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, $N = {N}_{0}{e}^{- \lambdat} . . . \left(1\right), Where,$N$=number of decayed nuclei in the sample at time$t and {N}_{0} =$initial number of nuclei.</p><p>Hence, total number of undecayed nuclei${N}_{0} - N$</p><p>Thus,${N}_{0} - N = {N}_{0}\left(1 - {e}^{- \lambdat}\right)$</p><p>This shows that the total number of undecayed nuclei decays exponentially with time. </p><p>Taking log both sides of equation$\left(1\right)$</p><p>$lnN = - \lambdat + ln{N}_{0}$</p><p>So, slope$= - \lambda$(reciprocal of mean life time)</p><p>Half life time$T\dfrac{1}{2} = \dfrac{\ln 2}{\lambda}$</p><p>Then,$\lambda \times T\dfrac{1}{2} = \ln 2 = constant

Hence, (B) and (C) are correct only.