then-z3
slug: if-the-vertices-of-a-square-are-z1-z2-z3-and-z4-taken-in-the-anticlockwise-order-then-z3-56932
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If the vertices of a square are \(z_1, z_2, z_3\) and \(z_4\) taken in the anti-clockwise order, then \(z_3=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(-i \mathrm{z}_1-(1+i) z_2\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(\mathrm{z}_1-(1+\mathrm{i}) \mathrm{z}_2\)</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">\(z_1+(1+i) z_2\)</span> </li><li class="correct"> <span class="option-label">D</span> <span class="option-data">\(-i \mathrm{z}_1+(1+\mathrm{i}) \mathrm{z}_2\)</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-i \mathrm{z}_1+(1+\mathrm{i}) \mathrm{z}_2\)</span> </div>

answer:
<div class="solution"><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/NPtX_yH46AgvuV5xb5VGN8Q32N73evhadAS1g8SDay8.original.fullsize.png"/><br/><br/>Hint : In \(\triangle A B C \frac{z_1-z_2}{z_3-z_2}=\left|\frac{z_1-z_2}{z_3-z_2}\right|\mathrm{i}^{\pi / 2}=\left|\frac{z_1-z_2}{z_3-z_2}\right| \cdot i=\frac{A B}{B C} \cdot i=i\)<br/>\(\Rightarrow \mathrm{z}_1-\mathrm{z}_2=\mathrm{i}\left(\mathrm{z}_3-\mathrm{z}_2\right) \Rightarrow-i \mathrm{z}_1+\mathrm{i} \mathrm{z}_2=\mathrm{z}_3-\mathrm{z}_2 \Rightarrow \mathrm{z}_3=-\mathrm{i} \mathrm{z}_1+(\mathrm{i}+1) \mathrm{z}_2=-\mathrm{i} \mathrm{z}_1+(1+\mathrm{i}) \mathrm{z}_2\)</div>


tags: ['Mathematics', 'Vector Algebra', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
title:
if-the-volume-of-the-parallelopiped-with-veca-times-vecb-vecb-times-vecc-and-vecc-times-veca-as-coterminous-edges-is-9-cu-units-then-the-volume-of-the

slug:
if-the-volume-of-the-parallelopiped-with-veca-times-vecb-vecb-times-vecc-and-vecc-times-veca-as-coterminous-edges-is-9-cu-units-then-the-volume-of-the-27947

question:
<div class="question">If the volume of the parallelopiped with \(\vec{a} \times \vec{b}, \vec{b} \times \vec{c}\) and \(\vec{c} \times \vec{a}\) as coterminous edges is 9 cu. units., then the volume of the parallelopiped with \((\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c}),(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a})\) and \((\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b})\) as coterminous edges is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">9 cu. units</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">729 cu. units</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">81 cu. units</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">243 cu. units</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">81 cu. units</span> </div>

answer:
<div class="solution">Hint : Volume of Parallelopiped whose coterminous edges are \(\vec{a}, \vec{b}\) and \(\vec{c}=[\vec{a} \vec{b} \vec{c}]\)<br/>So, Given that<br/><img src="https://cdn-question-pool.getmarks.app/pyq/wbjee/GiRyI81GDMChYYwYlD0sOCRXmrG9nAT7af64lukGezQ.original.fullsize.png"/><br/>\(\therefore[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}]=3<br/>\quad \ldots \ldots (II)\)<br/>Now volume of parallelopiped whose coterminous edges are<br/>\(\begin{aligned} &amp; (\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c}),(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a}),(\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b}) \\ &amp; {[(\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c})(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a})(\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b})]} \\ &amp; {[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]^2}\end{aligned}\)<br/>\(\begin{aligned}<br/>&amp; =(9)^2 \quad \text{(from (1))}\\<br/>&amp; =81<br/>\end{aligned}\)</div>


title: if-thetaphifracpi4-then-1tan-theta1tan-phi-is-equal-to
slug: if-thetaphifracpi4-then-1tan-theta1tan-phi-is-equal-to-49821
tags: ['Mathematics', 'Trigonometric Ratios & Identities', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(\theta+\phi=\frac{\pi}{4}\), then \((1+\tan \theta)(1+\tan \phi)\) is equal to</div>

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

answer:
<div class="solution">\(\begin{aligned}<br/>&amp; \text { Hints: }(1+\tan \theta)\left(1+\frac{(1-\tan \theta)}{1+\tan \theta}\right) \\<br/>&amp; =(1+\tan \theta) \frac{2}{1+\tan \theta}=2<br/>\end{aligned}\)</div>


title: if-three-positive-real-numbers-a-b-c-are-in-ap-and-ab-c-4-then-minimum-possible-value-of-b-is
slug: if-three-positive-real-numbers-a-b-c-are-in-ap-and-ab-c-4-then-minimum-possible-value-of-b-is-64464
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If three positive real numbers $a, b, c$ are in A.P. and $a b c=4$ then minimum possible value of $b$ is</div>

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

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

answer:
<div class="solution">$$<br/>\text { Hints } \begin{aligned}<br/>&amp; (b-d) b(b+d)=4 \\<br/>&amp; \left(b^2-d^2\right) b=4 \\<br/>&amp; b^3=4+d^2 b \\<br/>&amp; b^3 \geq 4 \Rightarrow b \geq(2)^{2 / 3}<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
if-u-x-and-v-x-are-two-independent-solutions-of-the-differential-equation-d-x-2-d-2-y-b-d-x-d-y-cy-0-then-additional-solutions-of-the-given-differenti

slug:
if-u-x-and-v-x-are-two-independent-solutions-of-the-differential-equation-d-x-2-d-2-y-b-d-x-d-y-cy-0-then-additional-solutions-of-the-given-differenti-47448

question:
<div class="question">If $u(x)$ and $v(x)$ are two independent solutions of the differential equation $\frac{d^{2} y}{d x^{2}}+b \frac{d y}{d x}+c y=0$ then additional solution(s) of the given differential equation is(are)</div>

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

correct_option:
<div class="correct-answer">
The correct answers are:
<span class="option-value">$y=5 u(x)+8 v(x)$, $y=c_{1}\{u(x)-v(x)\}+c_{2} v(x), c_{1}$ and $c_{2}$ are arbitrary constants</span> </div>

answer:
<div class="solution">We know that $u(x)$ and $v(x)$ are two independent solutions of the given differential equation, then their linear combination is also the solution of the given equation.
<br/>Here, we see that $y=5 u(x)+8 v(x)$ is a linear combination and $y=c_{1}\{u(x)-v(x)\}+c_{2} v(x)$ is also a linear combination of two independent solutions.</div>


title: if-x-1-2-1-1-4-2-1-8-3-1-and-y-1-1-x-2-2-x-4-3-x-6-then-the-value-of-loge-y-is
slug: if-x-1-2-1-1-4-2-1-8-3-1-and-y-1-1-x-2-2-x-4-3-x-6-then-the-value-of-loge-y-is-92617
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2013']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">$e$</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x=1+\frac{1}{2 \times 1 !}+\frac{1}{4 \times 2 !}+\frac{1}{8 \times 3 !}+\ldots$ and
<br/>$y=1+\frac{x^{2}}{1 !}+\frac{x^{4}}{2 !}+\frac{x^{6}}{3 !}+\ldots$
<br/>Then, the value of loge $y$ is</div>

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

answer:
<div class="solution">Given,
<br/>$x=1+\frac{1}{2 \times 1 !}+\frac{1}{4 \times 2 !}+\frac{1}{8 \times 3 !}+\ldots$
<br/>$\Rightarrow \quad x=1+\frac{(1 / 2)^{\prime}}{1 !}+\frac{(1 / 2)^{2}}{2 !}+\frac{(1 / 2)^{3}}{3 !}+\ldots$
<br/>$\Rightarrow \quad x=e^{1 / 2} \Rightarrow x^{2}=e$
<br/>and $\quad y=1+\frac{x^{2}}{1 !}+\frac{x^{4}}{2 !}+\frac{x^{6}}{3 !}+\ldots$
<br/>$\Rightarrow \quad y=1+\frac{\left(x^{2}\right)^{1}}{1 !}+\frac{\left(x^{2}\right)^{2}}{2 !}+\frac{\left(x^{2}\right)^{3}}{3 !}+\ldots$
<br/>$\Rightarrow \quad y=e^{x^{2}}=e^{e} \quad$ [from Eq. (i)]
<br/>Taking log on both sides, we get
<br/>log$y=e$</div>


tags: ['Physics', 'Experimental Physics', 'WBJEE', 'WBJEE 2010']
answer: <div class="solution">No Solution Available</div>
source_db: db\MarksBatch2_P1.db
title:
if-x-1-be-the-size-of-the-magnified-image-and-x-2-the-size-of-the-diminished-image-in-lens-displacement-method-then-the-size-of-the-object-is

slug:
if-x-1-be-the-size-of-the-magnified-image-and-x-2-the-size-of-the-diminished-image-in-lens-displacement-method-then-the-size-of-the-object-is-22196

question:
<div class="question">If $x_1$ be the size of the magnified image and $x_2$ the size of the diminished image in Lens Displacement Method, then the size of the object is :</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$\sqrt{x_1 x_2}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$x_1 x_2$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$x_1^2 x_2$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$x_1 x_2^2$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\sqrt{x_1 x_2}$</span> </div>


title: if-x-2-y-2-1-then
slug: if-x-2-y-2-1-then-78952
question: <div class="question">If $x^2+y^2=1$ then</div>
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2009']
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$y y^{\prime \prime}-\left(2 y^{\prime}\right)^2+1=0$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$y y^{\prime \prime}+\left(y^{\prime}\right)^2+1=0$</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">$y y^{\prime \prime}-\left(y^{\prime}\right)^2-1=0$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$y y^{\prime \prime}+\left(2 y^{\prime}\right)^2+1=0$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$y y^{\prime \prime}+\left(y^{\prime}\right)^2+1=0$</span> </div>

answer:
<div class="solution">Hints: $2 x+2 y y^{\prime}=0$<br/>$$<br/>\begin{aligned}<br/>&amp; x+y y^{\prime}=0 \\<br/>&amp; 1+y y^{\prime \prime}+\left(y^{\prime}\right)^2=0<br/>\end{aligned}<br/>$$</div>


title: if-x-2-y-2-a-2-then-0-a-1-d-x-d-y-2-d-x
slug: if-x-2-y-2-a-2-then-0-a-1-d-x-d-y-2-d-x-72996
tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x^{2}+y^{2}=a^{2}$, then $\int_{0}^{a} \sqrt{1+\left(\frac{d y}{d x}\right)^{2}} d x=$</div>

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

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

answer:
<div class="solution">Hint:
<br/>$y_{1}=-\frac{x}{y}$
<br/>$\int_{0}^{a} \sqrt{1+\frac{x^{2}}{y^{2}}} d x=a \int_{0}^{a} \frac{1}{y} \cdot d x=a \int_{0}^{a} \frac{d x}{\sqrt{a^{2}-x^{2}}}$
<br/>$=a\left[\sin ^{-1}\left(\frac{x}{a}\right)\right]_{0}^{]}=a \pi / 2$</div>


title: if-x-a-t-b-t-2-where-x-is-in-metre-m-and-t-is-in-hour-h-then-unit-of-b-will-be
slug: if-x-a-t-b-t-2-where-x-is-in-metre-m-and-t-is-in-hour-h-then-unit-of-b-will-be-86780
tags: ['Physics', 'Units and Dimensions', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x=a t+b t^{2},$ where $x$ is in metre $(\mathrm{m})$ and $t$ is in hour $(\mathrm{h})$, then unit of $b$ will be</div>

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

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

answer:
<div class="solution">Given, $x=a t+b t^{2}$
<br/>$\begin{array}{l}
<br/>{[x]=\left[b t^{2}\right]} \\
<br/>\text { Unit of } b=\frac{x}{t^{2}}=\frac{\text { metre }}{(h)^{2}}=\frac{m}{h^{2}}
<br/>\end{array}$</div>


title: if-x-and-y-are-digits-such-that-17-3556-x-y-428096000-then-x-y-equals
slug: if-x-and-y-are-digits-such-that-17-3556-x-y-428096000-then-x-y-equals-38304
tags: ['Mathematics', 'Permutation Combination', 'WBJEE', 'WBJEE 2015']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">15</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x$ and $y$ are digits such that $17 !=3556xy428096000,$ then $x+y$ equals</div>

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

answer:
<div class="solution">Given, $171=3556 \times 4428096000$
<br/>Since, 17! is divisible by $9,$ so sum of the digits $(48+x+y)$ must be divisible by 9
<br/>So, $x+y$ can be 15 or 6 . Also, 17! is divisible by $11,$ so $|10+x-y \mid$ must be multiple of 11 or $0 .$ The only possibility is
<br/>$$
<br/>\begin{array}{l}
<br/>|x-y|=1 \\
<br/>\therefore \quad x+y=15
<br/>\end{array}
<br/>$$</div>


title: if-x-d-x-d-y-y-f-x-y-x-f-x-y-then-f-x-y-is-equal-to
slug: if-x-d-x-d-y-y-f-x-y-x-f-x-y-then-f-x-y-is-equal-to-99686
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2021']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x \frac{d y}{d x}+y=\frac{x f(x y)}{f^{\prime}(x y)}$, then $|f(x y)|$ is equal to</div>

options:
<ul class="options"> <li class="correct"> <span class="option-label">A</span> <span class="option-data">$k e^{x^{2} / 2}$</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">$\mathrm{ke}^{\mathrm{y}^{2} / 2}$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\mathrm{ke}^{\mathrm{x}^{2}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{ke}^{\mathrm{y}^{2}}$</span> </li> </ul>

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

answer:
<div class="solution">$x \frac{d y}{d x}+y=\frac{x f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{x d y+y d x}{d x}=\frac{x f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{f^{\prime}(x y) d(x y)}{f(x y)}=x d x \Rightarrow \ln f(x y)=\frac{x^{2}}{2}+C \Rightarrow|f(x y)|=k e^{x^{2} / 2}$</div>


title: if-x-d-x-d-y-y-x-f-x-y-f-x-y-then-f-x-y-is-equal-to
slug: if-x-d-x-d-y-y-x-f-x-y-f-x-y-then-f-x-y-is-equal-to-69619
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x \frac{d y}{d x}+y=x \frac{f(x y)}{f^{\prime}(x y)}$, then $|f(x y)|$ is equal to</div>

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

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

answer:
<div class="solution">$x \frac{d y}{d x}+y=x \frac{f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{d(x y)}{d x}=x \frac{f(x y)}{f^{\prime}(x y)} \Rightarrow \frac{f^{\prime}(x y)}{f(x y)} d(x y)=x d x \Rightarrow \ln |f(x y)|=\frac{x^2}{2}+k \Rightarrow f(x y)=C e^{\frac{x^2}{2}}$</div>


tags: ['Mathematics', 'Definite Integration', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
title:
if-x-denotes-the-greatest-integer-less-than-or-equal-to-x-then-the-value-of-the-integral-0-2-x-2-x-d-x-equals

slug:
if-x-denotes-the-greatest-integer-less-than-or-equal-to-x-then-the-value-of-the-integral-0-2-x-2-x-d-x-equals-60308

question:
<div class="question">If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\int_{0}^{2} x^{2}[x] d x$ equals</div>

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

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{7}{3}$</span> </div>

answer:
<div class="solution">Let.
<br/>$$
<br/>\begin{aligned}
<br/>I &amp;=\int_{0}^{2} x^{2}[x] d x \\
<br/>&amp;=\int_{0}^{1} x^{2} \cdot 0 d x+\int_{1}^{2} x^{2} \times 1 d x \\
<br/>&amp;=\left[\frac{x^{3}}{3}\right]_{1}^{2}=\left[\frac{8}{3}-\frac{1}{3}\right]=\frac{7}{3}
<br/>\end{aligned}
<br/>$$</div>


title: if-x-is-a-positive-real-number-different-from-1-such-that-lo-g-a-x-lo-g-b-x-lo-g-c-x-are-in-ap-then
tags: ['Mathematics', 'Sequences and Series', 'WBJEE', 'WBJEE 2016']
source_db: db\MarksBatch2_P1.db
slug:
if-x-is-a-positive-real-number-different-from-1-such-that-lo-g-a-x-lo-g-b-x-lo-g-c-x-are-in-ap-then-59929

question:
<div class="question">If $x$ is a positive real number different from 1 such that $\log _{a} x, \log _{b} x, \log _{c} x$ are in $\mathrm{AP}$, then</div>

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

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

answer:
<div class="solution">Given, $\log _{a} x, \log _{b} x$ and $log_{c}$ $x$ are in $A P$.
<br/>$\therefore \quad 2 \log _{b} x=\log _{a} x+\log _{c} x$
<br/>$\begin{aligned} 2 \log _{b} x &amp;=\frac{1}{\log _{x} a}+\frac{1}{\log _{x} c} \\ \frac{2}{\log _{x} b} &amp;=\frac{\log _{x} a c}{\log _{x} a \log _{x} c} \\ 2 \log _{x} c &amp;=\frac{\log _{x} b}{\log _{x} a}\left(\log _{x} a c\right) \\ 2 \log _{x} c &amp;=\log _{a} b \cdot \log _{x} a c \\ \log _{x} c^{2} &amp;=\log _{x}(a c)^{\log _{a} b} \\ c^{2} &amp;=(a c)^{\log _{a} b} \end{aligned}$</div>


title: if-x-is-a-random-variable-such-that-x-26-then-1-4-x-is-equal-to
slug: if-x-is-a-random-variable-such-that-x-26-then-1-4-x-is-equal-to-51314
tags: ['Mathematics', 'Statistics', 'WBJEE', 'WBJEE 2019']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">10.4</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $X$ is a random variable such that $\sigma(X)=2.6$ then $\sigma(1-4 x)$ is equal to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">7.8</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">$-10.4$</span> </li><li class=""> <span class="option-label">C</span> <span class="option-data">13</span> </li><li class="correct"> <span class="option-label">D</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> </ul>

answer:
<div class="solution">Given that,
<br/>$$
<br/>\begin{aligned}
<br/>\sigma(X) &amp;=2.6=\sqrt{\operatorname{var}(x)} \\
<br/>\sigma(1-4 x) &amp;=\sqrt{\operatorname{var}(1-4 x)} \\
<br/>&amp;=\sqrt{16 \operatorname{var}(x)} \\
<br/>&amp;=4 \times \sqrt{\operatorname{var}(x)}=4 \times 2.6=10.4
<br/>\end{aligned}
<br/>$$</div>


title: if-x-lo-g-10-1-2-x-x-lo-g-10-5-lo-g-10-6-then-the-value-of-x-is
slug: if-x-lo-g-10-1-2-x-x-lo-g-10-5-lo-g-10-6-then-the-value-of-x-is-88198
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2018']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x+\log _{10}\left(1+2^{x}\right)=x \log _{10} 5+\log _{10} 6,$ then
<br/>the value of $x$ is</div>

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

answer:
<div class="solution">We have,
<br/>$\begin{aligned} x+\log _{10}\left(1+2^{x}\right) &amp;=x \log _{10} 5+\log _{10} 6 \\ \Rightarrow \log _{10}\left(1+2^{x}\right) &amp;=x \log _{10} 5+\log _{10} 6-x \\ &amp;=\log _{10} 5^{x}+\log _{10} 6-x \log _{10} 10 \\ &amp;=\log _{10}\left(5^{x} \cdot 6\right)-\log _{10} 10^{x} \\ \Rightarrow \quad &amp; \log _{10}\left(1+2^{x}\right)=\log _{10}\left(\frac{5^{x} \cdot 6}{10^{x}}\right) \end{aligned}$
<br/>$\Rightarrow \quad 1+2^{x}=\frac{5^{x} \cdot 6}{10^{x}}=\frac{6}{2^{x}}$
<br/>$\Rightarrow \quad 2^{x}\left(1+2^{x}\right)=6$
<br/>$\Rightarrow \quad t(1+t)=6$$\quad \quad \quad\left(\operatorname{let} 2^{x}=t\right)$
<br/>$\Rightarrow \quad t^{2}+t-6=0$
<br/>$\Rightarrow \quad(t+3)(t-2)=0$
<br/>$t+3=0$
<br/>or $t-2=0$
<br/>$\begin{array}{l}
<br/>\Rightarrow \quad t=2 \\
<br/>2^{x}=2 \Rightarrow x=1
<br/>\end{array}$</div>


title: if-x-lo-g-a-b-c-y-lo-g-b-c-a-z-lo-g-c-ab-then-the-value-of-1-x-1-1-y-1-1-z-1-will-be
slug: if-x-lo-g-a-b-c-y-lo-g-b-c-a-z-lo-g-c-ab-then-the-value-of-1-x-1-1-y-1-1-z-1-will-be-92347
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2009']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x=\log _a b c, y=\log _b c a, z=\log _c a b$, then the value of $\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}$ will be</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$x+y+z$</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">$a b+b c+c a$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\mathrm{abc}$</span> </li> </ul>

answer:
<div class="solution">Hints: $1+x=\log _a a+\log _a b c=\log _a a b c$<br/>$\begin{aligned}<br/>&amp; \frac{1}{1+x}=\log _{a b c} a, \text { Similarly } \frac{1}{1+y}=\log _{a b c} b \\<br/>&amp; \frac{1}{1+z}=\log _{a b c} c \text {, Ans. }=\log _{(a b c)} a b c=1<br/>\end{aligned}$</div>


title: if-x-satisfies-the-inequality-lo-g-25-x-2-lo-g-5-x-2-2-then-x-belongs-to
slug: if-x-satisfies-the-inequality-lo-g-25-x-2-lo-g-5-x-2-2-then-x-belongs-to-91289
tags: ['Mathematics', 'Basic of Mathematics', 'WBJEE', 'WBJEE 2022']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x$ satisfies the inequality $\log _{25} x^2+\left(\log _5 x\right)^2  &lt; 2$, then $x$ belongs to</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\left(\frac{1}{5}, 5\right)$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\left(\frac{1}{25}, 5\right)$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\left(\frac{1}{5}, 25\right)$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\left(\frac{1}{25}, 25\right)$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\left(\frac{1}{25}, 5\right)$</span> </div>

answer:
<div class="solution">Domain $=(0, \infty)$<br/>$\begin{aligned}<br/>&amp;\log _5 x+\left(\log _5 x\right)^2-2  &lt; 0 \\<br/>&amp;\Rightarrow\left(\log _5 x-1\right)\left(\log _5 x+2\right) &lt; 0 \\<br/>&amp;\therefore-2 &lt; \log _5 x &lt; 1 \\<br/>&amp;\therefore \frac{1}{25} &lt; x &lt; 5<br/>\end{aligned}$</div>


title: if-x-sin-x-y-d-y-y-sin-x-y-x-d-x-x-0-and-y-1-2-then-the-value-of-cos-x-y-is
slug: if-x-sin-x-y-d-y-y-sin-x-y-x-d-x-x-0-and-y-1-2-then-the-value-of-cos-x-y-is-25647
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2020']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $x \sin \left(\frac{y}{x}\right) d y=\left[y \sin \left(\frac{y}{x}\right)-x\right] d x, x&gt;0$ and $y(1)=\frac{\pi}{2}$ then the value of $\cos \left(\frac{y}{x}\right)$ is</div>

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

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

answer:
<div class="solution">Hint: 
<br/>$\operatorname{sinv}\left(v+x \cdot \frac{d v}{d x}\right)=v \sin v-1$
<br/>$\Rightarrow \mathrm{x} \sin \mathrm{v} \cdot \frac{\mathrm{dv}}{\mathrm{dx}}=-1$
<br/>$\Rightarrow \int \sin v \mathrm{~d} v=-\int \frac{\mathrm{dx}}{\mathrm{x}}$
<br/>$\Rightarrow-\cos v=-\log x+\mathrm{c}$
<br/>at $x=1 ; \mathrm{y}=\frac{\pi}{2} ; \mathrm{c}=0$
<br/>$\cos \left(\frac{\mathrm{y}}{\mathrm{x}}\right)=\log \mathrm{x}$</div>


title: if-x-x-2-x-1-x-x-1-2-x-2-x-1-2-x-3-x-1-3-x-2-3-then-coefficient-of-x-in-x-is
slug: if-x-x-2-x-1-x-x-1-2-x-2-x-1-2-x-3-x-1-3-x-2-3-then-coefficient-of-x-in-x-is-98338
tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2022']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">-2</span> </div>
answer: <div class="solution">co-eff of $x$ in $\Delta(x)=\frac{\Delta^{\prime}(0)}{1 !}=-2$</div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $\Delta(x)=\left|\begin{array}{ccc}x-2 &amp; (x-1)^2 &amp; x^3 \\ x-1 &amp; x^2 &amp; (x+1)^3 \\ x &amp; (x+1)^2 &amp; (x+2)^3\end{array}\right|$, then coefficient of $x$ in $\Delta(x)$ is</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">-2</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">3</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-4</span> </li> </ul>


tags: ['Mathematics', 'Determinants', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-x-y-and-z-are-greater-than-1-then-the-value-of-1-lo-g-y-x-lo-g-z-x-lo-g-x-y-1-lo-g-z-y-lo-g-x-z-lo-g-y-z-1-is

slug:
if-x-y-and-z-are-greater-than-1-then-the-value-of-1-lo-g-y-x-lo-g-z-x-lo-g-x-y-1-lo-g-z-y-lo-g-x-z-lo-g-y-z-1-is-72454

question:
<div class="question">If $x, y$ and $z$ are greater than 1 , then the value of $\left|\begin{array}{ccc}1 &amp; \log _{x} y &amp; \log _{x} z \\ \log _{y} x &amp; 1 &amp; \log _{y} z \\ \log _{z} x &amp; \log _{z} y &amp; 1\end{array}\right|$ is</div>

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

answer:
<div class="solution">Let $\Delta=\left|\begin{array}{ccc}1 &amp; \log _{x}y &amp; \log _{x} z \\ \log _{y} x &amp; 1 &amp; \log _{y} z \\ \log _{z} x &amp; \log _{z} y &amp; 1\end{array}\right|$
<br/>$=\left|\begin{array}{lll}\frac{\log x}{\log x} &amp; \frac{\log y}{\log x} &amp; \frac{\log z}{\log x} \\ \frac{\log x}{\log y} &amp; \frac{\log y}{\log y} &amp; \frac{\log z}{\log y} \\ \frac{\log x}{\log z} &amp; \frac{\log y}{\log z} &amp; \frac{\log 2}{\log z}\end{array}\right|$
<br/>On taking $\frac{1}{\log x} \frac{1}{\log y} \frac{1}{\log z}$ common from $R_{1}, R_{2}$ and $R_{3}$
<br/>we got
<br/>$$
<br/>=\frac{1}{\log x \cdot \log y \cdot \log z}\left|\begin{array}{lll}
<br/>\log x &amp; \log y &amp; \log z \\
<br/>\log x &amp; \log y &amp; \log z \\
<br/>\log x &amp; \log y &amp; \log z
<br/>\end{array}\right|
<br/>$$
<br/>$\therefore \Delta=0$</div>


title: if-x2y24-then-y-fracd-yd-xx
slug: if-x2y24-then-y-fracd-yd-xx-30423
question: <div class="question">If \(x^2+y^2=4\), then \(y \frac{d y}{d x}+x=\)</div>
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2011']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
answer: <div class="solution">Hints: \(x+y \frac{d y}{d x}=0\)</div>
source_db: db\MarksBatch2_P1.db
options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">4</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">1</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">-1</span> </li> </ul>


title: if-xfrac1x2-cos-theta-then-for-any-integer-n-xnfrac1xn
slug: if-xfrac1x2-cos-theta-then-for-any-integer-n-xnfrac1xn-63590
tags: ['Mathematics', 'Complex Number', 'WBJEE', 'WBJEE 2011']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(x+\frac{1}{x}=2 \cos \theta\), then for any integer \(n, x^n+\frac{1}{x^n}=\)</div>

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

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

answer:
<div class="solution">Hints: \(x+\frac{1}{x}=2 \cos \theta\)<br/>Let \(x=\cos \theta+1 \sin \theta\)<br/>\(\frac{1}{x}=\cos \theta-1 \sin \theta\)<br/>Thus \(x^n+\frac{1}{x^n}=2 \cos n \theta\)</div>


title: if-xsin-theta-and-ysin-k-theta-then-left1x2right-y2x-y1alpha-y0-for-alpha
slug: if-xsin-theta-and-ysin-k-theta-then-left1x2right-y2x-y1alpha-y0-for-alpha-34128
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2023']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If \(x=\sin \theta\) and \(y=\sin k \theta\), then \(\left(1-x^2\right) y_2-x y_1-\alpha y=0\), for \(\alpha=\)</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">\(\mathrm{k}\)</span> </li><li class=""> <span class="option-label">B</span> <span class="option-data">\(-\mathrm{k}\)</span> </li><li class="correct"> <span class="option-label">C</span> <span class="option-data">\(-\mathrm{k}^2\)</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">\(\mathrm{k}^2\)</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">\(-\mathrm{k}^2\)</span> </div>

answer:
<div class="solution">Hint : \(x=\sin \theta\)<br/>\(y=\sin k \theta\)<br/>\(\frac{d y}{d x}=\frac{d y / d \theta}{d x / d \theta}=\frac{k \cos k \theta}{\cos \theta}=y_1 \ldots\ldots (I)\)<br/>\(\begin{aligned}<br/>&amp; y_2=\frac{d^2 y}{d x^2}=\frac{d}{d \theta}\left(\frac{d y}{d x}\right) \times \frac{d \theta}{d x}=\left[\frac{-k^2 \sin (k \theta) \cdot \cos \theta+k \cos (k \theta) \sin \theta}{\cos ^2 \theta}\right] \times \frac{1}{\cos \theta} \\<br/>&amp; y_2=\frac{k \cos (k \theta) \sin \theta-k^2 \sin (k \theta) \cos \theta}{\cos ^3 \theta} \ldots \ldots \text { (II) }<br/>\end{aligned}\)<br/>also, \(1-x^2=1-\sin ^2 \theta=\cos ^2 \theta \ldots \ldots \text { (III) }\)<br/>So, putting the values of \(1-x^2, y_1\) and \(y_2\) from (I), (II), (III) we get<br/>\(\begin{aligned}<br/>&amp; \left(1-x^2\right) y_2-x y_1=\alpha y \\<br/>&amp; \Rightarrow\left(\cos ^2 \theta\right)\left[\frac{k \cos (k \theta) \sin \theta-k^2 \sin (k \theta) \cdot \cos \theta}{\cos ^3 \theta}\right]-\frac{k \sin \theta \cos (k \theta)}{\cos \theta}=\alpha(\sin k \theta) \\<br/>&amp; \Rightarrow \frac{k \cos (k \theta) \sin \theta-k^2 \sin (k \theta) \cos \theta-k \sin \theta \cos (k \theta)}{\cos \theta}=\alpha \sin (k \theta) \\<br/>&amp; \Rightarrow-k^2 \sin (k \theta)=\alpha \sin (k \theta)<br/>\end{aligned}\)<br/>So, \(\alpha=-k^2\)</div>


title: if-y-1-x-1-x-2-1-x-4-1-x-2-n-then-the-value-of-d-x-d-y-at-x-0-is
slug: if-y-1-x-1-x-2-1-x-4-1-x-2-n-then-the-value-of-d-x-d-y-at-x-0-is-82232
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2016']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $y=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) \ldots\left(1+x^{2 n}\right),$ then the value of $\left(\frac{d y}{d x}\right)$ at $x=0$ 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">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Given,
<br/>$y=(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right)+\ldots+\left(1+x^{2 n}\right)$
<br/>$\Rightarrow \log y=\log (1+x)+\log \left(1+x^{2}\right)$ $+\log \left(1+x^{4}\right)+\ldots+\log \left(1+x^{2n}\right)$
<br/>$\Rightarrow \frac{1}{y} \frac{d y}{d x}=\frac{1}{1+x}+\frac{2 x}{1+x^{2}}+\frac{3 x^{2}}{1+x^{3}}+\ldots+\frac{2 n x^{2 n-1}}{1+x^{2 n}}$
<br/>$\Rightarrow \frac{d y}{d x}=y\left[\frac{1}{1+x}+\frac{2 x}{1+x^{2}}+\frac{3 x^{2}}{1+x^{3}}+\ldots+\frac{2 n x^{2 n-1}}{1+x^{2 n}}\right]$
<br/>$\therefore \quad\left(\frac{d y}{d x}\right)_{x=0}=y\left(\frac{1}{1+0}\right)=1$</div>


title: if-y-1-x-1-x-2-1-x-4-1-x-2-n-then-the-value-of-d-x-d-y-x-0-is
slug: if-y-1-x-1-x-2-1-x-4-1-x-2-n-then-the-value-of-d-x-d-y-x-0-is-13580
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">1</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $y=(1+x)\left(1+x^2\right)\left(1+x^4\right) \ldots\left(1+x^{2 n}\right)$ then the value of $\left(\frac{d y}{d x}\right)_{x=0}$ 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">1</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">D</span> <span class="option-data">2</span> </li> </ul>

answer:
<div class="solution">Hints: T-log \&amp; Differentiate<br/>$$<br/>\begin{aligned}<br/>&amp; \frac{d y}{d x}=y\left[\frac{1}{1+x}+\frac{2 x}{1+x^2}+\ldots\right] \text { Put } x=0 \\<br/>&amp; \frac{d y}{d x}=1<br/>\end{aligned}<br/>$$</div>


title: if-y-3-y-2-y-0-where-y-0-1-y-0-0-then-the-value-of-y-at-x-lo-g-2-is
slug: if-y-3-y-2-y-0-where-y-0-1-y-0-0-then-the-value-of-y-at-x-lo-g-2-is-75891
tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2010']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">0</span> </div>
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $y^{\prime \prime}-3 y^{\prime}+2 y=0$ where $y(0)=1, y^{\prime}(0)=0$, then the value of $y$ at $x=\log , 2$ 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">-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">0</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li> </ul>

answer:
<div class="solution">$$<br/>\begin{aligned}<br/>&amp; \text { Hints: } \frac{d^2 y}{d x^2}-3 \frac{d y}{d x}+2 y=0 \\<br/>&amp; m^2-3 m+2=0, \quad y=A e^x+B e^{2 x} \\<br/>&amp; m=1, m=2, y^1=A e^x+2 B e^{2 x} \\<br/>&amp; y=0, A+B=1 \quad A+2 B=0, A=2, B=-1 \\<br/>&amp; y=2 e^x-e^{2 x} \\<br/>&amp; y=0 \quad \text { at } x=\ell n 2<br/>\end{aligned}<br/>$$</div>


tags: ['Mathematics', 'Parabola', 'WBJEE', 'WBJEE 2014']
source_db: db\MarksBatch2_P1.db
title:
if-y-4-x-3-is-parallel-to-a-tangent-to-the-parabola-y-2-12-x-then-its-distance-from-the-normal-parallel-to-the-given-line-is

slug:
if-y-4-x-3-is-parallel-to-a-tangent-to-the-parabola-y-2-12-x-then-its-distance-from-the-normal-parallel-to-the-given-line-is-72233

question:
<div class="question">If $y=4 x+3$ is parallel to a tangent to the parabola $y^{2}=12 x$, then its distance from the normal parallel to the given line is</div>

options:
<ul class="options"> <li class=""> <span class="option-label">A</span> <span class="option-data">$\frac{213}{\sqrt{17}}$</span> </li><li class="correct"> <span class="option-label">B</span> <span class="option-data">$\frac{219}{\sqrt{17}}$</span> <svg fill="none" height="24" viewbox="0 0 24 24" width="24" xmlns="http://www.w3.org/2000/svg"> <path d="M12 2.25C10.0716 2.25 8.18657 2.82183 6.58319 3.89317C4.97982 4.96452 3.73013 6.48726 2.99218 8.26884C2.25422 10.0504 2.06114 12.0108 2.43735 13.9021C2.81355 15.7934 3.74215 17.5307 5.10571 18.8943C6.46928 20.2579 8.20656 21.1865 10.0979 21.5627C11.9892 21.9389 13.9496 21.7458 15.7312 21.0078C17.5127 20.2699 19.0355 19.0202 20.1068 17.4168C21.1782 15.8134 21.75 13.9284 21.75 12C21.7473 9.41498 20.7192 6.93661 18.8913 5.10872C17.0634 3.28084 14.585 2.25273 12 2.25ZM16.2806 10.2806L11.0306 15.5306C10.961 15.6004 10.8783 15.6557 10.7872 15.6934C10.6962 15.7312 10.5986 15.7506 10.5 15.7506C10.4014 15.7506 10.3038 15.7312 10.2128 15.6934C10.1218 15.6557 10.039 15.6004 9.96938 15.5306L7.71938 13.2806C7.57865 13.1399 7.49959 12.949 7.49959 12.75C7.49959 12.551 7.57865 12.3601 7.71938 12.2194C7.86011 12.0786 8.05098 11.9996 8.25 11.9996C8.44903 11.9996 8.6399 12.0786 8.78063 12.2194L10.5 13.9397L15.2194 9.21937C15.2891 9.14969 15.3718 9.09442 15.4628 9.0567C15.5539 9.01899 15.6515 8.99958 15.75 8.99958C15.8486 8.99958 15.9461 9.01899 16.0372 9.0567C16.1282 9.09442 16.2109 9.14969 16.2806 9.21937C16.3503 9.28906 16.4056 9.37178 16.4433 9.46283C16.481 9.55387 16.5004 9.65145 16.5004 9.75C16.5004 9.84855 16.481 9.94613 16.4433 10.0372C16.4056 10.1282 16.3503 10.2109 16.2806 10.2806Z" fill="#24A865"></path> </svg> </li><li class=""> <span class="option-label">C</span> <span class="option-data">$\frac{211}{\sqrt{17}}$</span> </li><li class=""> <span class="option-label">D</span> <span class="option-data">$\frac{210}{\sqrt{17}}$</span> </li> </ul>

correct_option:
<div class="correct-answer">
The correct answer is:
<span class="option-value">$\frac{219}{\sqrt{17}}$</span> </div>

answer:
<div class="solution">Given equation of parabola is
<br/>$$
<br/>y^{2}=12 x
<br/>$$
<br/>On differentiating both sides w.r.t. $x,$ we get
<br/>$$
<br/>2 y \frac{d y}{d x}=12 \Rightarrow \frac{d y}{d x}=\frac{6}{y}
<br/>$$
<br/>Since, the normal to the curve is parallel to the line $y=4 x+3$
<br/>$\therefore$ Slope of normal curve $=$ Slope of line
<br/>$\Rightarrow \quad-\frac{Y}{6}=4$
<br/>$\Rightarrow \quad y=-24$
<br/>From Eq. (i), we get $(-24)^{2}=12 x$
<br/>$\Rightarrow \quad 24 \times 24=12 x$
<br/>$\Rightarrow \quad x=48$
<br/>$\therefore$ Normal point on a curve is (48,-24)
<br/>$\therefore$ Distance from (48,-24) to the line $4 x-y+3=0$ is
<br/>$\frac{4 \times 48+24+3}{\sqrt{4^{2}+1^{2}}}=\frac{219}{\sqrt{17}}$</div>


tags: ['Mathematics', 'Differential Equations', 'WBJEE', 'WBJEE 2014']
correct_option: <div class="correct-answer"> The correct answer is: <span class="option-value">2</span> </div>
source_db: db\MarksBatch2_P1.db
title:
if-y-cos-1-x-then-it-satisfies-the-differential-equation-1-x-2-d-x-2-d-2-y-x-d-x-d-y-c-where-c-is-equal-to

slug:
if-y-cos-1-x-then-it-satisfies-the-differential-equation-1-x-2-d-x-2-d-2-y-x-d-x-d-y-c-where-c-is-equal-to-49748

question:
<div class="question">If $\sqrt{y}=\cos ^{-1} x$, then it satisfies the differential equation $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=c,$ where $c$ is equal to</div>

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

answer:
<div class="solution">Given, $\sqrt{y}=\cos ^{-1} x \Rightarrow y=\left(\cos ^{-1} x\right)^{2}$
<br/>On differentiating both sides w.r.t. $x$, we get
<br/>$$
<br/>\frac{d y}{d x}=2\left(\cos ^{-1} x\right) \times \frac{-1}{\sqrt{1-x^{2}}}
<br/>$$
<br/>Again, differentiating both sides w.r.t. $x$, we get
<br/>$\frac{d^{2} y}{d x^{2}}=-2\left[\begin{array}{c}\sqrt{1-x^{2}} \times \frac{-1}{\sqrt{1-x^{2}}} \\ -\cos ^{-1} x \times\left(\frac{1}{2}\right) \frac{(-2 x)}{\left(1-x^{2}\right)^{1 / 2}} \\ \hline\left(\sqrt{1-x^{2}}\right)^{2}\end{array} \right ].$
<br/>$=-2\left[\frac{-1+\frac{x \cos ^{-1} x}{\left(1-x^{2}\right)^{1 / 2}}}{\left(1-x^{2}\right)}\right]$
<br/>$\frac{d^{2} y}{d x^{2}}=\left[\frac{2-\frac{2 x \cos ^{-1} x}{\left(1-x^{2}\right)^{1 / 2}}}{\left(1-x^{2}\right)}\right]$
<br/>$\Rightarrow \quad\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}=2+x \frac{d y}{d x}$
<br/>$\Rightarrow \quad\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=2$
<br/>But, it is given
<br/>$\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=c$
<br/>$\therefore c=2$</div>


title: if-y-e-m-s-i-n-1-x-then-1-x-2-d-x-2-d-2-y-x-d-x-d-y-k-y-0-where-k-is-equal-to
slug: if-y-e-m-s-i-n-1-x-then-1-x-2-d-x-2-d-2-y-x-d-x-d-y-k-y-0-where-k-is-equal-to-36731
tags: ['Mathematics', 'Differentiation', 'WBJEE', 'WBJEE 2017']
source_db: db\MarksBatch2_P1.db
question:
<div class="question">If $y=e^{m \sin ^{-1} x}$ then $\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-k y=0,$ where $k$ is
<br/>equal to</div>

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

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

answer:
<div class="solution">$y=e^{msin^{-1}x}$ $\Rightarrow \quad \frac{d y}{d x}=e^{msin^{-1}x} \cdot \frac{m}{\sqrt{1-x^{2}}}$
<br/>$\Rightarrow \sqrt{1-x^{2}} \frac{d y}{d x}=m e^{msin^{-1}x}$
<br/>$\Rightarrow \sqrt{1-x^{2}} \frac{d y}{d x}=m y$
<br/>$\Rightarrow \sqrt{1-x^{2}} \frac{d^{2} y}{d x^{2}}+\frac{1}{2 \sqrt{1-x^{2}}}(-2 x) \frac{d y}{d x}=m \frac{d y}{d x}$
<br/>$\Rightarrow\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=m \sqrt{1-x^{2}} \frac{d y}{d x}$
<br/>$\Rightarrow\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}=m \cdot(m y)$
<br/>$\Rightarrow\left(1-x^{2}\right) \frac{d^{2} y}{d x^{2}}-x \frac{d y}{d x}-m^{2} y=0$
<br/>$\therefore$
<br/>$k=m^{2}$</div>


