| Exam Board | Edexcel |
|---|---|
| Module | FP2 AS (Further Pure 2 AS) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Cartesian equation from argument condition |
| Difficulty | Challenging +1.2 This is a standard Further Maths locus problem requiring recognition that arg((z-a)/(z-b))=π/2 gives a semicircular arc, then finding the maximum distance from origin to points on this locus. While it requires geometric insight about argument loci and optimization, these are well-practiced FP2 techniques with straightforward execution once the setup is recognized. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Draws part of a circle with end points \((4,1)\) and \((2,7)\) | M1 | 1.1b |
| A semi-circle with end points \((4,1)\) and \((2,7)\) to the right | A1 | 1.1b |
| (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Centre \((3, 4)\) | B1 | 2.2a - Deduces the correct centre coordinates |
| \(r = \sqrt{(4-\text{"3"})^2 + (1-\text{"4"})^2}\) or \(r = \sqrt{(2-\text{"3"})^2 + (7-\text{"4"})^2}\) or \(r = \frac{1}{2}\sqrt{(4-2)^2 + (1-7)^2}\) | M1 | 1.1b - Uses Pythagoras and their centre to find the radius |
| \(d = \sqrt{(\text{"3"}-0)^2 + (\text{"4"}-0)^2}\) | M1 | 1.1b - Finds the distance from the origin to their centre using Pythagoras |
| Adds this distance to their radius | dM1 | 3.1a - Dependent on previous M mark; complete method to find maximum \( |
| \( | z | = 5 + \sqrt{10}\) |
| (5 marks) |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Draws part of a circle with end points $(4,1)$ and $(2,7)$ | M1 | 1.1b |
| A semi-circle with end points $(4,1)$ and $(2,7)$ to the right | A1 | 1.1b |
| **(2 marks)** | | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Centre $(3, 4)$ | B1 | 2.2a - Deduces the correct centre coordinates |
| $r = \sqrt{(4-\text{"3"})^2 + (1-\text{"4"})^2}$ or $r = \sqrt{(2-\text{"3"})^2 + (7-\text{"4"})^2}$ or $r = \frac{1}{2}\sqrt{(4-2)^2 + (1-7)^2}$ | M1 | 1.1b - Uses Pythagoras and their centre to find the radius |
| $d = \sqrt{(\text{"3"}-0)^2 + (\text{"4"}-0)^2}$ | M1 | 1.1b - Finds the distance from the origin to their centre using Pythagoras |
| Adds this distance to their radius | dM1 | 3.1a - Dependent on previous M mark; complete method to find maximum $|z|$ |
| $|z| = 5 + \sqrt{10}$ | A1 | 1.1b |
| **(5 marks)** | | |
---\begin{enumerate}
\item A complex number $z$ is represented by the point $P$ on an Argand diagram.
\end{enumerate}
Given that
$$\arg \left( \frac { z - 4 - i } { z - 2 - 7 i } \right) = \frac { \pi } { 2 }$$
(a) sketch the locus of $P$ as $z$ varies,\\
(b) determine the exact maximum possible value of $| z |$
\hfill \mbox{\textit{Edexcel FP2 AS 2023 Q3 [7]}}