| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Complex number arithmetic and simplification |
| Difficulty | Standard +0.3 This question involves standard complex number operations (converting from modulus-argument form, multiplication, division) and geometric interpretation on an Argand diagram. While it requires multiple steps and proving the triangle is equilateral needs some geometric insight about arguments and distances, these are all core A-level techniques without requiring novel problem-solving approaches. The proof in part (iii) is straightforward once the coordinates are found. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02c Complex notation: z, z*, Re(z), Im(z), |z|, arg(z)4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02l Geometrical effects: conjugate, addition, subtraction |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(w = \cos \frac{2}{3}\pi + i\sin \frac{2}{3}\pi\) (allow decimals) | B1 | Obtain answer \(uw = -\sqrt{3} - i\) (allow decimals) |
| (ii) Show U on an Argand diagram correctly | B1 | Show A and B in relatively correct positions |
| (iii) Prove that \(AB = UA\) (or \(UB\)), or prove that angle \(AUB\) = angle \(ABU\) (or angle \(BAU\)) or prove, for example, that \(AO = OB\) and angle \(AOB = 120°\), or prove that one angle of triangle \(UAB\) equals 60° | B1 | Complete a proof that triangle \(UAB\) is equilateral |
**(i)** State or imply $w = \cos \frac{2}{3}\pi + i\sin \frac{2}{3}\pi$ (allow decimals) | B1 | Obtain answer $uw = -\sqrt{3} - i$ (allow decimals) | B1√ | Multiply numerator and denominator of $\frac{u}{w}$ by $-1 - i\sqrt{3}$, or equivalent | M1 | Obtain answer $\frac{u}{w} = \sqrt{3} - i$ (allow decimals) | A1 | **[4]**
**(ii)** Show U on an Argand diagram correctly | B1 | Show A and B in relatively correct positions | B1√ | **[2]**
**(iii)** Prove that $AB = UA$ (or $UB$), or prove that angle $AUB$ = angle $ABU$ (or angle $BAU$) or prove, for example, that $AO = OB$ and angle $AOB = 120°$, or prove that one angle of triangle $UAB$ equals 60° | B1 | Complete a proof that triangle $UAB$ is equilateral | B1 | **[2]**5 The complex number 2 i is denoted by $u$. The complex number with modulus 1 and argument $\frac { 2 } { 3 } \pi$ is denoted by $w$.\\
(i) Find in the form $x + \mathrm { i } y$, where $x$ and $y$ are real, the complex numbers $w , u w$ and $\frac { u } { w }$.\\
(ii) Sketch an Argand diagram showing the points $U , A$ and $B$ representing the complex numbers $u$, $u w$ and $\frac { u } { w }$ respectively.\\
(iii) Prove that triangle $U A B$ is equilateral.
\hfill \mbox{\textit{CAIE P3 2003 Q5 [8]}}