| Exam Board | WJEC |
|---|---|
| Module | Further Unit 1 (Further Unit 1) |
| Year | 2019 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Complex transformations and mappings |
| Difficulty | Standard +0.3 This is a straightforward complex transformation question requiring basic algebraic manipulation. Part (a) involves expanding z² = (x+iy)² and separating real/imaginary parts—a standard technique. Part (b) substitutes y=3x into the expressions from (a) to eliminate the parameter, which is routine. No geometric insight or novel problem-solving is required, making this slightly easier than average. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation |
| Answer | Marks |
|---|---|
| \(u + iv = (x + iy)^2 - 1 = x^2 - y^2 + 2ixy - 1\). Comparing coefficients: Imaginary parts: \(v = 2xy\) (given); Real parts: \(u = x^2 - y^2 - 1\) | M1, A1, m1 |
| Answer | Marks | Guidance |
|---|---|---|
| Putting \(v = 3x\): \(v = 2x \times 3x = 6x^2\), \(u = x^2 - 9x^2 - 1 (= -8x^2 - 1)\) | A1, M1 | A1 for both \(u\) and \(v\); FT (a) |
| Eliminating \(x\), the equation of the locus \(Q\) is: \(u = -8\left(\frac{v}{6}\right) - 1\) oe simplified | M1, A1 | cao |
**Part (a):**
$u + iv = (x + iy)^2 - 1 = x^2 - y^2 + 2ixy - 1$. Comparing coefficients: Imaginary parts: $v = 2xy$ (given); Real parts: $u = x^2 - y^2 - 1$ | M1, A1, m1 |
**Part (b):**
Putting $v = 3x$: $v = 2x \times 3x = 6x^2$, $u = x^2 - 9x^2 - 1 (= -8x^2 - 1)$ | A1, M1 | A1 for both $u$ and $v$; FT (a)
Eliminating $x$, the equation of the locus $Q$ is: $u = -8\left(\frac{v}{6}\right) - 1$ oe simplified | M1, A1 | cao
---9. The complex numbers $z$ and $w$ are represented by the points $P ( x , y )$ and $Q ( u , v )$ respectively in Argand diagrams and
$$w = z ^ { 2 } - 1$$
\begin{enumerate}[label=(\alph*)]
\item Show that $v = 2 x y$ and obtain an expression for $u$ in terms of $x$ and $y$.
\item The point $P$ moves along the line $y = 3 x$. Find the equation of the locus of $Q$.
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 1 2019 Q9 [8]}}