| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | February |
| Marks | 9 |
| Topic | Complex Numbers Argand & Loci |
| Type | Circle equations in complex form |
| Difficulty | Standard +0.3 This is a straightforward Further Maths complex numbers question requiring standard techniques: finding circle center/radius from diameter endpoints (midpoint formula and distance), then solving simultaneous circle equations. The intersection calculation involves algebraic manipulation but follows a routine method. While it's a multi-part question worth 9 marks, it requires no novel insight—just careful application of well-practiced techniques, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation |
In an Argand diagram, the points $A$ and $B$ are represented by the complex numbers $-3 + 2i$ and $5 - 4i$ respectively. The points $A$ and $B$ are the end points of a diameter of a circle $C$.
\begin{enumerate}[label=(\alph*)]
\item Find the equation of $C$, giving your answer in the form
$$|z - a| = b \quad a \in \mathbb{C}, \quad b \in \mathbb{R}$$
[3]
\end{enumerate}
The circle $D$, with equation $|z - 2 - 3i| = 2$, intersects $C$ at the points representing the complex numbers $z_1$ and $z_2$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the complex numbers $z_1$ and $z_2$. [6]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q5 [9]}}