A radioactive nucleus \(n_2\) has 3 times the decay constant as compared to the decay constant of another radioactive nucleus \(n_1\). If initial number of both nuclei are the same, what is the ratio of number of nuclei of \(n_2\) to the number of nuclei of \(n_1\), after one half-life of \(n_1\) ?

\(\begin{aligned} & \mathrm{N}_2=\mathrm{N}_0 \mathrm{e}^{-3 \lambda t} \\ & \mathrm{~N}_1=\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}} \\ & \frac{\mathrm{N}_2}{\mathrm{~N}_1}=\mathrm{e}^{-2 \lambda \mathrm{t}} \\ & \mathrm{t}_{\text {half lifeofN }} \mathrm{t} \\ & \mathrm{t}=\frac{\ln 2}{\lambda}{ }_{\mathrm{n}} 1 \\ & \frac{\mathrm{~N}_0}{2}=\mathrm{N}_0 \mathrm{e}^{-\lambda \mathrm{t}}\end{aligned}\)
\(\begin{aligned} & \lambda \mathrm{t}=\ln 2 \\ & \mathrm{t}=\frac{\ln 2}{\lambda} \\ & =\mathrm{e}^{-2 \lambda \frac{\ln 2}{\lambda}} \\ & \frac{\mathrm{~N}_2}{\mathrm{~N}_1}=\frac{1}{4}\end{aligned}\)