N2012 P1 Q6

Do not use a calculator in answering this question.

The complex number \(z\) is given by \(z=1+\mathrm ic\), where \(c\) is a non-zero real number.

1. Find \(z^3\) in the form \(x+\mathrm iy\).[2]
2. Given that \(z^3\) is real, find the possible values of \(z\).[2]
3. For the value of \(z\) found in part (ii) for which \(c<0\), find the smallest positive integer \(n\) such that \(|z^n|>1000\). State the modulus and argument of \(z^n\) when \(n\) takes this value.[4]