(a)[4]
The equation $2x^3 - x^2 + 2x + 12 = 0$ has one real root and two complex roots. By showing your working, confirm that $1 + i\sqrt{3}$ is one of the complex roots. State the other complex root.
(b)[5]
On a sketch of an Argand diagram, mark the point corresponding to the complex number $1 + i\sqrt{3}$. On the same diagram, shade the set of points representing the complex numbers $z$ that satisfy both $|z - 1 - i\sqrt{3}| \leq 1$ and $\arg z \leq \frac{\pi}{3}$.