Highly energetic electrons are bombarded on a target of an element containing 30 neutrons. The ratio of radii of nucleus to that of Helium nucleus is $$(14)^{\frac{1}{3}}$$. Find

(A) Atomic number of the nucleus;

(B) the frequency of $$\mathrm{K}_{\alpha}$$ line of the X-ray produced.

$$\left(\mathrm{R}=1.1 \times 10^{7} \mathrm{~m}^{-1}\right.$$ and $$\left.c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)$$

A small body attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency $$\omega$$ and amplitude $$a$$. If at a height $$y^{\prime}$$ from the mean position, the body gets detached from the spring, calculate the value of $$y^{\prime}$$ so that the height $$\mathrm{H}$$ attained by the mass is maximum. The body does not interact with the spring during its subsequent motion after detachment $$\left(a \omega^{2}>g\right)$$

In the given circuit, the switch S is closed at time $$t=0$$. The charge Q on the capacitor at any instant t is given by $$Q(t)=Q_0(1-e^{-\alpha t})$$. Find the value of Q$$_0$$ and $$\alpha$$ in terms of given parameters shown in the circuit.

Two identical prisms of refractive index $$\sqrt{3}$$ are kept as shown in the figure. A light ray strikes the first prism at face AB. Find

(A) the angle of incidence so that the emergent ray from the first prism has minimum deviation;

(B) through what angle, the prism DCE should be rotated about C so that the final emergent ray also has minimum deviation?