(a)[4]
Solve the equation $(1 + 2i)w + i w^* = 3 + 5i$. Express your answer in the form $x + iy$, with $x$ and $y$ real.
(b(i))[4]
On a sketch of an Argand diagram, shade the set of points for complex numbers $z$ that satisfy $|z - 2 - 2i| \leq 1$ and $\arg(z - 4i) \geq -\frac{1}{4}\pi$.
(b(ii))[2]
Find the least value of $\operatorname{Im} z$ for points in this region, and give the result exactly.