| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Single locus sketching |
| Difficulty | Moderate -0.3 This question requires understanding that |z|=2 gives a circle of radius 2, restricted to a sector by the argument bounds, and that z² doubles the argument and squares the modulus. While it involves multiple concepts (modulus, argument, transformation), these are standard operations with straightforward geometric interpretations that students practice routinely in P3/Further Pure courses. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show an arc of a circle, centre the origin and radius 2. Only need 2 on \(\text{Re}(z)\) or \(2i\) on \(\text{Im}(z)\) or \(r = 2\) to show correct radius | B1 | |
| Show an arc centre the origin for \(0 \leqslant \arg z \leqslant \frac{1}{4}\pi\) with any radius | B1 | |
| Max B1 if sector shaded | ||
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show an arc of a circle, centre the origin and radius 4. Only need 4 on \(\text{Re}(z)\) or \(4i\) on \(\text{Im}(z)\) or \(r = 4\) to show correct radius | B1 | |
| Show an arc centre the origin for \(0 \leqslant \arg z \leqslant \frac{1}{2}\pi\) with any radius | B1 | |
| Max B1 if sector shaded | ||
| Total | 2 |
## Question 1:
### Part (a):
*Note: For all 4 marks, scales must be approximately equal, dashes can replace numbers. Arcs don't have to be perfectly circular, mark intention.*
| Answer | Mark | Guidance |
|--------|------|----------|
| Show an arc of a circle, centre the origin and radius 2. Only need 2 on $\text{Re}(z)$ or $2i$ on $\text{Im}(z)$ or $r = 2$ to show correct radius | B1 | |
| Show an arc centre the origin for $0 \leqslant \arg z \leqslant \frac{1}{4}\pi$ with any radius | B1 | |
| Max B1 if sector shaded | | |
| **Total** | **2** | |
---
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show an arc of a circle, centre the origin and radius 4. Only need 4 on $\text{Re}(z)$ or $4i$ on $\text{Im}(z)$ or $r = 4$ to show correct radius | B1 | |
| Show an arc centre the origin for $0 \leqslant \arg z \leqslant \frac{1}{2}\pi$ with any radius | B1 | |
| Max B1 if sector shaded | | |
| **Total** | **2** | |1 The complex number $z$ satisfies $| z | = 2$ and $0 \leqslant \arg z \leqslant \frac { 1 } { 4 } \pi$.
\begin{enumerate}[label=(\alph*)]
\item On the Argand diagram below, sketch the locus of the points representing $z$.
\item On the same diagram, sketch the locus of the points representing $z ^ { 2 }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1074_1363_628_351}\\
\includegraphics[max width=\textwidth, alt={}, center]{6280ab81-0bdb-47b4-8651-bff1261a0adf-02_1002_26_1820_2017}
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q1 [4]}}