(a)[5]
The complex number $w$ has $\Re w > 0$ and satisfies $w + 3w^* = iw^2$, where $w^*$ is the complex conjugate of $w$. Find $w$, giving the answer in the form $x + iy$, where $x$ and $y$ are real.
(b)[6]
On a sketch of an Argand diagram, shade the set of points for complex numbers $z$ that satisfy both $|z - 2i| \leq 2$ and $0 \leq \arg(z + 2i) \leq \tfrac{1}{4}\pi$. Calculate the greatest value of $|z|$ for points in this set, and give your answer correct to 2 decimal places.