Half life of a radioactive element is T. Then what fraction of initial number of active nuclei will decay in the time interval t = T to t = 2T?

If $N_0$ be the initial number of active nuclei of the radio active sample, then 

$N_1=\frac{N_0}{{\left(2\right)}^{\raisebox{1ex}{$T$}\!\left/ \!\raisebox{-1ex}{$T$}\right.}}=\raisebox{1ex}{$N_0$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

and $N_2=\frac{N_0}{{\left(2\right)}^{\raisebox{1ex}{$2T$}\!\left/ \!\raisebox{-1ex}{$T$}\right.}}=\raisebox{1ex}{$N_0$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

$\therefore$ Required fraction $=\frac{N_1-N_2}{N_0}=\frac{\frac{N_0}{2}-\frac{N_0}{4}}{N_0}=\frac{1}{4}$