| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | March |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.8 This question requires understanding of loci in the complex plane (circle and perpendicular bisector), finding their intersection, and then optimizing arg(z) over the region. While the individual components are standard A-level topics, combining geometric visualization with the optimization of argument requires solid spatial reasoning and is more demanding than routine loci questions. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle with centre \(4 + 2\)i | B1 | |
| Show a circle with radius 3 and centre not at the origin | B1 | |
| Show the straight line \(\text{Re}(z) = 5\) | B1 | |
| Shade the correct region. Allow even if radius 3 mark not gained or shown incorrectly | B1 | If 4 and 6 seen on diagram and line is at mid point, but 5 not marked, allow final two B1 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out a complete method for finding the greatest value of \(\arg z\) | M1 | e.g. \(\tan^{-1}\frac{2 + 2\sqrt{2}}{5}\). Allow \(2\sqrt{2}\) as \(\sqrt{3^2 - 1^2}\) |
| Obtain answer \(0.768\) radians or \(44.0°\) | A1 | |
| SC B1: \(\tan^{-1}(2/4) + \sin^{-1}(3/\sqrt{4^2+2^2}) = 26.565° + 42.130° = 68.695°\); \(68.7°\) or \([1.19896]\) \(1.20\) radians |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $4 + 2$i | B1 | |
| Show a circle with radius 3 and centre not at the origin | B1 | |
| Show the straight line $\text{Re}(z) = 5$ | B1 | |
| Shade the correct region. Allow even if radius 3 mark not gained or shown incorrectly | B1 | If 4 and 6 seen on diagram and line is at mid point, but 5 not marked, allow final two B1 marks |
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out a complete method for finding the greatest value of $\arg z$ | M1 | e.g. $\tan^{-1}\frac{2 + 2\sqrt{2}}{5}$. Allow $2\sqrt{2}$ as $\sqrt{3^2 - 1^2}$ |
| Obtain answer $0.768$ radians or $44.0°$ | A1 | |
| | | **SC** B1: $\tan^{-1}(2/4) + \sin^{-1}(3/\sqrt{4^2+2^2}) = 26.565° + 42.130° = 68.695°$; $68.7°$ or $[1.19896]$ $1.20$ radians |
---5
\begin{enumerate}[label=(\alph*)]
\item On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $| z - 4 - 2 i | \leqslant 3$ and $| z | \geqslant | 10 - z |$.
\item Find the greatest value of $\arg z$ for points in this region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q5 [6]}}