Two radioactive substances \(A\) and \(B\) have decay constants ' \(5 \lambda\) ' and ' \(\lambda\) ' respectively. At \(\mathrm{t}=0\), they have the same number of nuclei. The ratio of number of nuclei of A to those of B will be \(\left(\frac{1}{\mathrm{e}}\right)^2\) after a time interval

Number of nuclei remained after time \(t\) can be written as \(\mathrm{N}=\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}}\)
\(\mathrm{N}_1=\mathrm{N}_0 \mathrm{e}^{-5 \lambda \mathrm{t}}...(i)\)
and \(\mathrm{N}_2=\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}}\)...(ii)
Dividing equation (i) by equation (ii), we get,
\(\begin{aligned}
& \frac{N_1}{N_2}=e^{(-5 \lambda+\lambda) t}=e^{-4 \lambda t}=\frac{1}{e^{4 \lambda t}} \\
& \frac{N_1}{N_2}=\left(\frac{1}{e}\right)^2=\frac{1}{e^2} ...[Given]\\
\therefore \quad & \frac{1}{e^2}=\frac{1}{e^{4 \lambda t}} \\
\therefore \quad & 2=4 \lambda t \Rightarrow t=\frac{2}{4 \lambda}=\frac{1}{2 \lambda}
\end{aligned}\)