| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Regions under complex transformations |
| Difficulty | Challenging +1.2 This is a standard Further Maths FP2 question on Möbius transformations mapping circles to circles. Part (a) requires algebraic manipulation (substituting z = w(z-i), using |z|=3, and completing the square) which is methodical but involves several steps. Part (b) requires understanding which side of the circle maps to which region, typically done by testing a point. While this is a multi-step problem requiring FP2 knowledge, it follows a well-established technique taught explicitly in the syllabus with minimal novel insight needed. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation |
A transformation $T$ from the $z$-plane to the $w$-plane is given by
$$w = \frac{z}{z-i}, \quad z \neq i.$$
The circle with equation $|z| = 3$ is mapped by $T$ onto the curve $C$.
\begin{enumerate}[label=(\alph*)]
\item Show that $C$ is a circle and find its centre and radius. [8]
\end{enumerate}
The region $|z| < 3$ in the $z$-plane is mapped by $T$ onto the region $R$ in the $w$-plane.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Shade the region $R$ on an Argand diagram. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q6 [10]}}