In terms of potential difference V, electric current I, permittivity $${\varepsilon _0}$$, permeability $${\mu _0}$$ and speed of light c, the dimensionally correct equation(s) is(are) :

Four harmonic waves of equal frequencies and equal intensities I₀ have phase angles 0, $${\pi \over 3},{{2\pi } \over 3}$$ and $$\pi$$. When they are superposed, the intensity of the resulting wave is nI₀. The value of n is

For a radioactive material, its activity A and rate of change of its activity R are defined as $$A = - {{dN} \over {dt}}$$ and $$R = - {{dA} \over {dt}}$$, where N(t) is the number of nuclei at time t. Two radioactive source P(mean life $$\tau $$) and Q (mean life 2$$\tau $$) have the same activity at t = 0. Their rate of change of activities at t = 2$$\tau $$ are R_P and R_Q, respectively. If $${{{R_P}} \over {{R_Q}}} = {n \over e}$$, then the value of n is

A monochromatic beam of light is incident at 60$$^\circ$$ on one face of an equilateral prism of refractive index n and emerges from the opposite face making an angle $$\theta$$(n) with the normal (see figure). For n = $$\sqrt 3 $$ the value of $$\theta$$ is 60$$^\circ$$ and $${{d\theta } \over {dn}} = m$$. The value of m is