| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Circle equations in complex form |
| Difficulty | Standard +0.8 This is a Further Maths FP2 question requiring understanding of arg loci (half-line from a point), geometric reasoning about a circle tangent to a horizontal line with center on that half-line, and coordinate geometry in the complex plane. The multi-step nature (visualizing the locus, determining tangency conditions, solving for the center) and the need to integrate geometric constraints makes this moderately challenging, though it follows standard FP2 techniques. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
(a) Sketch, on the Argand diagram below, the locus $L$ of points satisfying $\arg(z - 2i) = \frac{2\pi}{3}$
[3 marks]
(b)(i) A circle $C$, of radius $3$, has its centre lying on $L$ and touches the line $\text{Im}(z) = 2$.
Sketch $C$ on the Argand diagram used in part (a).
[2 marks]
(ii) Find the centre of $C$, giving your answer in the form $a + bi$.
[3 marks]2
\begin{enumerate}[label=(\alph*)]
\item Sketch, on the Argand diagram below, the locus $L$ of points satisfying
$$\arg ( z - 2 \mathrm { i } ) = \frac { 2 \pi } { 3 }$$
\item \begin{enumerate}[label=(\roman*)]
\item A circle $C$, of radius 3, has its centre lying on $L$ and touches the line $\operatorname { Im } ( z ) = 2$. Sketch $C$ on the Argand diagram used in part (a).
\item Find the centre of $C$, giving your answer in the form $a + b \mathrm { i }$.\\[0pt]
[3 marks]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2014 Q2 [8]}}