A spherical body of radius R consists of a fluid of constant density and is in equilibrium under its own gravity. If P(r) is the pressure at r (r < R), then the correct option(s) is(are)

Consider a uniform spherical charge distribution of radius $${R_1}$$ centred at the origin $$O.$$ In this distribution, a spherical cavity of radius $${R_2},$$ centred at $$P$$ with distance $$OP=a$$ $$ = {R_1} - {R_2}$$ (see figure) is made. If the electric field inside the cavity at position $$\overrightarrow r $$ is $$\overrightarrow E \overrightarrow {\left( r \right)} ,$$ then the correct statement(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