10 Throughout this question the use of a calculator is not permitted.
The complex number \(( \sqrt { } 3 ) + \mathrm { i }\) is denoted by \(u\).
- Express \(u\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\), giving the exact values of \(r\) and \(\theta\). Hence or otherwise state the exact values of the modulus and argument of \(u ^ { 4 }\).
- Verify that \(u\) is a root of the equation \(z ^ { 3 } - 8 z + 8 \sqrt { } 3 = 0\) and state the other complex root of this equation.
- On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(| z - u | \leqslant 2\) and \(\operatorname { Im } z \geqslant 2\), where \(\operatorname { Im } z\) denotes the imaginary part of \(z\).
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