| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2015 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Perpendicular bisector locus |
| Difficulty | Standard +0.3 This is a straightforward Further Maths locus question requiring standard techniques: recognizing |z-a|=|z-b| gives a perpendicular bisector, finding axis intercepts by substitution, calculating a midpoint, and finding a circle through three points. All steps are routine applications of well-practiced methods with no novel insight required. While it's Further Maths content, the execution is mechanical and below average difficulty even for FP2. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| \[ | z - 2 + 4i | = |
| Answer | Marks | Guidance |
|---|---|---|
| The point \(O\) is the origin of the Argand diagram. Find the equation of the circle that passes through the points \(O\), \(A\) and \(B\), giving your answer in the form \( | z - a | = k\). |
The locus of points, $L$, satisfies the equation
$$|z - 2 + 4i| = |z|$$
**(a) [3 marks]**
Sketch $L$ on the Argand diagram.
**(b) [6 marks total]**
The locus $L$ cuts the real axis at $A$ and the imaginary axis at $B$.
**(i) [4 marks]**
Show that the complex number represented by $C$, the midpoint of $AB$, is $\frac{5}{2} - \frac{5}{4}i$
**(ii) [2 marks]**
The point $O$ is the origin of the Argand diagram. Find the equation of the circle that passes through the points $O$, $A$ and $B$, giving your answer in the form $|z - a| = k$.5 The locus of points, $L$, satisfies the equation
$$| z - 2 + 4 \mathrm { i } | = | z |$$
\begin{enumerate}[label=(\alph*)]
\item Sketch $L$ on the Argand diagram below.
\item The locus $L$ cuts the real axis at $A$ and the imaginary axis at $B$.
\begin{enumerate}[label=(\roman*)]
\item Show that the complex number represented by $C$, the midpoint of $A B$, is
$$\frac { 5 } { 2 } - \frac { 5 } { 4 } \mathrm { i }$$
\item The point $O$ is the origin of the Argand diagram. Find the equation of the circle that passes through the points $O , A$ and $B$, giving your answer in the form $| z - \alpha | = k$.\\[0pt]
[2 marks]
\section*{(a)}
\begin{center}
\includegraphics[max width=\textwidth, alt={}]{bc3aaed2-4aef-4aec-b657-098b1e581e55-10_1173_1242_1217_463}
\end{center}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2015 Q5 [9]}}