If `arg z = pi/4` ,then

Options

A. `Re(z^(2))=9lm(z ^(2))`
B. `lm(z^(2)=0`
C. `Re(z^(2))=0`
D. `Re(z)=0`

Answer

To solve the problem where we are given that the argument of \( z \) is \( \frac{\pi}{4} \), we can follow these steps:

### Step 1: Express \( z \) in terms of its components
Given that \( \arg z = \frac{\pi}{4} \), we can express \( z \) in terms of its real and imaginary parts:
\[ z = x + iy \]
where \( x \) is the real part and \( y \) is the imaginary part.

### Step 2: Relate the argument to the components
The argument of a complex number is given by:
\[ \arg z = \tan^{-1}\left(\frac{y}{x}\right) \]
Since \( \arg z = \frac{\pi}{4} \), we have:
\[ \tan^{-1}\left(\frac{y}{x}\right) = \frac{\pi}{4} \]

### Step 3: Take the tangent of both sides
Taking the tangent of both sides gives us:
\[ \frac{y}{x} = \tan\left(\frac{\pi}{4}\right) = 1 \]
This implies:
\[ y = x \]

### Step 4: Substitute \( y \) back into \( z \)
Now substituting \( y \) back into the expression for \( z \):
\[ z = x + ix = x(1 + i) \]

### Step 5: Calculate \( z^2 \)
Next, we calculate \( z^2 \):
\[ z^2 = (x(1 + i))^2 = x^2(1 + 2i + i^2) = x^2(1 + 2i - 1) = x^2(2i) \]
Thus,
\[ z^2 = 2x^2 i \]

### Step 6: Identify the real and imaginary parts of \( z^2 \)
From the expression \( z^2 = 2x^2 i \), we can identify:
- Real part of \( z^2 = 0 \)
- Imaginary part of \( z^2 = 2x^2 \)

### Step 7: Analyze the options given in the problem
1. **Real part of \( z^2 = 9 \times \) Imaginary part of \( z \)**:
- Real part of \( z^2 = 0 \), so this is incorrect.

2. **Imaginary part of \( z^2 = 0 \)**:
- Imaginary part of \( z^2 = 2x^2 \), which is not zero unless \( x = 0 \), so this is incorrect.

3. **Real part of \( z^2 = 0 \)**:
- This is correct since we found that the real part of \( z^2 \) is indeed 0.

4. **Real part of \( z = 0 \)**:
- This would imply \( z \) is purely imaginary, which contradicts our earlier findings since \( z = x(1 + i) \) where \( x \) is not necessarily zero. Therefore, this is incorrect.

### Conclusion
The correct option is that the real part of \( z^2 \) is 0.