Sample of two radioactive nuclides `A` and `B` are taken. `lambda_(A)` and `lambda_(B)` are the disintergration constants of `A` and `B` respectively. In which of the following cases, the two sample can simultaneously have the same decay rate at any time ?

Options

A. initial rate of decay of A is twice the initial rate of decay of B and `lambda_(A) =lambda_(B)`
B. initial rate of decay of A is twice the initial rate of decay of B and `lambda_(A) gt lambda_(B)`
C. Initial rate of decay of B is twice the initial rate of decay of A and `lambda_(A) gt lambda_(B)`
D. Initial rate of decay of B is same as the rate of decay of A at t =2h and `lambda_(B) lt lambda_(A)`

Explanation

To determine when two radioactive nuclides A and B can simultaneously have the same decay rate at any time, we need to analyze the relationship between their decay rates and disintegration constants.

### Step-by-Step Solution:

1. **Understanding Decay Rate**:
The decay rate (activity) of a radioactive sample is given by the formula:
\[ R = \lambda N \]
where \( R \) is the decay rate, \( \lambda \) is the disintegration constant, and \( N \) is the number of undecayed nuclei at a given time.

2. **Setting Up the Condition**:
For nuclides A and B to have the same decay rate at any time, we can set their decay rates equal to each other:
\[ R_A = R_B \]
This implies:
\[ \lambda_A N_A = \lambda_B N_B \]

3. **Analyzing the Relationship**:
Rearranging the equation gives:
\[ \frac{N_A}{N_B} = \frac{\lambda_B}{\lambda_A} \]
This means that the ratio of the number of undecayed nuclei of A to that of B must equal the ratio of their disintegration constants.

4. **Considering Initial Conditions**:
If we consider the initial conditions where \( N_A \) and \( N_B \) are the initial amounts of A and B respectively, we can express the relationship at \( t=0 \):
\[ N_A(0) = N_{A0}, \quad N_B(0) = N_{B0} \]
Therefore, at \( t=0 \):
\[ \lambda_A N_{A0} = \lambda_B N_{B0} \]

5. **Conclusion**:
For the two samples A and B to have the same decay rate at any time, the ratio of their initial amounts must be inversely proportional to their disintegration constants. Thus, if the initial amount of A is twice that of B, and the disintegration constant of A is greater than that of B, they can have the same decay rate.

### Final Answer:
The two samples A and B can simultaneously have the same decay rate at any time if:
- The initial amount of A is appropriately related to the initial amount of B based on their disintegration constants.