Question

If \( z = 1 + i \), find the modulus of \( z^2 \).

• A. \( \sqrt{2} \)
• B. \( 2 \)
• C. \( 2\sqrt{2} \)
• D. \( 4 \)

Hint:

For a complex number \( z \), the modulus of \( z^n \) is \( |z|^n \), and for \( z = a + bi \), the modulus is \( \sqrt{a^2 + b^2} \).

Answer 1

The Correct Option is B

Given \( z = 1 + i \), compute \( z^2 \): \[ z^2 = (1 + i)^2 = 1^2 + 2 \cdot 1 \cdot i + i^2 = 1 + 2i + (-1) = 2i \] The modulus of a complex number \( a + bi \) is \( \sqrt{a^2 + b^2} \). For \( z^2 = 0 + 2i \): \[ |z^2| = \sqrt{0^2 + 2^2} = \sqrt{4} = 2 \] Alternatively, since \( |z| = \sqrt{1^2 + 1^2} = \sqrt{2} \), the modulus of \( z^2 \) is: \[ |z^2| = |z|^2 = (\sqrt{2})^2 = 2 \] The modulus is: \[ \boxed{2} \]