JEE Main · 2023 · 30 Jan 2023 · Shift 2 · Modern Physics

When a nucleus decays through two independent channels the decay constants add, $\lambda ={\lambda }_1+{\lambda }_2$, and since $\lambda =\frac{\ln 2}{t_{1/2}}$ the reciprocals of the half-lives add: $\frac{1}{t}=\frac{1}{t_1}+\frac{1}{t_2}$. Converting the first half-life, $t_1=5$ min $=300$ s, with $t_2=30$ s, $\frac{1}{t}=\frac{1}{300}+\frac{1}{30}=\frac{1+10}{300}=\frac{11}{300}$, so the effective half-life is $t=\frac{300}{11}$ s. Comparing with $\frac{\alpha }{11}$ s gives $\alpha =300$.