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

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

Consider a hydrogen atom with its electron in the nth orbital. An electromagnetic radiation of wavelength 90 nm is used to ionize the atom. If the kinetic energy of the ejected electron is 10.4 eV, then the value of n is (hc = 1242 eV nm)

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life T years as the fuel.

The amount of fuel at the beginning is such that the total power requirement of the village is 12.5 % of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of nT years, then the value of n is