At `t=0`, number of radioactive nuclei of a radioactive substance are x and its radioactivity is y. Half-life of radioactive substance is T. Then,

Options

A. (a) `x/y` is constant throughout
B. (b) `x/ygtT`
C. (c) value of `xy` remains half after one half-life
D. (d) value of `xy` remains one fourth after one half-life

Explanation

To solve the problem, we need to analyze the relationship between the number of radioactive nuclei (x), radioactivity (y), and half-life (T) of a radioactive substance.

### Step-by-Step Solution:

1. **Understanding Radioactivity**:
The radioactivity (y) of a substance is given by the formula:
\[ y = \lambda \cdot N \]
where \( \lambda \) is the decay constant and \( N \) is the number of radioactive nuclei.

2. **Decay Constant and Half-Life**:
The decay constant \( \lambda \) is related to the half-life \( T \) by the equation:
\[ \lambda = \frac{\ln(2)}{T} \]

3. **Initial Conditions**:
At \( t = 0 \), we have:
- Number of radioactive nuclei, \( N = x \)
- Radioactivity, \( y = y_0 \)

4. **Relating x and y**:
Substituting \( N \) into the radioactivity formula:
\[ y_0 = \lambda \cdot x \]
Rearranging gives:
\[ \frac{x}{y_0} = \frac{1}{\lambda} \]

5. **Substituting for λ**:
Now substituting \( \lambda \):
\[ \frac{x}{y_0} = \frac{T}{\ln(2)} \]
This implies:
\[ x = y_0 \cdot \frac{T}{\ln(2)} \]
Therefore, we can express the ratio:
\[ \frac{x}{y_0} = \frac{T}{0.693} \approx 1.44T \]

6. **Conclusion on x/y**:
This means that \( \frac{x}{y_0} \) is a constant value (1.44T) and does not change over time. Thus, \( \frac{x}{y} \) is constant.

7. **After One Half-Life**:
After one half-life \( T \):
- The number of radioactive nuclei becomes \( \frac{x}{2} \)
- The radioactivity becomes \( \frac{y_0}{2} \)
- Therefore, the product \( xy \) becomes:
\[ xy = x \cdot y_0 = \frac{x}{2} \cdot \frac{y_0}{2} = \frac{xy_0}{4} \]
This shows that after one half-life, the product \( xy \) is reduced to one-fourth of its original value.

### Summary of Results:
- \( \frac{x}{y} \) is constant.
- The product \( xy \) is reduced to one-fourth after one half-life.