1
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2

Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$, and the remaining one has charge $x$. The perpendicular from each charge to the nearest hexagon side passes through the center 0 of the hexagon and is bisected by the side.

Which of the following statement(s) is(are) correct in SI units?

A
When $x=q$, the magnitude of the electric field at 0 is zero.
B
When $x=-q$, the magnitude of the electric field at 0 is $\frac{q}{6 \pi \epsilon_{o} a^{2}}$.
C
When $x=2 q$, the potential at 0 is $\frac{7 q}{4 \sqrt{3} \pi \epsilon_{o} a}$.
D
When $x=-3 q$, the potential at 0 is $-\frac{3 q}{4 \sqrt{3} \pi \epsilon_{o} a}$.
2
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2

The binding energy of nucleons in a nucleus can be affected by the pairwise Coulomb repulsion. Assume that all nucleons are uniformly distributed inside the nucleus. Let the binding energy of a proton be $E_{b}^{p}$ and the binding energy of a neutron be $E_{b}^{n}$ in the nucleus.

Which of the following statement(s) is(are) correct?

A
$E_{b}^{p}-E_{b}^{n}$ is proportional to $Z(Z-1)$ where $Z$ is the atomic number of the nucleus.
B
$E_{b}^{p}-E_{b}^{n}$ is proportional to $A^{-\frac{1}{3}}$ where $A$ is the mass number of the nucleus.
C
$E_{b}^{p}-E_{b}^{n}$ is positive.
D
$E_{b}^{p}$ increases if the nucleus undergoes a beta decay emitting a positron.
3
JEE Advanced 2022 Paper 1 Online
+3
-1

A small circular loop of area $A$ and resistance $R$ is fixed on a horizontal $x y$-plane with the center of the loop always on the axis $\hat{n}$ of a long solenoid. The solenoid has $m$ turns per unit length and carries current $I$ counterclockwise as shown in the figure. The magnetic field due to the solenoid is in $\hat{n}$ direction. List-I gives time dependences of $\hat{n}$ in terms of a constant angular frequency $\omega$. List-II gives the torques experienced by the circular loop at time $t=\frac{\pi}{6 \omega}$. Let $\alpha=\frac{A^{2} \mu_{0}^{2} m^{2} I^{2} \omega}{2 R}$.

List-I List-II
(I) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\jmath}+\cos \omega t \hat{k})$ (P) 0
(II) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\imath}+\cos \omega t \hat{\jmath})$ (Q) $-\frac{\alpha}{4} \hat{\imath}$
(III) $\frac{1}{\sqrt{2}}(\sin \omega t \hat{\imath}+\cos \omega t \hat{k})$ (R) $\frac{3 \alpha}{4} \hat{\imath}$
(IV) $\frac{1}{\sqrt{2}}(\cos \omega t \hat{\jmath}+\sin \omega t \hat{k})$ (S) $\frac{\alpha}{4} \hat{\jmath}$
(T) $-\frac{3 \alpha}{4} \hat{\imath}$

Which one of the following options is correct?

A
I $\rightarrow$ Q, II $\rightarrow$ P, III $\rightarrow$ S, IV $\rightarrow$ T
B
$\mathrm{I} \rightarrow \mathrm{S}, \mathrm{II} \rightarrow \mathrm{T}$, III $\rightarrow \mathrm{Q}$, IV $\rightarrow \mathrm{P}$
C
$\mathrm{I} \rightarrow \mathrm{Q}, \mathrm{II} \rightarrow \mathrm{P}$, III $\rightarrow \mathrm{S}$, IV $\rightarrow \mathrm{R}$
D
$\mathrm{I} \rightarrow \mathrm{T}$, II $\rightarrow \mathrm{Q}$, III $\rightarrow \mathrm{P}$, IV $\rightarrow \mathrm{R}$
4
JEE Advanced 2022 Paper 1 Online
+3
-1

List I describes four systems, each with two particles $A$ and $B$ in relative motion as shown in figures. List II gives possible magnitudes of their relative velocities (in $m s^{-1}$ ) at time $t=\frac{\pi}{3} s$.

List-I List-II
(I) $A$ and $B$ are moving on a horizontal circle of radius $1 \mathrm{~m}$ with uniform angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$. The initial angular positions of $A$ and $B$ at time $t=0$ are $\theta=0$ and $\theta=\frac{\pi}{2}$, respectively.
(P) $\frac{\sqrt{3}+1}{2}$
(II) Projectiles $A$ and $B$ are fired (in the same vertical plane) at $t=0$ and $t=0.1 \mathrm{~s}$ respectively, with the same speed $v=\frac{5 \pi}{\sqrt{2}} \mathrm{~m} \mathrm{~s}^{-1}$ and at $45^{\circ}$ from the horizontal plane. The initial separation between $A$ and $B$ is large enough so that they do not collide. $\left(g=10 \mathrm{~ms}^{-2}\right)$.
(Q) $\frac{\sqrt{3}-1}{\sqrt{2}}$
(III) Two harmonic oscillators $A$ and $B$ moving in the $x$ direction according to $x_{A}=x_{0} \sin \frac{t}{t_{0}}$ and $x_{B}=x_{0} \sin \left(\frac{t}{t_{0}}+\frac{\pi}{2}\right)$ respectively, starting from $t=0$. Take $x_{0}=1 \mathrm{~m}, t_{0}=1 \mathrm{~s}$.
(R) $\sqrt{10}$
(IV) Particle $A$ is rotating in a horizontal circular path of radius $1 \mathrm{~m}$ on the $x y$ plane, with constant angular speed $\omega=1 \mathrm{rad} \mathrm{s}^{-1}$. Particle $B$ is moving up at a constant speed $3 \mathrm{~m} \mathrm{~s}^{-1}$ in the vertical direction as shown in the figure. (Ignore gravity.)
(S) $\sqrt{2}$
(T) $\sqrt{25\pi^{2}+1}$

Which one of the following options is correct?

A
I $\rightarrow$ R, II $\rightarrow$ T, III $\rightarrow$ P, IV $\rightarrow$ S
B
I $\rightarrow$ S, II $\rightarrow$ P, III $\rightarrow$ Q, IV $\rightarrow$ R
C
I $\rightarrow$ S, II $\rightarrow$ T, III $\rightarrow$ P, IV $\rightarrow$ R
D
I $\rightarrow$ T, II $\rightarrow$ P, III $\rightarrow$ R, IV $\rightarrow$ S
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