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

In the following circuit, the current through the resistor R(=2$$\Omega$$) is I amperes. The value of I is :

An electron in an excited state of Li²⁺ ion has angular momentum $${{3h} \over {2\pi }}$$. The de Broglie wavelength of the electron in this state is p$$\pi$$a₀ (where a₀ is the Bohr radius). The value of p is