| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| 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 multi-part question requiring standard techniques: (i) multiplying by conjugate to simplify a complex fraction, (ii) finding modulus and argument from Cartesian form, and (iii) sketching standard loci (circle centered at origin and perpendicular bisector). All parts are routine A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Multiply numerator and denominator by \(1 + \sqrt{3}\,i\), or equivalent | M1 | |
| \(4i - 4\sqrt{3}\) and \(3 + 1\) | A1 | |
| Obtain final answer \(-\sqrt{3} + i\) | A1 | |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State that the modulus of \(u\) is \(2\) | B1 | |
| State that the argument of \(u\) is \(\frac{5}{6}\pi\) (or \(150°\)) | B1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle with centre the origin and radius \(2\) | B1 | |
| Show \(u\) in a relatively correct position | B1 | FT |
| Show the perpendicular bisector of the line joining \(u\) and the origin | B1 | FT |
| Shade the correct region | B1 | |
| Total: 4 |
## Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Multiply numerator and denominator by $1 + \sqrt{3}\,i$, or equivalent | M1 | |
| $4i - 4\sqrt{3}$ and $3 + 1$ | A1 | |
| Obtain final answer $-\sqrt{3} + i$ | A1 | |
| **Total: 3** | | |
## Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| State that the modulus of $u$ is $2$ | B1 | |
| State that the argument of $u$ is $\frac{5}{6}\pi$ (or $150°$) | B1 | |
| **Total: 2** | | |
## Question 8(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre the origin and radius $2$ | B1 | |
| Show $u$ in a relatively correct position | B1 | FT |
| Show the perpendicular bisector of the line joining $u$ and the origin | B1 | FT |
| Shade the correct region | B1 | |
| **Total: 4** | | |8 Throughout this question the use of a calculator is not permitted.\\
The complex number $u$ is defined by
$$u = \frac { 4 \mathrm { i } } { 1 - ( \sqrt { } 3 ) \mathrm { i } }$$
(i) Express $u$ in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact.\\
(ii) Find the exact modulus and argument of $u$.\\
(iii) On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $| z | < 2$ and $| z - u | < | z |$.
\hfill \mbox{\textit{CAIE P3 2019 Q8 [9]}}