2016 MJC P1 Q10

Do not use a calculator in answering this question.

1. Solve the simultaneous equations
   \(z=w+2\mathrm{i}-1\) and \({{z}^{2}}-\mathrm{i}w+\frac{5}{2}=0\),
   giving \(z\) and \(w\) in the form \(x+y\mathrm{i}\) where \(x\) and \(y\) are real.[5]
2. 1. Given that \(z=w-\frac{1}{w}\) where \(w=2(\cos \theta +\mathrm{i}\sin \theta )\), \(-\pi <\theta \le \pi \), express the real and imaginary parts \(z\) in terms of \(\theta \).[3]
   2. Hence show that the locus of \(z\) on an Argand diagram lies on the curve with cartesian equation \(\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\) where \(a\) and \(b\) are constants.[3]