| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Intersection of two loci |
| Difficulty | Standard +0.8 This question requires understanding two different locus types (circle and perpendicular bisector), sketching them accurately, then using geometric reasoning to find the minimum distance between points on the loci. While the individual concepts are standard A-level, the synthesis of visualization and optimization through geometric insight elevates it above routine exercises. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle centre \((3, -2)\) | B1 | |
| Show a circle with radius \(2\), FT centre not at origin | B1FT | |
| Show the point representing \((-3, 4)\) or midpoint \((0, 1)\) | B1 | |
| Show the perpendicular bisector of the line joining \((-3, 4)\) and centre of circle; FT on position of \((-3,4)\) and centre | B1FT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out a correct method for finding the least value of \( | z - w | \) |
| Obtain \(\sqrt{18} - 2\) or \(3\sqrt{2} - 2\) | A1 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle centre $(3, -2)$ | B1 | |
| Show a circle with radius $2$, FT centre not at origin | B1FT | |
| Show the point representing $(-3, 4)$ or midpoint $(0, 1)$ | B1 | |
| Show the perpendicular bisector of the line joining $(-3, 4)$ and centre of circle; FT on position of $(-3,4)$ and centre | B1FT | |
## Question 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out a correct method for finding the least value of $|z - w|$ | M1 | Distance $(3,-2)$ to $(0,1)$ $- 2$ |
| Obtain $\sqrt{18} - 2$ or $3\sqrt{2} - 2$ | A1 | |
---7
\begin{enumerate}[label=(\alph*)]
\item On a single Argand diagram sketch the loci given by the equations $| z - 3 + 2 i | = 2$ and $| w - 3 + 2 \mathrm { i } | = | w + 3 - 4 \mathrm { i } |$ where z and $w$ are complex numbers.
\item Hence find the least value of $| \mathbf { z } - \mathbf { w } |$ for points on these loci. Give your answer in an exact form.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q7 [6]}}