| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 6 |
| 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 loci question requiring students to sketch a circle centered at -2 with radius 2 and a horizontal line Im(z)=1, then find their intersection. Part (b) involves basic geometry to identify the point with maximum argument. Standard technique with minimal problem-solving beyond visualization. |
| Spec | 4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)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 \(-2\) | B1 | |
| Show a circle with radius 2 and centre not the origin | B1 | |
| Show the line \(y = 1\) | B1 | |
| Shade the correct region | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identify the correct point and carry out a correct method to find the argument | M1 | |
| Obtain answer \(\dfrac{11}{12}\pi\) | A1 | 2.88 radians or 165° |
## Question 5(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $-2$ | B1 | |
| Show a circle with radius 2 and centre not the origin | B1 | |
| Show the line $y = 1$ | B1 | |
| Shade the correct region | B1 | |
## Question 5(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Identify the correct point and carry out a correct method to find the argument | M1 | |
| Obtain answer $\dfrac{11}{12}\pi$ | A1 | 2.88 radians or 165° |
---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 $| z + 2 | \leqslant 2$ and $\operatorname { Im } z \geqslant 1$.
\item Find the greatest value of $\arg z$ for points in the shaded region.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q5 [6]}}