The figure shows an arbitrarily shaped planar conducting loop $A$ in the $X Y$ plane. Two nonintersecting regions with areas $a_1$ and $a_2$ within the loop are subjected to magnetic fields $\vec{B}_1=\frac{m}{\sqrt{2}} \sin (\omega t)(1 \hat{x}+0 \hat{y}+1 \hat{z})$, and $\vec{B}_2=-\frac{n}{\sqrt{2}} \cos \left(2 \omega t+\frac{\pi}{4}\right)(0 \hat{x}+1 \hat{y}+1 \hat{z})$, respectively.

What is the expression for the induced rms voltage in loop $A$ ?

A uniform spherical volume charge distribution of radius 2 m , centered at the origin, has a strength of $\frac{3}{\pi} \times 10^{-6} \mathrm{C} / \mathrm{m}^3$. A point charge of strength $\pi \times 8.854 \times 10^{-12} \mathrm{C}$ is moved from $(-3,0,-4)$ to $(0,0,4)$ in Cartesian coordinate system. The relative permittivity of the medium is 1 and the coordinate values are in meters. The work done during the process is $\_\_\_\_$ $\mu \mathrm{J}$. (Round off to two decimal places)

Two $n \times n$ matrices $A$ and $B$ have a common eigenvalue 2 , and the same corresponding nonzero eigenvector. Which of the following options is/are correct?

(Note: $I$ is the $n \times n$ identity matrix.)

Given that $\vec{F}(x, y, z)=\sin (y) \hat{x}+\cos (x) \hat{y}+5 \hat{z}$, the integral $\iint_S \vec{F}(x, y, z) \cdot \overrightarrow{d s}$ over the unit sphere $S$ centered at the origin evaluates to $\_\_\_\_$ . (Round off to one decimal place)