FP2 June 2009 Q6
6. A transformation \(T\) from the \(z\)-plane to the \(w\)-plane is given by \[w = \frac{z}{z + \mathrm{i}}, \quad z \neq -\mathrm{i}\]
The circle with equation \(|z| = 3\) is mapped by \(T\) onto the curve \(C\).
(a) Show that \(C\) is a circle and find its centre and radius. (8)
The region \(|z| < 3\) in the \(z\)-plane is mapped by \(T\) onto the region \(R\) in the \(w\)-plane.
(b) Shade the region \(R\) on an Argand diagram. (2)
| Scheme | Marks |
|---|---|
| \(w = \dfrac{z}{z + \mathrm{i}},\ z \neq -\mathrm{i}\) | |
| \(w(z + \mathrm{i}) = z \Rightarrow wz + \mathrm{i}w = z \Rightarrow \mathrm{i}w = z - wz\) \(\Rightarrow \mathrm{i}w = z(1 - w) \Rightarrow z = \dfrac{\mathrm{i}w}{(1 - w)}\) Complete method of rearranging to make \(z\) the subject. \(z = \frac{\mathrm{i}w}{(1 - w)}\) | M1 A1 aef |
| \(|z| = 3 \Rightarrow \left|\dfrac{\mathrm{i}w}{1 - w}\right| = 3\) Putting \(|z\) in terms of their \(w| = 3\) | dM1 |
| \(\left\{\begin{array}{l}|\mathrm{i}w| = 3|1 - w| \Rightarrow |w| = 3|w - 1| \Rightarrow |w|^2 = 9|w - 1|^2 \\ \Rightarrow |u + \mathrm{i}v|^2 = 9|u + \mathrm{i}v - 1|^2\end{array}\right\}\) | |
| \(\Rightarrow u^2 + v^2 = 9\left[(u - 1)^2 + v^2\right]\) Applies \(w = u + \mathrm{i}v\), and uses Pythagoras correctly to get an equation in terms of \(u\) and \(v\) without any i’s. Correct equation. | ddM1 A1 |
| \(\left\{\begin{array}{l}\Rightarrow u^2 + v^2 = 9u^2 - 18u + 9 + 9v^2 \\ \Rightarrow 0 = 8u^2 - 18u + 8v^2 + 9\end{array}\right\}\) | |
| \(\Rightarrow 0 = u^2 - \frac{9}{4}u + v^2 + \frac{9}{8}\) Simplifies down to \(u^2 + v^2 \pm \alpha u \pm \beta v \pm \delta = 0\). | dddM1 |
| \(\Rightarrow \left(u - \frac{9}{8}\right)^2 - \frac{81}{64} + v^2 + \frac{9}{8} = 0\) \(\Rightarrow \left(u - \frac{9}{8}\right)^2 + v^2 = \frac{9}{64}\) | |
| {Circle} centre \(\left(\frac{9}{8}, 0\right)\), radius \(\frac{3}{8}\) One of centre or radius correct. Both centre and radius correct. | A1 A1 |
| (8) |
Notes
(corrected from the printed mark scheme: the first line is printed “\(z = -\mathrm{i}\)”, a lost \(\neq\) sign)
| Scheme | Marks |
|---|---|
| Circle indicated on the Argand diagram in the correct position in follow through quadrants. Ignore plotted coordinates. | B1ft |
| Region outside a circle indicated only. | B1 |
| (2) | |
| (10 marks) |