Two radioactive substances A and B of mass numbers 200 and 212 respectively, shows spontaneous $\alpha$-decay with same $Q$ value of 1 MeV . The ratio of energies of $\alpha$-rays produced by A and B is $\_\_\_\_$ .

$$ \text { The output } Y \text { for the given inputs } A \text { and } B \text { to the circuit is : } $$

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)