1
JEE Advanced 2024 Paper 2 Online
Numerical
+3
-0
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Let $f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]$ be the function defined by $f(x)=\sin ^2 x$ and let $g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^2}$.

The value of $2 \int\limits_0^{\frac{\pi}{2}} f(x) g(x) d x-\int\limits_0^{\frac{\pi}{2}} g(x) d x$ is ____________.
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2
JEE Advanced 2024 Paper 2 Online
Numerical
+3
-0
Change Language

Let $f:\left[0, \frac{\pi}{2}\right] \rightarrow[0,1]$ be the function defined by $f(x)=\sin ^2 x$ and let $g:\left[0, \frac{\pi}{2}\right] \rightarrow[0, \infty)$ be the function defined by $g(x)=\sqrt{\frac{\pi x}{2}-x^2}$.

The value of $\frac{16}{\pi^3} \int\limits_0^{\frac{\pi}{2}} f(x) g(x) d x$ is ______.
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3
JEE Advanced 2024 Paper 2 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

A region in the form of an equilateral triangle (in $x-y$ plane) of height $L$ has a uniform magnetic field $\vec{B}$ pointing in the $+z$-direction. A conducting loop $\mathrm{PQR}$, in the form of an equilateral triangle of the same height $L$, is placed in the $x-y$ plane with its vertex $\mathrm{P}$ at $x=0$ in the orientation shown in the figure. At $t=0$, the loop starts entering the region of the magnetic field with a uniform velocity $\vec{v}$ along the $+x$-direction. The plane of the loop and its orientation remain unchanged throughout its motion.

JEE Advanced 2024 Paper 2 Online Physics - Electromagnetic Induction Question 1 English

Which of the following graph best depicts the variation of the induced emf $(E)$ in the loop as a function of the distance $(x)$ starting from $x=0$ ?

A
JEE Advanced 2024 Paper 2 Online Physics - Electromagnetic Induction Question 1 English Option 1
B
JEE Advanced 2024 Paper 2 Online Physics - Electromagnetic Induction Question 1 English Option 2
C
JEE Advanced 2024 Paper 2 Online Physics - Electromagnetic Induction Question 1 English Option 3
D
JEE Advanced 2024 Paper 2 Online Physics - Electromagnetic Induction Question 1 English Option 4
4
JEE Advanced 2024 Paper 2 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

A particle of mass $m$ is under the influence of the gravitational field of a body of mass $M(\gg m)$. The particle is moving in a circular orbit of radius $r_0$ with time period $T_0$ around the mass $M$. Then, the particle is subjected to an additional central force, corresponding to the potential energy $V_{\mathrm{c}}(r)=m \alpha / r^3$, where $\alpha$ is a positive constant of suitable dimensions and $r$ is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius $r_0$ in the combined gravitational potential due to $M$ and $V_{\mathrm{c}}(r)$, but with a new time period $T_1$, then $\left(T_1^2-T_0^2\right) / T_1^2$ is given by

[G is the gravitational constant.]

A
$\frac{3 \alpha}{G M r_0^2}$
B
$\frac{\alpha}{2 G M r_0^2}$
C
$\frac{\alpha}{G M r_0^2}$
D
$\frac{2 \alpha}{G M r_0^2}$
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