| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Challenging +1.2 This is a standard FP2 loci question with routine techniques: (a-b) require basic circle identification and sketching (straightforward), while (c) involves algebraic manipulation of a complex transformation to show w is real implies |z-i|=2. The transformation work requires careful algebra but follows standard methods taught in FP2. More challenging than typical A-level pure maths due to Further Maths content, but represents a standard textbook exercise rather than requiring novel insight. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
The point $P$ represents the complex number $z$ on an Argand diagram, where
$$|z - i| = 2.$$
The locus of $P$ as $z$ varies is the curve $C$.
\begin{enumerate}[label=(\alph*)]
\item Find a cartesian equation of $C$. [2]
\item Sketch the curve $C$. [2]
\end{enumerate}
A transformation $T$ from the $z$-plane to the $w$-plane is given by
$$w = \frac{z + i}{3 + iz}, \quad z \neq 3i.$$
The point $Q$ is mapped by $T$ onto the point $R$. Given that $R$ lies on the real axis,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item show that $Q$ lies on $C$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q5 [9]}}