Following statements related to radioactivity are given below:
(A) Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions.
(B) The number of undecayed nuclei in the radioactive sample decays exponentially with time.
(C) Slope of the graph of $$\log_e$$ (no. of undecayed nuclei) vs. time represents the reciprocal of mean life time ($$\tau$$).
(D) Product of decay constant ($$\lambda$$) and half-life time $$\left(T_{\frac{1}{2}}\right)$$ is not constant.
Choose the most appropriate answer from the options given below:

Let us analyze each statement:

Statement (A): "Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions." Radioactivity is indeed random and spontaneous, but it is independent of physical and chemical conditions (temperature, pressure, chemical state do not affect nuclear decay). So statement (A) is incorrect.

Statement (B): "The number of undecayed nuclei in the radioactive sample decays exponentially with time." The number of undecayed nuclei follows $$N = N_0 e^{-\lambda t}$$, which is an exponential decay. So statement (B) is correct.

Statement (C): "Slope of the graph of $$\log_e$$(no. of undecayed nuclei) vs. time represents the reciprocal of mean life time ($$\tau$$)." Taking the natural log of $$N = N_0 e^{-\lambda t}$$ gives $$\ln N = \ln N_0 - \lambda t$$. The slope of $$\ln N$$ vs. $$t$$ is $$-\lambda$$. Since mean life $$\tau = \frac{1}{\lambda}$$, the slope equals $$-\frac{1}{\tau}$$, which represents the (negative) reciprocal of mean life time. So statement (C) is correct.

Statement (D): "Product of decay constant ($$\lambda$$) and half-life time ($$T_{1/2}$$) is not constant." We know $$T_{1/2} = \frac{\ln 2}{\lambda}$$, so $$\lambda \times T_{1/2} = \ln 2$$, which is a constant. So statement (D) is incorrect.

The correct statements are (B) and (C) only. Hence, the correct answer is Option C.