| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Geometric relationships on Argand diagram |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring complex division, Argand diagram interpretation, angle calculation using arguments, and finding the circumcircle of a right triangle. While each individual step uses standard techniques (complex division by multiplying by conjugate, argument properties, circle geometry), the combination of multiple concepts and the need to recognize the right-angle property and apply circle theorems elevates this above average difficulty for A-level, though it remains a structured question with clear progression. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z_2 = \dfrac{12-5i}{3+2i} \times \dfrac{3-2i}{3-2i} = \dfrac{36-24i-15i-10}{13} = 2-3i\) | M1 A1 (2) | Multiply by \(\dfrac{3-2i}{3-2i}\) for M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Plot \(P(3, 2)\) and \(Q(2, -3)\) on Argand diagram | B1, B1ft (2) | Position of points not clear: award B1B0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| grad. \(OP \times\) grad. \(OQ = \dfrac{2}{3} \times -\dfrac{3}{2} = -1 \Rightarrow \angle POQ = \dfrac{\pi}{2}\) | M1 A1 (2) | Use of calculator/decimals: award M1A0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(z = \dfrac{3+2}{2} + \dfrac{2+(-3)}{2}i = \dfrac{5}{2} - \dfrac{1}{2}i\) | M1 A1 (2) | Final answer must be in complex form for A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(r = \sqrt{\left(\dfrac{5}{2}\right)^2 + \left(-\dfrac{1}{2}\right)^2} = \dfrac{\sqrt{26}}{2}\) or exact equivalent | M1 A1 (2) [10] | Radius or diameter for M1 |
## Question 9:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z_2 = \dfrac{12-5i}{3+2i} \times \dfrac{3-2i}{3-2i} = \dfrac{36-24i-15i-10}{13} = 2-3i$ | M1 A1 (2) | Multiply by $\dfrac{3-2i}{3-2i}$ for M1 |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Plot $P(3, 2)$ and $Q(2, -3)$ on Argand diagram | B1, B1ft (2) | Position of points not clear: award B1B0 |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| grad. $OP \times$ grad. $OQ = \dfrac{2}{3} \times -\dfrac{3}{2} = -1 \Rightarrow \angle POQ = \dfrac{\pi}{2}$ | M1 A1 (2) | Use of calculator/decimals: award M1A0 |
### Part (d):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $z = \dfrac{3+2}{2} + \dfrac{2+(-3)}{2}i = \dfrac{5}{2} - \dfrac{1}{2}i$ | M1 A1 (2) | Final answer must be in complex form for A1 |
### Part (e):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $r = \sqrt{\left(\dfrac{5}{2}\right)^2 + \left(-\dfrac{1}{2}\right)^2} = \dfrac{\sqrt{26}}{2}$ or exact equivalent | M1 A1 (2) [10] | Radius or diameter for M1 |
---9. Given that $z _ { 1 } = 3 + 2 i$ and $z _ { 2 } = \frac { 12 - 5 i } { z _ { 1 } }$,
\begin{enumerate}[label=(\alph*)]
\item find $z _ { 2 }$ in the form $a + i b$, where $a$ and $b$ are real.
\item Show on an Argand diagram the point $P$ representing $z _ { 1 }$ and the point $Q$ representing $z _ { 2 }$.
\item Given that $O$ is the origin, show that $\angle P O Q = \frac { \pi } { 2 }$.
The circle passing through the points $O , P$ and $Q$ has centre $C$. Find
\item the complex number represented by C,
\item the exact value of the radius of the circle.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2009 Q9 [10]}}