| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex Numbers Argand & Loci |
| Type | Locus with parameter variation |
| Difficulty | Standard +0.8 This question requires understanding of complex number loci and algebraic manipulation. Part (i) involves proving a modulus relationship (straightforward algebra leading to a circle) and sketching. Part (ii) requires rationalizing a complex fraction with a parameter, then proving the real part is constant—this demands careful algebraic manipulation and trigonometric identity work. While systematic, it's more demanding than standard textbook exercises and requires multi-step reasoning across both geometric and algebraic representations. |
| Spec | 4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Find modulus of \(2\cos\theta - 2i\sin\theta\) and show it is equal to 2 | B1 | |
| Show a circle with centre at the point representing \(i\) | B1 | |
| Show a circle with radius 2 | B1 | [3 marks] |
| (ii) Substitute for \(z\) and multiply numerator and denominator by the conjugate of \(z + 2 - i\), or equivalent | M1 | |
| Obtain correct real denominator in any form | A1 | |
| Identify and obtain correct unsimplified real part in terms of \(\cos\theta\), e.g. \((2\cos\theta + 2)/(8\cos\theta + 8)\) | A1 | |
| State that real part equals \(\frac{1}{4}\) | A1 | [4 marks] |
**(i)** Find modulus of $2\cos\theta - 2i\sin\theta$ and show it is equal to 2 | B1 |
Show a circle with centre at the point representing $i$ | B1 |
Show a circle with radius 2 | B1 | [3 marks]
**(ii)** Substitute for $z$ and multiply numerator and denominator by the conjugate of $z + 2 - i$, or equivalent | M1 |
Obtain correct real denominator in any form | A1 |
Identify and obtain correct unsimplified real part in terms of $\cos\theta$, e.g. $(2\cos\theta + 2)/(8\cos\theta + 8)$ | A1 |
State that real part equals $\frac{1}{4}$ | A1 | [4 marks]
---5 The variable complex number $z$ is given by
$$z = 2 \cos \theta + \mathrm { i } ( 1 - 2 \sin \theta ) ,$$
where $\theta$ takes all values in the interval $- \pi < \theta \leqslant \pi$.\\
(i) Show that $| z - \mathrm { i } | = 2$, for all values of $\theta$. Hence sketch, in an Argand diagram, the locus of the point representing $z$.\\
(ii) Prove that the real part of $\frac { 1 } { z + 2 - \mathrm { i } }$ is constant for $- \pi < \theta < \pi$.
\hfill \mbox{\textit{CAIE P3 2008 Q5 [7]}}