Question

If \( z \) is a complex number such that \( |z| = 5 \) and \( \text{Re}(z) = 3 \), then the value of \( z^2 \) is:

• A. \( 9 + 40i \)
• B. \( 9 - 40i \)
• C. 25
• D. \( -7 + 24i \)

Hint:

For complex numbers, use the modulus and real part to find the imaginary part, then compute powers directly or use polar form for clarity.

Answer 1

The Correct Option is D

- Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Given \( \text{Re}(z) = 3 \), so \( x = 3 \), and \( |z| = 5 \), so: \[ |z| = \sqrt{x^2 + y^2} = 5 \implies \sqrt{3^2 + y^2} = 5 \implies 9 + y^2 = 25 \implies y^2 = 16 \implies y = \pm 4 \]
- Thus, \( z = 3 + 4i \) or \( z = 3 - 4i \). Both give the same \( z^2 \). Compute for \( z = 3 + 4i \): \[ z^2 = (3 + 4i)^2 = 9 + 2(3)(4i) + (4i)^2 = 9 + 24i + 16(-1) = 9 + 24i - 16 = -7 + 24i \]