2025 VJC P1 Q7

Do not use a calculator in answering this question.

1. Find the complex number \(z\) which satisfies the equation \(\frac{4|z|}{15-z^*} = 5i\).[3]
2. The complex number \(w\) is such that \((w-i)^3 = -i\).
   1. Given that one possible value of \(w\) is \(2i\), find the two other possible values of \(w\). Give your answers in cartesian form \(a+bi\).[4]

The points \(W_1\), \(W_2\) and \(W_3\) on the Argand diagram represent the three roots of the equation \((w-i)^3 = -i\), and the point \(A\) represents the complex number \(ki\), where \(k\) is a positive real number.

2. Show that the points \(W_1\), \(W_2\) and \(W_3\) lie on a circle with centre \(A\) for some value of \(k\), stating the value of \(k\).[2]