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 un-decayed 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 `(T_(1//2))` is not constant.
Choose the most appropriate answer from the options given below :

Options

A. (A) and (B) only
B. (B) and (D) only
C. (B) and (C) only
D. (C) and (D) only

Answer

To solve the question, we need to analyze each of the statements related to radioactivity and determine their validity.

### Step-by-Step Solution:

1. **Statement (A)**: "Radioactivity is a random and spontaneous process and is dependent on physical and chemical conditions."
- **Analysis**: This statement is **false**. Radioactivity is indeed a random and spontaneous process, but it is **not dependent** on physical or chemical conditions. The decay of radioactive nuclei is intrinsic to the nuclei themselves.

2. **Statement (B)**: "The number of un-decayed nuclei in the radioactive sample decays exponentially with time."
- **Analysis**: This statement is **true**. The decay of radioactive nuclei follows an exponential decay law, which can be expressed mathematically as \( N(t) = N_0 e^{-\lambda t} \), where \( N(t) \) is the number of undecayed nuclei at time \( t \), \( N_0 \) is the initial number of undecayed nuclei, and \( \lambda \) is the decay constant.

3. **Statement (C)**: "Slope of the graph of \( \log_e \) (no. of undecayed nuclei) Vs. time represents the reciprocal of mean lifetime \( \tau \)."
- **Analysis**: This statement is **true**. When we plot \( \log_e(N) \) against time, the slope of the resulting straight line is equal to \(-\lambda\), and the mean lifetime \( \tau \) is related to the decay constant by \( \tau = \frac{1}{\lambda} \). Therefore, the slope represents \(-\frac{1}{\tau}\).

4. **Statement (D)**: "Product of decay constant \( \lambda \) and half-life time \( T_{1/2} \) is not constant."
- **Analysis**: This statement is **false**. The product of the decay constant \( \lambda \) and the half-life \( T_{1/2} \) is a constant, specifically \( \lambda T_{1/2} = \ln(2) \).

### Conclusion:
- Valid Statements: (B) and (C)
- Invalid Statements: (A) and (D)

### Final Answer:
The most appropriate answer is that statements (B) and (C) are true, while statements (A) and (D) are false.