| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | June |
| Marks | 11 |
| 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 multi-part question covering standard complex number techniques: converting to modulus-argument form (routine), verifying a root and finding its conjugate (straightforward substitution), and sketching a region defined by simple inequalities (a circle and a half-line). All parts are textbook exercises requiring no novel insight, though the combination of multiple standard techniques places it slightly above average difficulty. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02i Quadratic equations: with complex roots4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply that \(r = 2\) | B1 | |
| State or imply that \(\theta = \frac{1}{3}\pi\) (allow \(1.05\) radians or \(60°\)) | B1 | |
| Obtain modulus \(4\), and argument \(\frac{2}{3}\pi\) of \(u^2\) (allow \(2^2 : 2.09\) or \(2.10\) radians or \(120°\)) | B1 + B1 | |
| Obtain modulus \(8\) and argument \(\pi\) of \(u^3\) (allow \(2^3\); \(3.14\) or \(3.15\) radians or \(180°\)) | B1 | Total: 5 marks |
| Guidance: Follow through on wrong \(r\) and \(\theta\) | ||
| SR: if \(u^2\) and \(u^3\) are only given in polar form, give B1 for \(u^2\) and B1 for \(u^3\). |
| Answer | Marks | Guidance |
|---|---|---|
| Deduce that \(u^2 - 2u + 4 = 0\) from \(u^3 + 8 = 0\) | B1 | |
| Verify that \(u^2 - 2u + 4 = 0\) by calculation | B1 | |
| State that the other root is \(1 - i\sqrt{3}\), or equivalent | B1 | Total: 2 marks |
| Guidance: NB: stating that the roots are \(1 \pm i\sqrt{3}\) is sufficient for both B marks. |
| Answer | Marks | Guidance |
|---|---|---|
| (No working required) | ||
| (iii) Show both points correctly on an Argand diagram | B1 | |
| Show the correct relevant circle | B1 | |
| Show line (segment) correctly | B1 | |
| Shade the correct region | B1 | |
| Guidance: SR: allow work on separate diagrams to be eligible for the first three B marks. | Total: 4 marks |
**(i)** State or imply that $r = 2$ | B1 |
State or imply that $\theta = \frac{1}{3}\pi$ (allow $1.05$ radians or $60°$) | B1 |
Obtain modulus $4$, and argument $\frac{2}{3}\pi$ of $u^2$ (allow $2^2 : 2.09$ or $2.10$ radians or $120°$) | B1 + B1 |
Obtain modulus $8$ and argument $\pi$ of $u^3$ (allow $2^3$; $3.14$ or $3.15$ radians or $180°$) | B1 | **Total: 5 marks**
**Guidance:** Follow through on wrong $r$ and $\theta$ | |
SR: if $u^2$ and $u^3$ are only given in polar form, give B1 for $u^2$ and B1 for $u^3$. | |
**(ii) EITHER route:**
Deduce that $u^2 - 2u + 4 = 0$ from $u^3 + 8 = 0$ | B1 |
Verify that $u^2 - 2u + 4 = 0$ by calculation | B1 |
State that the other root is $1 - i\sqrt{3}$, or equivalent | B1 | **Total: 2 marks**
**Guidance:** NB: stating that the roots are $1 \pm i\sqrt{3}$ is sufficient for both B marks. | |
**OR route:**
(No working required) | |
**(iii)** Show both points correctly on an Argand diagram | B1 |
Show the correct relevant circle | B1 |
Show line (segment) correctly | B1 |
Shade the correct region | B1 |
**Guidance:** SR: allow work on separate diagrams to be eligible for the first three B marks. | | **Total: 4 marks**
---9 The complex number $1 + i \sqrt { } 3$ is denoted by $u$.\\
(i) Express $u$ in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $r > 0$ and $- \pi < \theta \leqslant \pi$. Hence, or otherwise, find the modulus and argument of $u ^ { 2 }$ and $u ^ { 3 }$.\\
(ii) Show that $u$ is a root of the equation $z ^ { 2 } - 2 z + 4 = 0$, and state the other root of this equation.\\
(iii) Sketch an Argand diagram showing the points representing the complex numbers $i$ and $u$. Shade the region whose points represent every complex number $z$ satisfying both the inequalities
$$| z - \mathrm { i } | \leqslant 1 \quad \text { and } \quad \arg z \geqslant \arg u .$$
\hfill \mbox{\textit{CAIE P3 2002 Q9 [11]}}