| Exam Board | Edexcel |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2002 |
| Session | June |
| Marks | 7 |
| 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 Further Maths question with standard techniques: part (a) requires routine manipulation of modulus equations to find a circle of Apollonius (a standard FP2 exercise), and part (b) involves composing two simple transformations by direct substitution. Both parts are textbook-standard with no novel insight required, making it slightly easier than average even for Further Maths. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \( | x+(y-2)i | =2 |
| \(\therefore x^2+(y-2)^2=4(x^2+(y+1)^2)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3x^2+3y^2+12y=0\) (any correct form; 3 terms) | A1 | (2) |
| Sketch circle | B1 | |
| Centre \((0,-2)\) | B1 | |
| \(r=2\) or touches axis | B1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(w=3(z-7+11i)\) | B1 | |
| \(=3z-21+33i\) | B1 | (2) |
| (7 marks) |
# Question 9:
## Part (a)(i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $|x+(y-2)i|=2|x+(y+i)|$ | M1 | |
| $\therefore x^2+(y-2)^2=4(x^2+(y+1)^2)$ | | |
## Part (a)(ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3x^2+3y^2+12y=0$ (any correct form; 3 terms) | A1 | **(2)** |
| Sketch circle | B1 | |
| Centre $(0,-2)$ | B1 | |
| $r=2$ or touches axis | B1 | **(3)** |
## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $w=3(z-7+11i)$ | B1 | |
| $=3z-21+33i$ | B1 | **(2)** |
| | | **(7 marks)** |
---9. (a) The point $P$ represents a complex number $z$ in an Argand diagram. Given that
$$| z - 2 i | = 2 | z + i |$$
\begin{enumerate}[label=(\roman*)]
\item find a cartesian equation for the locus of $P$, simplifying your answer.
\item sketch the locus of $P$.\\
(b) A transformation $T$ from the $z$-plane to the $w$-plane is a translation $- 7 + 11$ i followed by an enlargement with centre the origin and scale factor 3 .
Write down the transformation $T$ in the form
$$w = a z + b , \quad a , b \in \mathbb { C }$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP2 2002 Q9 [7]}}