| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2012 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.8 This FP2 question requires converting a complex modulus equation to Cartesian form to identify a circle (involving algebraic manipulation and completing the square), sketching an argument locus (a half-line), then finding their intersection algebraically. While systematic, it demands multiple techniques, careful algebra, and coordinate geometry—more challenging than standard C3/C4 but typical for Further Maths loci problems. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
\begin{enumerate}
\item The point $P$ represents a complex number $z$ on an Argand diagram such that
\end{enumerate}
$$| z - 6 \mathrm { i } | = 2 | z - 3 |$$
(a) Show that, as $z$ varies, the locus of $P$ is a circle, stating the radius and the coordinates of the centre of this circle.
The point $Q$ represents a complex number $z$ on an Argand diagram such that
$$\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }$$
(b) Sketch, on the same Argand diagram, the locus of $P$ and the locus of $Q$ as $z$ varies.\\
(c) Find the complex number for which both $| z - 6 \mathrm { i } | = 2 | z - 3 |$ and $\arg ( z - 6 ) = - \frac { 3 \pi } { 4 }$
\hfill \mbox{\textit{Edexcel FP2 2012 Q8 [14]}}