The complex number $z$ is specified by $z = \dfrac{5a - 2i}{3 + ai}$, where $a$ is an integer. It is also given that $\arg z = -\dfrac{\pi}{4}$.
(a)[6]
Determine the value of $a$ and hence write $z$ in the form $x + iy$, where $x$ and $y$ are real.
(b)[3]
Write $z^3$ in the form $re^{i\theta}$, where $r > 0$ and $-\pi < \theta \leq \pi$. Provide the exact simplified values of $r$ and $\theta$.
Worked solution & mark scheme
This 9-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Multiply both the numerator and denominator by $(3-ai)” …