A radioactive sample has initial concentration `N_(0)` of nuclei. Then,

Options

A. the number of undecayed nuclei present in the sample decays exponentially with time
B. the activity (R ) of the sample at any instant is directly proportional to the number of undecayed nuclei present in the sample at that tiem
C. the number of decayed nuclei grows exponentially with time
D. the number of decayed nuclei grows lineraly with time

Explanation

To solve the question regarding the radioactive sample with an initial concentration \( N_0 \) of nuclei, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Radioactive Decay**:
The number of undecayed nuclei \( N(t) \) at time \( t \) can be expressed using the formula:
\[ N(t) = N_0 e^{-\lambda t} \]
where \( \lambda \) is the decay constant.

2. **Finding the Number of Decayed Nuclei**:
The number of decayed nuclei \( D(t) \) at time \( t \) can be calculated as:
\[ D(t) = N_0 - N(t) = N_0 - N_0 e^{-\lambda t} = N_0 (1 - e^{-\lambda t}) \]
This shows how many nuclei have decayed over time.

3. **Calculating Activity**:
The activity \( R \) of the sample, which is the rate of decay, is given by:
\[ R = -\frac{dN}{dt} = \lambda N(t) = \lambda N_0 e^{-\lambda t} \]
This indicates that the activity is proportional to the number of undecayed nuclei.

4. **Behavior of Decayed Nuclei**:
Since \( D(t) \) increases as time progresses, we can analyze its growth:
- As \( t \) increases, \( e^{-\lambda t} \) approaches 0, making \( D(t) \) approach \( N_0 \).
- Therefore, the number of decayed nuclei grows exponentially with time.

5. **Conclusion**:
- The number of undecayed nuclei decreases exponentially.
- The number of decayed nuclei increases exponentially.
- The activity is directly proportional to the number of undecayed nuclei.

### Final Answer:
The statements regarding the behavior of the radioactive sample can be summarized as:
- The number of undecayed nuclei decreases exponentially.
- The number of decayed nuclei increases exponentially.
- The activity is proportional to the number of undecayed nuclei.