2025 JPJC P2 Q4

Do not use a calculator in answering this question.

1. One of the roots of the equation \(5w^3 + pw^2 + 68w + q = 0\), where \(p\) and \(q\) are real, is \(3 - i\). Find the other roots of the equation and the values of \(p\) and \(q\).[5]
2. Two complex numbers are given by \(z_1 = -1 + 2i\) and \(z_2 = 2 + i\). Draw an Argand diagram showing \(z_1\) and \(z_2\), labelling the origin as \(O\) and the points representing \(z_1\) and \(z_2\) as \(Z_1\) and \(Z_2\) respectively. Given that \(z_3 = z_1 + z_2\), mark the corresponding point \(Z_3\) on your Argand diagram, showing clearly the geometrical relationship between \(Z_1\), \(Z_2\) and \(Z_3\).[2]
   1. Find \(\frac{z_1}{z_2}\) in the form \(ki\).[2]
   2. Hence, or otherwise, show that \(OZ_2Z_3Z_1\) is a square.[2]