| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Region shading with multiple inequalities |
| Difficulty | Standard +0.3 This is a straightforward Argand diagram question requiring students to sketch a circle (center 3+2i, radius 1) intersected with a half-plane (Im z ≥ 2), then find the maximum argument geometrically. While it involves multiple steps, each component is standard A-level material with no novel insight required—slightly easier than average due to the routine nature of loci sketching and basic geometric reasoning. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show circle with centre \(3 + 2i\) | B1 | |
| Show circle with radius 1. Must match *their* scales: if scales not identical should have an ellipse | B1 | |
| Show line \(y = 2\) in at least the diameter of a circle in the first quadrant | B1 | |
| Shade the correct region in a correct diagram | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identify the correct point | B1 | |
| Carry out a correct method for finding the argument | M1 | e.g. \(\arg x = \tan^{-1}\frac{2}{3} + \sin^{-1}\frac{1}{\sqrt{13}}\). Exact working required |
| Obtain answer \(49.8°\) | A1 | Or better. \(0.869\) radians scores B1M1A0. Special Case 1: B1M0 for \(45°\) if they have shaded the wrong half of the circle. Special Case 2: 3 out of 3 available if they identify the correct point on the correct circle and it is consistent with *their* shading |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show circle with centre $3 + 2i$ | **B1** | |
| Show circle with radius 1. Must match *their* scales: if scales not identical should have an ellipse | **B1** | |
| Show line $y = 2$ in at least the diameter of a circle in the first quadrant | **B1** | |
| Shade the correct region in a correct diagram | **B1** | |
**Total: 4 marks**
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Identify the correct point | **B1** | |
| Carry out a correct method for finding the argument | **M1** | e.g. $\arg x = \tan^{-1}\frac{2}{3} + \sin^{-1}\frac{1}{\sqrt{13}}$. Exact working required |
| Obtain answer $49.8°$ | **A1** | Or better. $0.869$ radians scores **B1M1A0**. Special Case 1: **B1M0** for $45°$ if they have shaded the wrong half of the circle. Special Case 2: 3 out of 3 available if they identify the correct point on the correct circle and it is consistent with *their* shading |
**Total: 3 marks**
---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 $| z - 3 - 2 \mathbf { i } | \leqslant 1$ and $\operatorname { Im } z \geqslant 2$.
\item Find the greatest value of $\arg z$ for points in the shaded region, giving your answer in degrees.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q5 [7]}}