AQA Further Paper 2 Specimen — Question 5 4 marks

Exam BoardAQA
ModuleFurther Paper 2 (Further Paper 2)
SessionSpecimen
Marks4
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeSolve equations using trigonometric identities
DifficultyStandard +0.3 This is a straightforward application of De Moivre's theorem requiring students to convert the RHS to polar form, apply the theorem, and solve for θ. While it's a Further Maths topic, the method is standard and mechanical with no conceptual challenges beyond recognizing the approach.
Spec4.02q De Moivre's theorem: multiple angle formulae
Find the smallest value \(\theta\) of for which \((\cos \theta + i \sin \theta)^5 = \frac{1}{\sqrt{2}}(1 - i)\) \(\{\theta \in \mathbb{R} : \theta > 0\}\) [4 marks]
Question 5:
AnswerMarks Guidance
5Uses de Moivre’s theorem AO3.1a
cosisin5  1i
2
1
cos5isin5 1i
2
1 1
cos5 sin5
2 2
7π
5
4
7π

20
Equates real and imaginary parts
AnswerMarks Guidance
and obtains two equationsAO1.1a A1
Deduces that the smallest possible
7π
value of 5 is
4
FT from ‘their’ equations provided
AnswerMarks Guidance
M1 has been awardedAO2.2a A1F
Obtains the smallest possible value
of (cid:2016) from fully correct reasoning
FT from ‘their’ 5θ provided M1 has
AnswerMarks Guidance
been awardedAO1.1b A1F
Total4
QMarking Instructions AO 
Question 5:
5 | Uses de Moivre’s theorem | AO3.1a | M1 | 1
cosisin5  1i
2
1
cos5isin5 1i
2
1 1
cos5 sin5
2 2
7π
5
4
7π

20
Equates real and imaginary parts
and obtains two equations | AO1.1a | A1
Deduces that the smallest possible
7π
value of 5 is
4
FT from ‘their’ equations provided
M1 has been awarded | AO2.2a | A1F
Obtains the smallest possible value
of (cid:2016) from fully correct reasoning
FT from ‘their’ 5θ provided M1 has
been awarded | AO1.1b | A1F
Total | 4
Q | Marking Instructions | AO | Marks | Typical Solution
Find the smallest value $\theta$ of for which

$(\cos \theta + i \sin \theta)^5 = \frac{1}{\sqrt{2}}(1 - i)$ $\{\theta \in \mathbb{R} : \theta > 0\}$
[4 marks]

\hfill \mbox{\textit{AQA Further Paper 2  Q5 [4]}}
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Q1 1 Q2 3 Q3 3 Q4 4 Q5 4 Q6 5 Q7 5 Q7 2 Q8 5 Q9 6 Q10 8 Q11 8 Q12 11 Q13 7 Q14 9 Q15 10 Q16 9