11 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 show that \(u ^ { 6 }\) is real and state its value.
- On a sketch of an Argand diagram, shade the region whose points represent complex numbers \(z\) satisfying the inequalities \(0 \leqslant \arg ( z - u ) \leqslant \frac { 1 } { 4 } \pi\) and \(\operatorname { Re } z \leqslant 2\).
- Find the greatest value of \(| z |\) for points in the shaded region. Give your answer correct to 3 significant figures.
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