2025 ACJC LCP2B Q2

Do not use a calculator in answering this question.

1. The complex number \(z\) has modulus \(1\) and argument \(\frac{\pi }{6}\). The complex number \(w\) is such that \(w=1+\sqrt{3}\,\mathbf{i}\). Indicate the points \(A\), \(B\) and \(C\) representing the complex numbers \(z\), \(w\) and \(z+w\) on an Argand diagram. Hence find \({{\left| z+w \right|}^{2}}\).[3]
2. It is given that \(x=1+\sqrt{3}\,\mathbf{i}\) is one of the roots of the equation \(3{{x}^{3}}+a{{x}^{2}}+16x+b=0\), where \(a\) and \(b\) are real. Find the other roots of the equation and the values of \(a\) and \(b\).[5]