A parallel plate capacitor is having separation between plates 0.885 mm . It has a capacitance of $1 \mu \mathrm{~F}$ when the space between the plates is filled with an insulating material of resistivity $1 \times 10^{13} \Omega \mathrm{~m}$ and resistance $17.7 \times 10^{14} \Omega$. Relative permittivity of the insulating material is $\alpha \times 10^7$. The value of $\alpha$ is $\_\_\_\_$ .

(Take permittivity of free space $=8.85 \times 10^{-12} \mathrm{~F} / \mathrm{m}$ )

Some distant star is to be observed by some telescope of diameter of objective lens $a$, at an angular resolution of $3.0 \times 10^{-7}$ radian. If the wavelength of light from the star reaching the telescope is 500 nm , the minimum diameter of the objective lens of the telescope is $\_\_\_\_$ cm. (nearest interger)

A 5 mg particle carrying a charge of $5 \pi \times 10^{-6} \mathrm{C}$ is moving with velocity of $(3 \hat{i}+2 \hat{k}) \times 10^{-2} \mathrm{~m} / \mathrm{s}$ in a region having magnetic field $\vec{B}=0.1 \hat{k} \mathrm{~Wb} / \mathrm{m}^2$. It moves a distance of $\alpha$ meter along $\hat{k}$ when it completes 5 revolutions. The value of $\alpha$ is $\_\_\_\_$。

The stored charge in the capacitor in steady state of the following circuit is $\_\_\_\_$ $\mu \mathrm{C}$.