Two radioactive substances A and B have decay constants 5 λ and λ...

Number of nuclei remained after time $t$ can be written as $N=N_0 e^{-\lambda t}$ where $N_0$ is initial number of nuclei of both the substances. and $$\begin{aligned} & N_1=N_0 e^{-5 \lambda t} \ldots(i)\\ & N_2=N_0 e^{-\lambda t} \ldots(ii) \end{aligned}$$ Dividing Eq. (i) by Eq. (ii), we obtain $\frac{N_1}{N_2}=e^{(-5 \lambda+\lambda) t}=e^{-4 \lambda t}=\frac{1}{e^{4 \lambda t}}$ But, we have given $\frac{N_1}{N_2}=$\left(\frac{1}{e}\right)^2=\frac{1}{e^2}$Hence,$\quad \frac{1}{e^2}=\frac{1}{e^{4 \lambda t}}$Comparing the powers, we get or$$\begin{aligned} & 2=4 \lambda t \\ & t=\frac{2}{4 \lambda}=\frac{1}{2 \lambda} \end{aligned}$$