Answer: (1)

Solution

Let $\lambda_1$ and $\lambda_2$ be the decay constants of two process. $N$ be the number of nuclei left undecayed after two process. From the law of radioactive decay we have $-\frac{dN}{dt} = \lambda_1 N + \lambda_2 N$ $\left[\because \frac{dN}{dt} = -\lambda N\right]$ $\Rightarrow -\frac{dN}{dt} = (\lambda_1 + \lambda_2)N$ $\Rightarrow \lambda_{\text{eq}} = (\lambda_1 + \lambda_2)$ $\frac{\ln 2}{T} = \frac{\ln 2}{T_1} + \frac{\ln 2}{T_2}$ $\left(\because \lambda = \frac{\ln 2}{T}\right)$ $\Rightarrow \frac{1}{T} = \frac{1}{T_1} + \frac{1}{T_2}$ $\frac{1}{T} = \frac{1}{10} + \frac{1}{100} = \frac{11}{100}$ [Given: $T_1 = 10$ s & $T_2 = 100$ s] $\Rightarrow T = \frac{100}{11} = 9$ sec.