If z is a purely imaginary number then arg (z) may be

Options

A. 0 or `pi`
B. `-pi` or 0
C. `pi or pi`
D. `-(pi)/(2)` or `(pi)/(2)`

Answer

To solve the question, we need to determine the argument of a purely imaginary number \( z \).

### Step-by-Step Solution:

1. **Understanding Purely Imaginary Numbers**:
A purely imaginary number can be expressed in the form:
\[ z = 0 + ib \]
where \( b \) is a real number and \( i \) is the imaginary unit.

2. **Identifying the Real and Imaginary Parts**:
From the expression \( z = ib \), we can identify:
- Real part \( \text{Re}(z) = 0 \)
- Imaginary part \( \text{Im}(z) = b \)

3. **Using the Argument Formula**:
The argument of a complex number \( z \) is given by:
\[ \text{arg}(z) = \theta \]
where \( \tan(\theta) = \frac{\text{Im}(z)}{\text{Re}(z)} \).

4. **Calculating the Argument**:
Substituting the real and imaginary parts into the formula:
\[ \tan(\theta) = \frac{b}{0} \]
Since the real part is zero, this expression is undefined, which implies that \( \tan(\theta) \) approaches infinity.

5. **Determining the Angles**:
The tangent function approaches infinity at the angles:
\[ \theta = \frac{\pi}{2} \quad \text{or} \quad \theta = -\frac{\pi}{2} \]
Therefore, the argument of a purely imaginary number can be either \( \frac{\pi}{2} \) or \( -\frac{\pi}{2} \).

### Final Answer:
Thus, the argument of \( z \) when \( z \) is a purely imaginary number may be:
\[ \text{arg}(z) = \frac{\pi}{2} \quad \text{or} \quad -\frac{\pi}{2} \]