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?
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 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?
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?