(a(i))[2]
Without a calculator, rewrite the complex number $\frac{2 + 6i}{1 - 2i}$ in the form $x + iy$, where $x$ and $y$ are real.
(a(ii))[3]
Hence, without a calculator, write $\frac{2 + 6i}{1 - 2i}$ in the form $r(\cos \theta + i \sin \theta)$, where $r > 0$ and $-\pi < \theta \leq \pi$, and give the exact values of $r$ and $\theta$.
(b)[5]
On a sketch Argand diagram, shade the set of points representing complex numbers $z$ that satisfy both $|z - 3i| \leq 1$ and $\Re z \leq 0$, where $\Re z$ is the real part of $z$. Determine the largest value of $\arg z$ for points in this region, and give your answer in radians correct to 2 decimal places.