Explanation

(d) 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.
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N_{1} = \ N_{0}e^{- 5\lambda t}\ \ \ \ \ \ \ \ldots..\ (i)$
and $\ \ \ \ \ \ N_{2} = \ N_{0}e^{- \lambda t}\ \ \ \ \ \ \ \ \ \ldots.(ii)$
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 ,$\frac{1}{e^{2}} = \frac{1}{e^{4\lambda t}}$
Comparing the powers, we get
$\ \ \ \ \ \ \ \ \ \ 2 = 4\lambda t$
or $\ \ \ \ \ t = \ \frac{2}{4\lambda}\ = \frac{1}{2\lambda}$