CAIE Further Paper 2 2023 November — Question 2 5 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2023
SessionNovember
Marks5
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Mark schemeDownload PDF ↗
TopicComplex Numbers Arithmetic
TypeRoots of unity and special equations
DifficultyStandard +0.3 This is a straightforward Further Maths complex numbers question requiring conversion to polar form, application of De Moivre's theorem to find cube roots, and expressing answers in the specified trigonometric form. While it involves multiple steps (convert RHS to polar, find three cube roots, adjust by -5i), these are standard techniques with no novel insight required. The 5-mark allocation and routine nature place it slightly easier than average.
Spec4.02b Express complex numbers: cartesian and modulus-argument forms4.02q De Moivre's theorem: multiple angle formulae4.02r nth roots: of complex numbers
Find the roots of the equation \((z + 5i)^3 = 4 + 4\sqrt{3}i\), giving your answers in the form \(r\cos\theta + ir\sin\theta - 5)\), where \(r > 0\) and \(0 \leq \theta < 2\pi\). [5]
Question 2:
AnswerMarks
21
π
AnswerMarks Guidance
(z+5i)3 =4+4i 3=8e3B1 Finds modulus and argument of 4+4i 3.
 1 1  1  1 
z =2cos π+isin π−5i = 2cos π+i2sin π−5
AnswerMarks Guidance
1  9 9  9  9 M1 A1 Finds one root.
7  7  13  13 
z =2cos π+i2sin π−5, z =2cos π+i2sin π−5
2 3
AnswerMarks Guidance
9  9  9  9 A1 FT Finds other two roots. FT on their modulus.
A1 FT
5
AnswerMarks Guidance
QuestionAnswer Marks 
Question 2:
2 | 1
π
(z+5i)3 =4+4i 3=8e3 | B1 | Finds modulus and argument of 4+4i 3.
 1 1  1  1 
z =2cos π+isin π−5i = 2cos π+i2sin π−5
1  9 9  9  9  | M1 A1 | Finds one root.
7  7  13  13 
z =2cos π+i2sin π−5, z =2cos π+i2sin π−5
2 3
9  9  9  9  | A1 FT | Finds other two roots. FT on their modulus.
A1 FT
5
Question | Answer | Marks | Guidance
Find the roots of the equation $(z + 5i)^3 = 4 + 4\sqrt{3}i$, giving your answers in the form $r\cos\theta + ir\sin\theta - 5)$, where $r > 0$ and $0 \leq \theta < 2\pi$. [5]

\hfill \mbox{\textit{CAIE Further Paper 2 2023 Q2 [5]}}
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Q1 4 Q2 5 Q3 6 Q4 10 Q5 10 Q6 14 Q7 11 Q8 15