Two samples A and B of same radioactive nuclide are prepared. Sample A has twice the initial activity of sample B. For this situation, mark out the correct statement (s).

Options

A. The half-lives of both the samples would be same.
B. The half-lives of the samples are different.
C. After each has passed through `5` half-lives, ratio of activity of `A` to `B` is `2:1`.
D. After each has passed through `5` half-lives, ratio of activity of `A` to` B` is `64:1`.

Explanation

To solve the problem, we need to analyze the information given about the two samples A and B of the same radioactive nuclide.

### Step-by-Step Solution:

1. **Understanding Initial Activity**:
- Let the initial activity of sample B be \( A_0 \).
- Since sample A has twice the initial activity of sample B, the initial activity of sample A will be \( 2A_0 \).

2. **Half-Life Concept**:
- The half-life of a radioactive nuclide is a characteristic property of the material and does not depend on the amount of the substance or its initial activity.
- Therefore, both samples A and B, being the same nuclide, will have the same half-life (\( T_{1/2} \)).

3. **Activity After Multiple Half-Lives**:
- The activity of a radioactive sample decreases over time according to the formula:
\[ A(t) = A_0 \left( \frac{1}{2} \right)^{\frac{t}{T_{1/2}}} \]
- After \( n \) half-lives, the activity becomes:
\[ A(nT_{1/2}) = A_0 \left( \frac{1}{2} \right)^{n} \]
- For sample A after 5 half-lives:
\[ A_A(5T_{1/2}) = 2A_0 \left( \frac{1}{2} \right)^{5} = 2A_0 \cdot \frac{1}{32} = \frac{2A_0}{32} = \frac{A_0}{16} \]
- For sample B after 5 half-lives:
\[ A_B(5T_{1/2}) = A_0 \left( \frac{1}{2} \right)^{5} = A_0 \cdot \frac{1}{32} = \frac{A_0}{32} \]

4. **Finding the Ratio of Activities**:
- The ratio of the activities after 5 half-lives is:
\[ \frac{A_A(5T_{1/2})}{A_B(5T_{1/2})} = \frac{\frac{A_0}{16}}{\frac{A_0}{32}} = \frac{32}{16} = 2 \]
- This shows that the ratio of activities of A to B after 5 half-lives is \( 2:1 \).

5. **Conclusion**:
- From the analysis, we can conclude:
- Statement A is correct: The half-lives of both samples are the same.
- Statement B is incorrect: The half-lives of the samples are not different.
- Statement C is correct: After 5 half-lives, the ratio of activity of A to B is indeed \( 2:1 \).
- Statement D is incorrect: The ratio of activities after 5 half-lives is not \( 64:1 \).

### Final Answer:
The correct statements are A and C.