| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | November |
| Marks | 8 |
| 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 complex numbers question requiring standard techniques: finding modulus/argument (routine), verifying a cube equation (direct substitution), and sketching a region defined by simple inequalities (circle and vertical line). All parts are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02d Exponential form: re^(i*theta)4.02k Argand diagrams: geometric interpretation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| State modulus \(2\) | B1 | |
| State argument \(-\frac{1}{3}\pi\) or \(-60°\) (\(\frac{5}{3}\pi\) or \(300°\)) | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| *EITHER:* Expand \((1-(\sqrt{3})i)^3\) completely and process \(i^2\) and \(i^3\) | M1 | |
| Verify that the given relation is satisfied | A1 | |
| *OR:* \(u^3 = 2^3(\cos(-\pi) + i\sin(-\pi))\) or equivalent: follow their answers to (i) | M1 | |
| Verify that the given relation is satisfied | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Show a circle with centre \(1-(\sqrt{3})i\) in a relatively correct position | B1 | |
| Show a circle with radius 2 passing through the origin | B1 | |
| Show the line \(\text{Re } z = 2\) | B1 | |
| Shade the correct region | B1 | |
| Total | 4 |
## Question 7(i):
| Answer | Mark | Notes |
|--------|------|-------|
| State modulus $2$ | B1 | |
| State argument $-\frac{1}{3}\pi$ or $-60°$ ($\frac{5}{3}\pi$ or $300°$) | B1 | |
## Question 7(ii):
| Answer | Mark | Notes |
|--------|------|-------|
| *EITHER:* Expand $(1-(\sqrt{3})i)^3$ completely and process $i^2$ and $i^3$ | M1 | |
| Verify that the given relation is satisfied | A1 | |
| *OR:* $u^3 = 2^3(\cos(-\pi) + i\sin(-\pi))$ or equivalent: follow their answers to (i) | M1 | |
| Verify that the given relation is satisfied | A1 | |
# Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Show a circle with centre $1-(\sqrt{3})i$ in a relatively correct position | B1 | |
| Show a circle with radius 2 passing through the origin | B1 | |
| Show the line $\text{Re } z = 2$ | B1 | |
| Shade the correct region | B1 | |
| **Total** | **4** | |
---7 Throughout this question the use of a calculator is not permitted.\\
The complex number $1 - ( \sqrt { } 3 ) \mathrm { i }$ is denoted by $u$.\\
(i) Find the modulus and argument of $u$.\\
(ii) Show that $u ^ { 3 } + 8 = 0$.\\
(iii) On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying both the inequalities $| z - u | \leqslant 2$ and $\operatorname { Re } z \geqslant 2$, where $\operatorname { Re } z$ denotes the real part of $z$.\\[0pt]
[4]\\
$8 \quad$ Let $\mathrm { f } ( x ) = \frac { 8 x ^ { 2 } + 9 x + 8 } { ( 1 - x ) ( 2 x + 3 ) ^ { 2 } }$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q7 [8]}}