AQA Further Paper 2 Specimen — Question 4 4 marks

Exam BoardAQA
ModuleFurther Paper 2 (Further Paper 2)
SessionSpecimen
Marks4
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Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeDirect nth roots: find and express roots
DifficultyStandard +0.3 This is a straightforward application of De Moivre's theorem to find cube roots of a complex number. Students need to express i in exponential form (e^(iπ/2)), then find the three roots by dividing the argument by 3 and adding 2πk/3. While it's a Further Maths topic, it's a standard textbook exercise requiring routine application of a well-practiced technique with no novel insight needed.
Spec4.02r nth roots: of complex numbers
Solve the equation \(z^3 = i\), giving your answers in the form \(e^{i\theta}\), where \(-\pi < \theta \leq \pi\) [4 marks]
Question 4:
AnswerMarks Guidance
4Expresses i or z in polar form AO1.2
i
i = e 2
1
 π 3  π 2nπ 
i( 2nπ) i(  )
z = e 2   e 6 3 
   
π 5π π 3 
, , or etc
6 6 2 2 
π π 5π
i i i
z  e 2,e 6,e 6
ALT
z ei z cosisin
z3 icos3isin3i
cos30 and sin31
π 5π π 3π 
 , , or etc
6 6 2 2 
π π 5π
i i i
z  e 2,e 6,e 6
AnswerMarks Guidance
Uses de Moivre’s TheoremAO3.1a M1
Finds three consecutive values for
AnswerMarks Guidance
(cid:2016)AO1.1a A1
Finds all three correct solutions for
AnswerMarks Guidance
zAO1.1b A1
Total4
QMarking Instructions AO 
Question 4:
4 | Expresses i or z in polar form | AO1.2 | B1 | π
i
i = e 2
1
 π 3  π 2nπ 
i( 2nπ) i(  )
z = e 2   e 6 3 
   
π 5π π 3 
, , or etc
6 6 2 2 
π π 5π
i i i
z  e 2,e 6,e 6
ALT
z ei z cosisin
z3 icos3isin3i
cos30 and sin31
π 5π π 3π 
 , , or etc
6 6 2 2 
π π 5π
i i i
z  e 2,e 6,e 6
Uses de Moivre’s Theorem | AO3.1a | M1
Finds three consecutive values for
(cid:2016) | AO1.1a | A1
Finds all three correct solutions for
z | AO1.1b | A1
Total | 4
Q | Marking Instructions | AO | Marks | Typical Solution
Solve the equation $z^3 = i$, giving your answers in the form $e^{i\theta}$, where $-\pi < \theta \leq \pi$
[4 marks]

\hfill \mbox{\textit{AQA Further Paper 2  Q4 [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