Without using a calculator, solve the equation \(\mathrm { i } w ^ { 2 } = ( 2 - 2 \mathrm { i } ) ^ { 2 }\).
Sketch an Argand diagram showing the region \(R\) consisting of points representing the complex numbers \(z\) where
$$| z - 4 - 4 i | \leqslant 2$$
For the complex numbers represented by points in the region \(R\), it is given that
$$p \leqslant | z | \leqslant q \quad \text { and } \quad \alpha \leqslant \arg z \leqslant \beta$$
Find the values of \(p , q , \alpha\) and \(\beta\), giving your answers correct to 3 significant figures.
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