CAIE Further Paper 2 2024 June — Question 1 5 marks

Exam BoardCAIE
ModuleFurther Paper 2 (Further Paper 2)
Year2024
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeDirect nth roots: general complex RHS
DifficultyStandard +0.3 This is a straightforward application of De Moivre's theorem to find cube roots. Students must convert to modulus-argument form, divide the argument by 3, and find three roots by adding 2π/3. While it's a Further Maths topic, it's a standard textbook exercise requiring only methodical application of a well-practiced technique with no novel insight needed.
Spec4.02b Express complex numbers: cartesian and modulus-argument forms4.02r nth roots: of complex numbers
1 Find the roots of the equation \(z ^ { 3 } = - 108 \sqrt { 3 } + 108\) i, giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 < \theta < 2 \pi\).
Question 1:
AnswerMarks Guidance
AnswerMarks Guidance
\(z^3 = 216\cos\frac{5\pi}{6} + i216\sin\frac{5\pi}{6}\)B1 Finds modulus and argument of \(-108\sqrt{3} + 108i\)
\(z_1 = 6\cos\frac{5\pi}{18} + i6\sin\frac{5\pi}{18}\)M1 A1FT Finds one root. FT on their modulus. For M1 must have divided their argument of \(-108\sqrt{3} + 108i\) by 3.
\(z_2 = 6\cos\frac{17\pi}{18} + i6\sin\frac{17\pi}{18}\), \(z_3 = 6\cos\frac{29\pi}{18} + i6\sin\frac{29\pi}{18}\)M1 A1 Finds other two roots.
Total: 5 
**Question 1:**

| Answer | Marks | Guidance |
|--------|-------|----------|
| $z^3 = 216\cos\frac{5\pi}{6} + i216\sin\frac{5\pi}{6}$ | **B1** | Finds modulus and argument of $-108\sqrt{3} + 108i$ |
| $z_1 = 6\cos\frac{5\pi}{18} + i6\sin\frac{5\pi}{18}$ | **M1 A1FT** | Finds one root. FT on their modulus. For M1 must have divided their argument of $-108\sqrt{3} + 108i$ by 3. |
| $z_2 = 6\cos\frac{17\pi}{18} + i6\sin\frac{17\pi}{18}$, $z_3 = 6\cos\frac{29\pi}{18} + i6\sin\frac{29\pi}{18}$ | **M1 A1** | Finds other two roots. |
| | **Total: 5** | |
1 Find the roots of the equation $z ^ { 3 } = - 108 \sqrt { 3 } + 108$ i, giving your answers in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $r > 0$ and $0 < \theta < 2 \pi$.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2024 Q1 [5]}}
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Q1 5 Q2 4 Q3 7 Q4 8 Q5 11 Q6 12 Q7 12 Q8 16