Problem

6. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$z_1 = 3 + 2i$, $z_2 = 2 + 3i$, $z_3 = a + bi$, $a, b \in \mathbb{R}$

(a) Determine the exact value of $|z_1 + z_2|$

Given that $w = \dfrac{z_2 z_3}{z_1}$

(b) determine $w$ in terms of $a$ and $b$, giving your answer in the form $x + iy$, where $x, y \in \mathbb{R}$

Given also that $w = \dfrac{4}{13} - \dfrac{58}{13}i$

(c) determine the value of $a$ and the value of $b$

(d) determine $\arg w$, giving your answer in radians to 4 significant figures.