| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Arithmetic |
| Type | Geometric properties using complex numbers |
| Difficulty | Standard +0.3 This is a standard FP1 complex numbers question requiring routine techniques: complex division by multiplying by conjugate, plotting on Argand diagram, verifying perpendicularity using arguments or dot product, and finding circle properties. While it has multiple parts (10 marks total), each step follows textbook methods without requiring novel insight. The perpendicularity result and circle through origin make the geometry straightforward. Slightly easier than average A-level due to its structured, procedural nature. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
Given that $z_1 = 3 + 2i$ and $z_2 = \frac{12 - 5i}{z_1}$.
\begin{enumerate}[label=(\alph*)]
\item Find $z_2$ in the form $a + ib$, where $a$ and $b$ are real.
[2]
\item Show, on an Argand diagram, the point $P$ representing $z_1$ and the point $Q$ representing $z_2$.
[2]
\item Given that $O$ is the origin, show that $\angle POQ = \frac{\pi}{2}$.
[2]
\end{enumerate}
The circle passing through the points $O$, $P$ and $Q$ has centre $C$. Find
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item the complex number represented by $C$,
[2]
\item the exact value of the radius of the circle.
[2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 Q9 [10]}}