IB AA HL May 2026 P2 TZA Q8

Consider the equation \(z^3=c\), where \(z\in\mathbb{C}\) and \(c\) is a positive constant.

One of the solutions of the equation is \(z=-2+2i\sqrt3\).

1. Find the value of \(c\).[2]
2. Write down the other two solutions of \(z^3=c\) giving your answers in the form \(a+bi\), where \(a,b\in\mathbb{R}\).[2]
3. Find the solutions of \((iz)^3=c\) giving your answers in the form \(a+bi\), where \(a,b\in\mathbb{R}\).[2]
4. The solutions of \((iz)^3=c\) can be represented on an Argand diagram by three points \(A\), \(B\) and \(C\).
   Find the area of the triangle \(ABC\).[2]