In a radioactive sample, $${}_{19}^{40}K$$ nuclei either decay into stable $${}_{20}^{40}Ca$$ nuclei with decay constant 4.5 $$ \times $$ 10^-10 per year or into stable $${}_{18}^{40}Ar$$ nuclei with decay constant 0.5 $$ \times $$ 10^-10 per year. Given that in this sample all the stable $${}_{20}^{40}Ca$$ and $${}_{18}^{40}Ar$$ nuclei are produced by the $${}_{19}^{40}K$$ nuclei only. In time t $$ \times $$ 10⁹ years, if the ratio of the sum of stable $${}_{20}^{40}Ca$$ and $${}_{18}^{40}Ar$$ nuclei to the radioactive $${}_{19}^{40}K$$ nuclei is 99, the value of t will be

[Given : In 10 = 2.3]

A current carrying wire heats a metal rod. The wire provides a constant power (P) to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature (T) in the metal rod changes with time (t) as $$T(t) = {T_0}\left( {1 + \beta {t^{{1 \over 4}}}} \right)$$, where $$\beta $$ is a constant with appropriate dimension while T₀ is a constant with dimension of temperature. The heat capacity of the metal is

A thin spherical insulating shell of radius R carries a uniformly distributed charge such that the potential at its surface is V₀. A hole with a small area $$\alpha $$4$$\pi $$R²($$\alpha $$ << 1) is made on the shell without affecting the rest of the shell. Which one of the following statements is correct?

Consider a spherical gaseous cloud of mass density $$\rho $$(r) in free space where r is the radial distance from its center. The gaseous cloud is made of particles of equal mass m moving in circular orbits about the common center with the same kinetic energy K. The force acting on the particles is their mutual gravitational force. If $$\rho $$(r) is constant in time, the particle number density n(r) = $$\rho $$(r)/m is [G is universal gravitational constant]