| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2008 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Geometric relationships on Argand diagram |
| Difficulty | Standard +0.8 This is a multi-part question requiring modulus/argument calculations, understanding of multiplication/division effects on complex numbers, geometric interpretation of equilateral triangles on Argand diagrams, and applying this to find specific vertices. Part (iii) requires conceptual insight about rotations creating equilateral triangles, and part (iv) demands synthesis of these ideas with accurate calculation. While individual steps are standard Further Maths content, the geometric reasoning and multi-step synthesis elevate this above routine exercises. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms4.02k Argand diagrams: geometric interpretation4.02m Geometrical effects: multiplication and division |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State that the modulus of \(w\) is 1 | B1 | |
| State that the argument of \(w\) is \(\frac{2}{3}\pi\) or \(120°\) (accept 2.09, or 2.1) | B1 | [2] |
| (ii) State that the modulus of \(wz\) is \(R\) | B1√ | |
| State that the argument of \(wz\) is \(\theta + \frac{2}{3}\pi\) | B1√ | |
| State that the modulus of \(z/w\) is \(R\) | B1√ | |
| State that the argument of \(z/w\) is \(\theta - \frac{2}{3}\pi\) | B1√ | [4] |
| (iii) State or imply the points are equidistant from the origin | B1 | |
| State or imply that two pairs of points subtend \(\frac{2}{3}\pi\) at the origin, or that all three pairs subtend equal angles at the origin | B1 | [2] |
| (iv) Multiply \(4 + 2i\) by \(w\) and use \(i^2 = -1\) | M1 | |
| Obtain \(-(2+\sqrt{3}) + (2\sqrt{3}-1)i\), or exact equivalent | A1 | |
| Divide \(4 + 2i\) by \(w\), multiplying numerator and denominator by the conjugate of \(w\), or equivalent | M1 | |
| Obtain \(-(2-\sqrt{3}) - (2\sqrt{3}+1)i\), or exact equivalent | A1 | [4] |
| [Use of polar form of \(4 + 2i\) can earn M marks and then A marks for obtaining exact \(x + iy\) answers.] | ||
| [SR: If answers only seen in polar form, allow B1+B1 in (i), B1√+B1√ in (ii), but A0+A0 in (iv).] |
**(i)** State that the modulus of $w$ is 1 | B1 |
State that the argument of $w$ is $\frac{2}{3}\pi$ or $120°$ (accept 2.09, or 2.1) | B1 | [2]
**(ii)** State that the modulus of $wz$ is $R$ | B1√ |
State that the argument of $wz$ is $\theta + \frac{2}{3}\pi$ | B1√ |
State that the modulus of $z/w$ is $R$ | B1√ |
State that the argument of $z/w$ is $\theta - \frac{2}{3}\pi$ | B1√ | [4]
**(iii)** State or imply the points are equidistant from the origin | B1 |
State or imply that two pairs of points subtend $\frac{2}{3}\pi$ at the origin, or that all three pairs subtend equal angles at the origin | B1 | [2]
**(iv)** Multiply $4 + 2i$ by $w$ and use $i^2 = -1$ | M1 |
Obtain $-(2+\sqrt{3}) + (2\sqrt{3}-1)i$, or exact equivalent | A1 |
Divide $4 + 2i$ by $w$, multiplying numerator and denominator by the conjugate of $w$, or equivalent | M1 |
Obtain $-(2-\sqrt{3}) - (2\sqrt{3}+1)i$, or exact equivalent | A1 | [4] |
[Use of polar form of $4 + 2i$ can earn M marks and then A marks for obtaining exact $x + iy$ answers.] |
[SR: If answers only seen in polar form, allow B1+B1 in (i), B1√+B1√ in (ii), but A0+A0 in (iv).] |10 The complex number $w$ is given by $w = - \frac { 1 } { 2 } + \mathrm { i } \frac { \sqrt { } 3 } { 2 }$.\\
(i) Find the modulus and argument of $w$.\\
(ii) The complex number $z$ has modulus $R$ and argument $\theta$, where $- \frac { 1 } { 3 } \pi < \theta < \frac { 1 } { 3 } \pi$. State the modulus and argument of $w z$ and the modulus and argument of $\frac { z } { w }$.\\
(iii) Hence explain why, in an Argand diagram, the points representing $z , w z$ and $\frac { z } { w }$ are the vertices of an equilateral triangle.\\
(iv) In an Argand diagram, the vertices of an equilateral triangle lie on a circle with centre at the origin. One of the vertices represents the complex number $4 + 2 \mathrm { i }$. Find the complex numbers represented by the other two vertices. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real and exact.
\hfill \mbox{\textit{CAIE P3 2008 Q10 [12]}}