| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | March |
| Marks | 5 |
| 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 argument loci (rays from a point), combining two inequalities to find an intersection region, and then optimizing arg(z) over that region—requiring geometric insight beyond routine shading. The calculation in part (b) demands finding a specific boundary point and computing its argument, which is non-trivial problem-solving rather than standard recall. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Show correct half-lines from \(1+2i\), symmetrical about \(y=2i\), drawn between \(\dfrac{\pi}{4}\) and \(\dfrac{5\pi}{12}\) | B1 | |
| Show the line \(x=3\) extending in both quadrants | B1 | |
| Shade the correct region. Allow dashes on axes as scale. FT if only error is one of: FULL lines or \(x \neq 3\) or one sign error in \(1+2i\) or angle outside tolerance or scale missing on one axis | B1 FT | SC No scale on either axis: allow B1 FT for otherwise correct figure in correct position |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out a complete method for finding the least value of arg \(z\) | M1 | e.g. \(-\tan^{-1}\frac{(2\sqrt{3}-2)}{3}\) or \(\tan^{-1}\frac{(-2\sqrt{3}+2)}{3}\) |
| Obtain answer \(-0.454\) (3dp) | A1 | SC B1 0.454 |
**Question 2(a):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Show correct half-lines from $1+2i$, symmetrical about $y=2i$, drawn between $\dfrac{\pi}{4}$ and $\dfrac{5\pi}{12}$ | **B1** | |
| Show the line $x=3$ extending in both quadrants | **B1** | |
| Shade the correct region. Allow dashes on axes as scale. FT if only error is one of: FULL lines or $x \neq 3$ or one sign error in $1+2i$ or angle outside tolerance or scale missing on one axis | **B1 FT** | **SC** No scale on either axis: allow **B1 FT** for otherwise correct figure in correct position |
| **Total: 3** | | |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out a complete method for finding the least value of arg $z$ | **M1** | e.g. $-\tan^{-1}\frac{(2\sqrt{3}-2)}{3}$ or $\tan^{-1}\frac{(-2\sqrt{3}+2)}{3}$ |
| Obtain answer $-0.454$ (3dp) | **A1** | **SC B1** 0.454 |
---2
\begin{enumerate}[label=(\alph*)]
\item On an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $- \frac { 1 } { 3 } \pi \leqslant \arg ( z - 1 - 2 \mathrm { i } ) \leqslant \frac { 1 } { 3 } \pi$ and $\operatorname { Re } z \leqslant 3$.
\item Calculate the least value of $\arg z$ for points in the region from (a). Give your answer in radians correct to 3 decimal places.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q2 [5]}}