| Exam Board | AQA |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Optimization of argument on loci |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question requiring visualization of loci and optimization of argument on a circle. While it involves several steps (drawing loci, finding circle equation, optimizing argument), each component uses standard FP2 techniques: basic locus sketching, geometric reasoning about tangency, and finding extreme arguments. The optimization in part (b)(iii) requires some geometric insight but follows a predictable pattern for this topic. Slightly above average difficulty due to the multi-step nature and the optimization component, but well within standard FP2 scope. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
2
\begin{enumerate}[label=(\alph*)]
\item Draw on an Argand diagram the locus $L$ of points satisfying the equation $\arg z = \frac { \pi } { 6 }$.\\
(1 mark)
\item \begin{enumerate}[label=(\roman*)]
\item A circle $C$, of radius 6, has its centre lying on $L$ and touches the line $\operatorname { Re } ( z ) = 0$.
Draw $C$ on your Argand diagram from part (a).
\item Find the equation of $C$, giving your answer in the form $\left| z - z _ { 0 } \right| = k$.
\item The complex number $z _ { 1 }$ lies on $C$ and is such that $\arg z _ { 1 }$ has its least possible value. Find $\arg z _ { 1 }$, giving your answer in the form $p \pi$, where $- 1 < p \leqslant 1$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP2 2012 Q2 [8]}}