Two radioactive materials $$X_1$$ and $$X_2$$ have decay constants '$$5 \lambda$$' and '$$\lambda$$' respectively. Initially, they have the same number of nuclei. After time '$$t$$', the ratio of number of nuclei of $$X_1$$ to that of $$\mathrm{X}_2$$ is $$\frac{1}{\mathrm{e}}$$. Then $$\mathrm{t}$$ is equal to

A nucleus breaks into two nuclear parts, which have their velocity ratio $$2: 1$$. The ratio of their nuclear radii will be

Ratio centripetal acceleration for an electron revolving in 3^rd and 5^th Bohr orbit of hydrogen atom is

When an electron in hydrogen atom jumps from third excited state to the ground state, the de-Broglie wavelength associated with the electron becomes