Loci of complex numbers - Complex numbers 2

9 marks Standard +0.3

8

Showing all necessary working, express the complex number \(\frac { 2 + 3 \mathrm { i } } { 1 - 2 \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\). Give the values of \(r\) and \(\theta\) correct to 3 significant figures.
On an Argand diagram sketch the locus of points representing complex numbers \(z\) satisfying the equation \(| z - 3 + 2 i | = 1\). Find the least value of \(| z |\) for points on this locus, giving your answer in an exact form.

3 marks Moderate -0.8

1

By writing \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), show that \(z z ^ { * } = | z | ^ { 2 }\).
Given that \(z z ^ { * } = 9\), describe the locus of \(z\).

18 marks Standard +0.3

1

1. Differentiate with respect to \(x\) the equation \(a \tan y = x\) (where \(a\) is a constant), and hence show that the derivative of \(\arctan \frac { x } { a }\) is \(\frac { a } { a ^ { 2 } + x ^ { 2 } }\).
2. By first expressing \(x ^ { 2 } - 4 x + 8\) in completed square form, evaluate the integral \(\int _ { 0 } ^ { 4 } \frac { 1 } { x ^ { 2 } - 4 x + 8 } \mathrm {~d} x\), giving your answer exactly.
3. Use integration by parts to find \(\int \arctan x \mathrm {~d} x\).
1. A curve has polar equation \(r = 2 \cos \theta\), for \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Show, by considering its cartesian equation, that the curve is a circle. State the centre and radius of the circle.
2. Another circle has radius 2 and its centre, in cartesian coordinates, is ( 0,2 ). Find the polar equation of this circle.