| Exam Board | WJEC |
|---|---|
| Module | Further Unit 1 (Further Unit 1) |
| Year | 2024 |
| 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 is a Further Maths question requiring algebraic manipulation of complex fractions to find real/imaginary parts, then substituting a constraint to find a locus. It involves multiple steps including rationalizing complex denominators and algebraic simplification, but follows standard techniques for complex transformations without requiring exceptional insight. |
| 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 |
\begin{enumerate}
\item The complex numbers $z$ and $w$ are represented, respectively, by the points $P ( x , y )$ and $Q ( u , v )$ in Argand diagrams and
\end{enumerate}
$$w = \frac { z } { 1 - z }$$
(a) Show that $v = \frac { y } { ( 1 - x ) ^ { 2 } + y ^ { 2 } }$ and obtain an expression for $u$ in terms of $x$ and $y$.\\
(b) The point $P$ moves along the line $y = 1 - x$. Find and simplify the equation of the locus of $Q$.\\
\hfill \mbox{\textit{WJEC Further Unit 1 2024 Q4 [10]}}