| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Complex numbers 2 |
| Type | Modulus and argument calculations |
| Difficulty | Standard +0.8 This question requires applying double-angle formulas (cos 2θ = 2cos²θ - 1), converting to modulus-argument form, and manipulating complex fractions. Part (i) needs trigonometric insight to simplify the modulus expression, while part (ii) requires algebraic manipulation of 1/z using the result from (i). The multi-step reasoning and need to recognize the trigonometric identities elevates this above routine complex number exercises, though it remains within standard A-level techniques. |
| Spec | 4.02a Complex numbers: real/imaginary parts, modulus, argument4.02b Express complex numbers: cartesian and modulus-argument forms |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: State a correct expression for \( | z | \) or \( |
| Use double angle formulae throughout or Pythagoras | M1 | |
| Obtain given answer \(2\cos \theta\) correctly | A1 | |
| State a correct expression for tangent of argument, e.g. \((\sin 2\theta/(1 + \cos 2\theta))\) | B1 | |
| Use double angle formulae to express it in terms of \(\cos \theta\) and \(\sin \theta\) | M1 | |
| Obtain \(\tan \theta\) and state that the argument is \(\theta\) | A1 | |
| OR: Use double angle formulae to express \(z\) in terms of \(\cos \theta\) and \(\sin \theta\) | M1 | |
| Obtain a correct expression, e.g. \(1 + \cos^2 \theta - \sin^2 \theta + 2i \sin \theta \cos \theta\) | A1 | |
| Convert the expression to polar form | M1 | |
| Obtain \(2\cos\theta(\cos \theta + i\sin \theta)\) | A1 | |
| State that the modulus is \(2\cos \theta\) | A1 | |
| State that the argument is \(\theta\) | A1 | [6] |
| (ii) Substitute for \(z\) and multiply numerator and denominator by the conjugate of \(z\), or equivalent | M1 | |
| Obtain correct real denominator in any form | A1 | |
| Identify and obtain real part equal to \(\frac{2}{5}\) | A1 | [3] |
**(i)** **EITHER:** State a correct expression for $|z|$ or $|z|^2$, e.g. $(1 + \cos 2\theta)^2 + (\sin 2\theta)^2$ | B1 |
Use double angle formulae throughout or Pythagoras | M1 |
Obtain given answer $2\cos \theta$ correctly | A1 |
State a correct expression for tangent of argument, e.g. $(\sin 2\theta/(1 + \cos 2\theta))$ | B1 |
Use double angle formulae to express it in terms of $\cos \theta$ and $\sin \theta$ | M1 |
Obtain $\tan \theta$ and state that the argument is $\theta$ | A1 |
**OR:** Use double angle formulae to express $z$ in terms of $\cos \theta$ and $\sin \theta$ | M1 |
Obtain a correct expression, e.g. $1 + \cos^2 \theta - \sin^2 \theta + 2i \sin \theta \cos \theta$ | A1 |
Convert the expression to polar form | M1 |
Obtain $2\cos\theta(\cos \theta + i\sin \theta)$ | A1 |
State that the modulus is $2\cos \theta$ | A1 |
State that the argument is $\theta$ | A1 | [6]
**(ii)** Substitute for $z$ and multiply numerator and denominator by the conjugate of $z$, or equivalent | M1 |
Obtain correct real denominator in any form | A1 |
Identify and obtain real part equal to $\frac{2}{5}$ | A1 | [3]8 The variable complex number $z$ is given by
$$z = 1 + \cos 2 \theta + i \sin 2 \theta$$
where $\theta$ takes all values in the interval $- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi$.\\
(i) Show that the modulus of $z$ is $2 \cos \theta$ and the argument of $z$ is $\theta$.\\
(ii) Prove that the real part of $\frac { 1 } { z }$ is constant.
\hfill \mbox{\textit{CAIE P3 2010 Q8 [9]}}