9 The complex number \(1 + i \sqrt { } 3\) is denoted by \(u\).
- Express \(u\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Hence, or otherwise, find the modulus and argument of \(u ^ { 2 }\) and \(u ^ { 3 }\).
- Show that \(u\) is a root of the equation \(z ^ { 2 } - 2 z + 4 = 0\), and state the other root of this equation.
- Sketch an Argand diagram showing the points representing the complex numbers \(i\) and \(u\). Shade the region whose points represent every complex number \(z\) satisfying both the inequalities
$$| z - \mathrm { i } | \leqslant 1 \quad \text { and } \quad \arg z \geqslant \arg u .$$