These question consists of two statements each linked as Assertion and Reason. While answering these question you are required to choose any one of the following five responses.
Assertion: Average value of current in half the cycle an AC circuit can't be zero.
Reason: For positive half cycle average value of current is `2/pi` `i_(0)`, where `i_(0)` is the peak value current. In time interval from `t_(1)` to `t_(2)` average value of current will be zero.

Options

A. If both Assertion and Reason are true and Reason is the correct explanation of Assertion.
B. If both Assertion and Reason are true but Reason is not the correct explanation of Assertion.
C. If Assertion is true but Reason is false.
D. If Assertion is false but Reason is true.

Explanation

To solve the problem, we need to analyze the Assertion and Reason provided in the question.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that the average value of current in half the cycle of an AC circuit cannot be zero.
- In an AC circuit, the current varies sinusoidally. The average value over a complete cycle is zero, but we need to analyze the average over just half a cycle.

2. **Calculating the Average Current**:
- For the positive half cycle of an AC current, the average value can be calculated using the formula:
\[ I_{avg} = \frac{1}{T/2} \int_0^{T/2} I_0 \sin(\omega t) \, dt \]
- Here, \(I_0\) is the peak current, \(\omega\) is the angular frequency, and \(T\) is the time period.

3. **Performing the Integration**:
- The integral of \(\sin(\omega t)\) is \(-\frac{1}{\omega} \cos(\omega t)\). Therefore, we have:
\[ I_{avg} = \frac{2}{T} \left[-\frac{I_0}{\omega} \cos(\omega t)\right]_0^{T/2} \]
- Substituting the limits:
\[ I_{avg} = \frac{2 I_0}{T} \left[-\frac{1}{\omega} \left(\cos(\frac{\pi}{2}) - \cos(0)\right)\right] \]
- Since \(\cos(0) = 1\) and \(\cos(\frac{\pi}{2}) = 0\):
\[ I_{avg} = \frac{2 I_0}{T} \left[-\frac{1}{\omega} (0 - 1)\right] = \frac{2 I_0}{T} \cdot \frac{1}{\omega} \]

4. **Simplifying the Expression**:
- Recognizing that \(\omega = \frac{2\pi}{T}\):
\[ I_{avg} = \frac{2 I_0}{T} \cdot \frac{T}{2\pi} = \frac{2 I_0}{2\pi} = \frac{I_0}{\pi} \]
- Therefore, the average value of the current for the positive half cycle is:
\[ I_{avg} = \frac{2 I_0}{\pi} \]
- This shows that the average current in the positive half cycle is not zero.

5. **Understanding the Reason**:
- The reason states that in the time interval from \(t_1\) to \(t_2\), the average value of the current will be zero. This is true if \(t_1\) and \(t_2\) are chosen such that they encompass both the positive and negative halves of the cycle, leading to cancellation.

6. **Conclusion**:
- The assertion is **false** because the average value of current in the positive half cycle is not zero; it is \(\frac{2 I_0}{\pi}\).
- The reason is **true** in the context of a full cycle.
- Therefore, the correct answer is that the assertion is false, and the reason is true.

### Final Answer:
- **Assertion**: False
- **Reason**: True
- **Correct Option**: D