| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2003 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Standard +0.8 This FP2 question requires sketching multiple loci (circle and half-lines), identifying a region, then proving a transformation maps a circle to a perpendicular bisector, and determining the image region. While systematic, it demands solid understanding of complex transformations, geometric interpretation, and multi-step reasoning beyond standard A-level, though it's a fairly typical FP2 question rather than exceptionally challenging. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
\begin{enumerate}
\item (i) (a) On the same Argand diagram sketch the loci given by the following equations.
\end{enumerate}
$$| z - 1 | = 1 , \quad , , \arg ( z + 1 ) = \frac { \pi } { 12 } , \quad , \arg ( z + 1 ) = \frac { \pi } { 2 }$$
(b) Shade on your diagram the region for which
$$| z - 1 | \leq 1 \quad \text { and } \quad \frac { \pi } { 12 } \leq \arg ( z + 1 ) \leq \frac { \pi } { 2 }$$
(ii) (a) Show that the transformation $\quad w = \frac { z - 1 } { z } , \quad z \neq 0$,
$$\text { maps } | z - 1 | = 1 \text { in the } \boldsymbol { z } \text {-plane onto } | w | = | w - 1 | \text { in the } \boldsymbol { w } \text {-plane. }$$
The region $| z - 1 | \leq 1$ in the $z$-plane is mapped onto the region $T$ in the $w$-plane.\\
(b) Shade the region $T$ on an Argand diagram.\\
\hfill \mbox{\textit{Edexcel FP2 2003 Q1 [10]}}