IAL 2025 May FP1 Q5

Question

[!problem]

The complex number $z_1$ is given by

$z_1 = r\left(\cos \frac{7\pi}{6} + i \sin \frac{7\pi}{6}\right)$

where $r$ is a positive constant.

The complex number $z_2$ has modulus 5.

Given that $|z_1 z_2| = 15$

(a) state the value of $r$.

(1)

Given further that $z_1 + z_2$ is a real number,

(b) determine the possible complex numbers $z_2$ in the form $a + bi$ where $a$ and $b$ are constants.

(5)

(c) Show $z_1$ and the possible complex numbers $z_2$ on a single Argand diagram.

(2)