Question

If \( -3 + ix^2y \) and \( x^2 + y + 4i \) are complex conjugates, then \( x = \):

• A. \( 0 \)
• B. \( \pm 1 \)
• C. \( \pm 3 \)
• D. \( \pm 4 \)

Hint:

When dealing with complex conjugates, always match real and imaginary parts separately.

Answer 1

The Correct Option is B

Step 1: Use the property of complex conjugates.
If \( z_1 = -3 + ix^2y \) and \( z_2 = x^2 + y + 4i \) are conjugates, then: \[ z_1 = \overline{z_2} \Rightarrow -3 + ix^2y = x^2 + y - 4i \]
Step 2: Compare real and imaginary parts
Equating real parts: \[ -3 = x^2 + y \quad \cdots (1) \] Equating imaginary parts: \[ x^2y = -4 \quad \cdots (2) \] From (1): \( y = -3 - x^2 \) Substitute into (2): \[ x^2(-3 - x^2) = -4 \Rightarrow -3x^2 - x^4 = -4 \Rightarrow x^4 + 3x^2 - 4 = 0 \] Solve this quadratic in \( x^2 \): \[ \text{Let } z = x^2 \Rightarrow z^2 + 3z - 4 = 0 \Rightarrow z = 1 \Rightarrow x = \pm 1 \]