| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex transformations (Möbius) |
| Difficulty | Challenging +1.3 This is a standard FP2 complex transformations question requiring systematic application of Möbius transformation techniques. Part (a) involves substituting z = re^(iπ/4) and showing |w| = 1 through algebraic manipulation. Part (b) requires finding the image of |z| = 1, typically yielding another circle or line, using standard methods (substitution or finding fixed points). Parts (c-d) are routine sketching. While requiring multiple techniques and careful algebra, this follows well-established FP2 patterns without requiring novel insight, making it moderately above average difficulty. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction4.02m Geometrical effects: multiplication and division |
The transformation $T$ from the complex $z$-plane to the complex $w$-plane is given by
$$w = \frac{z + 1}{z + i}, \quad z \neq -i.$$
\begin{enumerate}[label=(\alph*)]
\item Show that $T$ maps points on the half-line $\arg(z) = \frac{\pi}{4}$ in the $z$-plane into points on the circle $|w| = 1$ in the $w$-plane. [4]
\item Find the image under $T$ in the $w$-plane of the circle $|z| = 1$ in the $z$-plane. [6]
\item Sketch on separate diagrams the circle $|z| = 1$ in the $z$-plane and its image under $T$ in the $w$-plane. [2]
\item Mark on your sketches the point $P$, where $z = i$, and its image $Q$ under $T$ in the $w$-plane. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q35 [14]}}