| Exam Board | WJEC |
|---|---|
| Module | Further Unit 1 (Further Unit 1) |
| Year | 2023 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring algebraic manipulation of complex numbers (expanding z², separating real/imaginary parts), substituting a linear constraint to find a locus equation, and calculating perpendicular distance from a point to a line. While systematic, it demands careful algebra across multiple steps and understanding of complex-to-Cartesian transformations, placing it moderately above average difficulty. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
9. The complex numbers $z$ and $w$ are represented by the points $P ( x , y )$ and $Q ( u , v )$ respectively, in Argand diagrams, and $w = 1 - z ^ { 2 }$.
\begin{enumerate}[label=(\alph*)]
\item Find expressions for $u$ and $v$ in terms of $x$ and $y$.
\item The point $P$ moves along the line $y = 4 x$. Find the equation of the locus of $Q$.
\item Find the perpendicular distance of the point corresponding to $z = 2 + 5 \mathrm { i }$ in the $( u , v )$-plane, from the locus of $Q$.
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 1 2023 Q9 [12]}}