Paper 1, Section I, $1 \mathrm{C}$

(a) State de Moivre's theorem and use it to derive a formula for the roots of order $n$ of a complex number $z=a+i b$. Using this formula compute the cube roots of $z=-8$.

(b) Consider the equation $|z+3 i|=3|z|$ for $z \in \mathbb{C}$. Give a geometric description of the set $S$ of solutions and sketch $S$ as a subset of the complex plane.

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