Complex Numbers

If \( z = 3 + 4i \), what is the modulus of \( z \)?

Which of the following is the conjugate of \( 5 - 2i \)?

If \( z = 1 + i \), what is \( z^2 \)?

Find the argument of the complex number \( -1 + \sqrt{3}i \).

If \( z = 2(\cos 60^\circ + i \sin 60^\circ) \), find \( z^3 \) in polar form.

Solve \( z^2 + 1 = 0 \) for complex \( z \).

If \( z = x + iy \) satisfies \( |z| = 5 \) and \( \text{Re}(z) = 3 \), what is \( \text{Im}(z) \)?

Express \( \frac{1}{1 + i} \) in the form \( a + bi \), where \( a,b \in \mathbb{R} \).

If \( z = 4(\cos \theta + i \sin \theta) \) and \( z^4 = 16 \), find \( \theta \).

Find all cube roots of unity and identify the one which corresponds to \( \cos 120^\circ + i \sin 120^\circ \).