| Exam Board | Edexcel |
|---|---|
| Module | FP2 AS (Further Pure 2 AS) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Circle of Apollonius locus |
| Difficulty | Challenging +1.8 This FP2 question requires understanding circle of Apollonius loci, using the tangency condition to find α, then optimizing an argument constraint. Part (b) requires geometric insight about tangency to the imaginary axis, and part (c) involves finding the tangent line from the origin to determine the maximum argument—requiring multiple sophisticated techniques but following a clear logical path once the setup is understood. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Circle touching the imaginary axis | M1 | Any circle drawn tangentially to imaginary axis, any quadrant |
| Correct circle, centre on line \(y = 3\), in quadrant 1, tangent to imaginary axis | A1 | Must be in quadrant 1 with centre on \(y=3\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Circle must touch at \(3i\); \(\ | 3i-(-3+3i)\ | = \alpha\ |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\ | x+iy-(-3+3i)\ | = 3\ |
| Centre is \(\left(\frac{3}{2}, 3\right)\) | A1 | Correct centre stated or implied |
| Radius is \(\frac{3}{2}\) | A1 | Correct radius stated or implied |
| \(\beta = \theta - \phi\) where \(\theta = \arctan\!\left(\frac{3}{3/2}\right)\), or \(\phi = \arcsin\!\left(\dfrac{3/2}{\sqrt{(3/2)^2+3^2}}\right)\) | M1 | Realises right-angle triangle and uses it to find one relevant angle |
| Attempts both \(\theta = \arctan\!\left(\frac{3}{3/2}\right)\) and \(\phi = \arcsin\!\left(\dfrac{3/2}{\sqrt{(3/2)^2+3^2}}\right)\); \(\Rightarrow \beta = \arctan 2 - \arcsin\dfrac{\sqrt{5}}{5}\) | M1 | Full method: finds both angles and subtracts; or uses \(\frac{\pi}{2} - 2\arcsin\frac{1}{\sqrt{5}}\) |
| \(\beta \approx 0.644\) | A1 | Correct answer, awrt \(0.644\) |
# Question 4:
## Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Circle touching the imaginary axis | M1 | Any circle drawn tangentially to imaginary axis, any quadrant |
| Correct circle, centre on line $y = 3$, in quadrant 1, tangent to imaginary axis | A1 | Must be in quadrant 1 with centre on $y=3$ |
## Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Circle must touch at $3i$; $\|3i-(-3+3i)\| = \alpha\|3i-(1+3i)\| \Rightarrow 3 = \alpha\|-1\| \Rightarrow \alpha = 3$ | B1 | Correct explanation given |
## Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\|x+iy-(-3+3i)\| = 3\|x+iy-(1+3i)\| \Rightarrow (x+3)^2+(y-3)^2 = 9(x-1)^2+9(y-3)^2$ $\Rightarrow \left(x-\frac{3}{2}\right)^2+(y-3)^2 = \frac{9}{4}$ | M1 | Correct method to deduce centre or radius; may use geometry or substitute $z=x+iy$ |
| Centre is $\left(\frac{3}{2}, 3\right)$ | A1 | Correct centre stated or implied |
| Radius is $\frac{3}{2}$ | A1 | Correct radius stated or implied |
| $\beta = \theta - \phi$ where $\theta = \arctan\!\left(\frac{3}{3/2}\right)$, or $\phi = \arcsin\!\left(\dfrac{3/2}{\sqrt{(3/2)^2+3^2}}\right)$ | M1 | Realises right-angle triangle and uses it to find one relevant angle |
| Attempts both $\theta = \arctan\!\left(\frac{3}{3/2}\right)$ and $\phi = \arcsin\!\left(\dfrac{3/2}{\sqrt{(3/2)^2+3^2}}\right)$; $\Rightarrow \beta = \arctan 2 - \arcsin\dfrac{\sqrt{5}}{5}$ | M1 | Full method: finds both angles and subtracts; or uses $\frac{\pi}{2} - 2\arcsin\frac{1}{\sqrt{5}}$ |
| $\beta \approx 0.644$ | A1 | Correct answer, awrt $0.644$ |
---\begin{enumerate}
\item A circle $C$ in the complex plane has equation
\end{enumerate}
$$| z - ( - 3 + 3 i ) | = \alpha | z - ( 1 + 3 i ) |$$
where $\alpha$ is a real constant with $\alpha > 1$\\
Given that the imaginary axis is a tangent to $C$\\
(a) sketch, on an Argand diagram, the circle $C$\\
(b) explain why the value of $\alpha$ is 3
The circle $C$ is contained in the region
$$R = \left\{ z \in \mathbb { C } : \beta \leqslant \arg z \leqslant \frac { \pi } { 2 } \right\}$$
(c) Determine the maximum value of $\beta$
Give your answer in radians to 3 significant figures.
\hfill \mbox{\textit{Edexcel FP2 AS 2024 Q4 [9]}}