Two radioactive substances A and B have decay constants 5 λ and λ respectively. At t = 0, they have the same number of nuclei. The ratio of number of nuclei of A to that of B will be (1/e)² after a time interval of

• $\frac{1}{\mathrm{\lambda }}$

• $\frac{1}{2\mathrm{\lambda }}$

• $\frac{1}{3\mathrm{\lambda }}$

• $\frac{1}{4\mathrm{\lambda }}$

The half-life of a radioactive isotope X is 20 yr. It decays to another element Y which is stable. The two elements X and Y were found to be in the ratio 1 : 7 in a sample of a given rock. The age of the rock is estimated to be

• 40 yr

• 60 yr

• 80 yr

• 100 yr

A certain mass of hydrogen is changed to helium by the process of fusion. The mass defect in fusion reaction is 0.02866 u. The energy liberated per u is (given 1 u = 931 MeV)

• 2.67 MeV

• 26.7 MeV

• 6.675 MeV

• 13.35 MeV

For the radioactive nuclei that undergo either α or β decay, which one of the following cannot occur ?

• Isobar of original nucleus is produced

• Isotope of the original nucleus is produced

• Nuclei with higher atomic number than that of the original nucleus is produced

• Nuclei with lower atomic number than that of the original nucleus is produced