Following statements related to radioactivity are given below: (A)...

Radioactive decay is a random and spontaneous process, it depends on unbalancing of nucleus. This process is dependent on the law of conservation of charge. We know that, (N=N_0 e^{-\lambda t} \quad \ldots(1)), Where, $N=$ number of decayed nuclei in the sample at time $t$ and $N_0=$ initial number of nuclei. Hence, total number of undecayed nuclei $N_0-N$ Thus, (N_0-N=N_0\left(1-e^{-\lambda t}\right)) This shows that the total number of undecayed nuclei decays exponentially with time. Taking log both sides of equation (1) (\ln N=-\lambda t+\ln N_0) So, slope (=-\lambda) (reciprocal of mean life time) Half life time (T_{$\frac{1}{2}$}=$\frac{\ln 2}{\lambda}$) Then, (\lambda $\times$T_{$\frac{1}{2}$}=\ln 2=$\mathrm{constant}$) Hence, (B) and (C) are correct only.