| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Standard +0.3 This is a standard FP2 loci question with routine techniques: (a) sketching a circle is straightforward, (b) finding z from modulus and argument is a direct application of polar form, (c) the transformation mapping requires algebraic manipulation but follows a standard method (substitute z in terms of w, use the circle equation, simplify to get a line). All parts are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation |
A complex number $z$ is represented by the point $P$ in the Argand diagram. Given that
$$|z - 3i| = 3,$$
\begin{enumerate}[label=(\alph*)]
\item sketch the locus of $P$. [2]
\item Find the complex number $z$ which satisfies both $|z - 3i| = 3$ and $\arg (z - 3i) = \frac{3}{4}\pi$. [4]
The transformation $T$ from the $z$-plane to the $w$-plane is given by
$$w = \frac{2i}{w}.$$
\item Show that $T$ maps $|z - 3i| = 3$ to a line in the $w$-plane, and give the cartesian equation of this line. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 Q46 [11]}}