| Exam Board | WJEC |
|---|---|
| Module | Further Unit 1 (Further Unit 1) |
| Year | 2018 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Perpendicular bisector locus |
| Difficulty | Standard +0.3 This is a standard Further Maths locus question requiring students to convert a complex modulus equation to Cartesian form by substituting z = x + iy and equating distances. The geometric interpretation (perpendicular bisector) is straightforward once the equation is found. While it requires multiple algebraic steps, it follows a well-practiced technique with no novel insight needed. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer: Putting \(z = x + iy\): \( | x + iy - 4 - i | = |
| Answer: \( | (x-4) + i(y-1) | = |
| Answer: \((x-4)^2 + (y-1)^2 = (x+2)^2 + y^2\); \(x^2 - 8x + 16 + y^2 - 2y + 1 = x^2 + 4x + 4 + y^2\); \(12x + 2y - 13 = 0\) | A1 | oe |
| Answer | Marks |
|---|---|
| Answer: It is the perpendicular bisector of the line joining the points \((4, 1)\) and \((-2, 0)\) OR The locus of \(P\) is all the points which are equidistant from \((4,1)\) and \((-2,0)\). | B1 or (B1) |
**Part a)**
Answer: Putting $z = x + iy$: $|x + iy - 4 - i| = |x + iy + 2|$ | M1 A1 m1 |
Answer: $|(x-4) + i(y-1)| = |(x+2) + iy|$ |
Answer: $(x-4)^2 + (y-1)^2 = (x+2)^2 + y^2$; $x^2 - 8x + 16 + y^2 - 2y + 1 = x^2 + 4x + 4 + y^2$; $12x + 2y - 13 = 0$ | A1 | oe
**Part b)**
Answer: It is the perpendicular bisector of the line joining the points $(4, 1)$ and $(-2, 0)$ OR The locus of $P$ is all the points which are equidistant from $(4,1)$ and $(-2,0)$. | B1 or (B1) |The complex number $z$ is represented by the point $P(x, y)$ in the Argand diagram and
$$|z - 4 - \mathrm{i}| = |z + 2|.$$
\begin{enumerate}[label=(\alph*)]
\item Find the equation of the locus of $P$. [4]
\item Give a geometric interpretation of the locus of $P$. [1]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 1 2018 Q7 [5]}}