(a)[5]
The complex numbers $u$ and $w$ satisfy the equations $u - w = 4i$ and $uw = 5$. Find $u$ and $w$, giving every answer in the form $x + iy$, where $x$ and $y$ are real.
(b(i))[5]
On a sketch of an Argand diagram, shade the set of points representing complex numbers $z$ that satisfy the inequalities $|z - 2 + 2i| \leq 2$, $\arg z \leq -\frac{1}{4}\pi$ and $\operatorname{Re} z \geq 1$, where $\operatorname{Re} z$ is the real part of $z$.
(b(ii))[1]
Calculate the largest possible value of $\operatorname{Re} z$ for points in the shaded region.