2021 ASRJC JC2 MYA P1 Q3

Do not use a calculator in answering this question.

The point \(P\) represents the complex number \(z=-2+b\,\mathbf{i}\), where \(b>0\). The line \(OP\) makes an angle of \({{45}^{{}^\circ }}\) with the horizontal.

1. State the value of \(b\).[1]
2. Hence find the modulus and argument of \(\frac{1}{z}\).[2]
3. Show on an Argand diagram the points \(P\), \(Q\) and \(R\) representing the complex numbers \(z\), \(\frac{1}{z}\) and \(z+\frac{1}{z}\) respectively. Indicate on your diagram the value of \(\angle PRQ\).[3]