Let `R_(t)` represents activity of a sample at an insant and `N_(t)` represent number of active nuclei in the sample at the instant. `T_(1//2)` represents the half life.
`{:(,"Column I",,"Column II"),((A),t=T_(1//2),(p),R_(t)=(R_(0))/(2)),((B),t=(T_(1//2))/(ln2),(q),N_(0)-N_(t)=(N_(0))/(2)),((C),t=(3)/(2)T_(1//2),(r),(R_(t)-R_(0))/(R_(0)) = (1-e)/(e)),(,,(s),N_(t)=(N_(0))/(2sqrt(2))):}`

Answer

To solve the problem, we need to match the entries from Column I with those in Column II based on the relationships between activity \( R_t \), number of active nuclei \( N_t \), and half-life \( T_{1/2} \).

### Step-by-step Solution:

1. **Part A: \( t = T_{1/2} \)**
- At half-life, the activity \( R_t \) is half of the initial activity \( R_0 \).
- Therefore, we can write:
\[ R_t = \frac{R_0}{2} \]
- Similarly, the number of active nuclei \( N_t \) is also half of the initial number \( N_0 \):
\[ N_t = \frac{N_0}{2} \]
- Thus, we can match:
- \( A \) with \( p \) and \( q \).

2. **Part B: \( t = \frac{T_{1/2}}{\ln 2} \)**
- We know that the decay constant \( \lambda \) is given by:
\[ \lambda = \frac{\ln 2}{T_{1/2}} \]
- Using the equation for the number of active nuclei:
\[ N_t = N_0 e^{-\lambda t} \]
- Substituting \( t = \frac{T_{1/2}}{\ln 2} \):
\[ N_t = N_0 e^{-\left(\frac{\ln 2}{T_{1/2}}\right) \left(\frac{T_{1/2}}{\ln 2}\right)} = N_0 e^{-1} \]
- Thus, the change in the number of active nuclei is:
\[ N_0 - N_t = N_0 - \frac{N_0}{e} = N_0 \left(1 - \frac{1}{e}\right) \]
- Hence, we can match:
- \( B \) with \( r \).

3. **Part C: \( t = \frac{3}{2} T_{1/2} \)**
- Again using the decay equation:
\[ N_t = N_0 e^{-\lambda t} \]
- Substituting \( t = \frac{3}{2} T_{1/2} \):
\[ N_t = N_0 e^{-\left(\frac{\ln 2}{T_{1/2}}\right) \left(\frac{3}{2} T_{1/2}\right)} = N_0 e^{-\frac{3}{2} \ln 2} = N_0 \left(\frac{1}{2}\right)^{3/2} = \frac{N_0}{2\sqrt{2}} \]
- Therefore, we can match:
- \( C \) with \( s \).

### Final Matches:
- \( A \) matches with \( p \) and \( q \).
- \( B \) matches with \( r \).
- \( C \) matches with \( s \).