Question 3 - A-Level Maths

AQA FP2 2007 June — Question 3 5 marks

Exam Board	AQA
Module	FP2 (Further Pure Mathematics 2)
Year	2007
Session	June
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Complex numbers 2
Type	Express roots in trigonometric form
Difficulty	Standard +0.3 This is a straightforward application of De Moivre's theorem requiring students to equate (cos θ + i sin θ)^15 = cos 15θ + i sin 15θ to -i = cos 270° + i sin 270°, then solve 15θ = 270° to get θ = 18°. While it's Further Maths content, it's a direct, single-concept application with minimal steps, making it slightly easier than an average A-level question overall.
Spec	4.02q De Moivre's theorem: multiple angle formulae

3 Use De Moivre's Theorem to find the smallest positive angle $\theta$ for which $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { 15 } = - \mathrm { i }$$ (5 marks)

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Question 3:

Answer	Marks	Guidance
Working	Marks	Guidance
$(\cos\theta + i\sin\theta)^{15} = \cos 15\theta + i\sin 15\theta$	M1	or $= e^{15i\theta}$
$\cos 15\theta = 0$
$\sin 15\theta = -1$	m1A1	or $-i = e^{\frac{3\pi i}{2}}$; m1 for both R&I parts written down
$15\theta = \frac{3\pi}{2}$ or $270°$	A1F
$\theta = \frac{\pi}{10}$ or $18°$	A1F	ft provided value of $15\theta$ is correct
SC: $\cos 15\theta + i\sin 15\theta = i$; $\sin 15\theta = -1$	(M1)(B1)	or for $\cos 15\theta = 0$
$\theta = \frac{\pi}{10}$	(B1)	(3)

## Question 3:
| Working | Marks | Guidance |
|---------|-------|----------|
| $(\cos\theta + i\sin\theta)^{15} = \cos 15\theta + i\sin 15\theta$ | M1 | or $= e^{15i\theta}$ |
| $\cos 15\theta = 0$ | | |
| $\sin 15\theta = -1$ | m1A1 | or $-i = e^{\frac{3\pi i}{2}}$; m1 for **both** R&I parts written down |
| $15\theta = \frac{3\pi}{2}$ or $270°$ | A1F | |
| $\theta = \frac{\pi}{10}$ or $18°$ | A1F | ft provided value of $15\theta$ is correct |
| **SC**: $\cos 15\theta + i\sin 15\theta = i$; $\sin 15\theta = -1$ | (M1)(B1) | or for $\cos 15\theta = 0$ |
| $\theta = \frac{\pi}{10}$ | (B1) | (3) |

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3 Use De Moivre's Theorem to find the smallest positive angle $\theta$ for which

$$( \cos \theta + \mathrm { i } \sin \theta ) ^ { 15 } = - \mathrm { i }$$

(5 marks)

\hfill \mbox{\textit{AQA FP2 2007 Q3 [5]}}

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Q1 7 Q2 12 Q3 5 Q4 7 Q5 9 Q6 7 Q7 15 Q8 13