CAIE FP1 2010 June — Question 5 8 marks

Exam BoardCAIE
ModuleFP1 (Further Pure Mathematics 1)
Year2010
SessionJune
Marks8
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeExpress roots in trigonometric form
DifficultyChallenging +1.8 This is a challenging Further Maths question requiring multiple sophisticated steps: deriving the quintuple angle formula using de Moivre's theorem and binomial expansion, recognizing the substitution x = sin θ to transform the polynomial, and solving sin 5θ = -1/2 to find five distinct roots in the required form. While the techniques are standard for FP1, the multi-stage reasoning and algebraic manipulation place it well above average difficulty.
Spec4.02q De Moivre's theorem: multiple angle formulae
5 Use de Moivre's theorem to show that $$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ Hence find all the roots of the equation $$32 x ^ { 5 } - 40 x ^ { 3 } + 10 x + 1 = 0$$ in the form \(\sin ( q \pi )\), where \(q\) is a positive rational number.
AnswerMarks Guidance
\((c + is)^2 = c^2 + 5c^4(is) + 10c^3(is)^2 + 10c^2(is)^3 + 5c(is)^4 + (is)^5\)M1
\(\theta = 5c^4s - 10c^2s^3 + s^5\)M1
\(= 5s(1 - s^2)^2 - 10s^3(1 - s^2) + s^5\)M1
\(= 16s^5 - 20s^3 + 5s\) (AG)A1 OEW [4]
\(x = \sin \theta \Rightarrow \sin 5\theta = -1/2\)M1
Roots are \(\sin q\pi\) where \(q = 7/30, 11/30, 31/30, 35/30, 43/30\)A3
A1 for 1 root: + A1 for 2 further roots: + A1 for completion CWO[4]
Alternative answers:
AnswerMarks
\(q = \frac{11}{30}, \frac{23}{30}, \frac{35}{30}, \frac{47}{30}, \frac{59}{30}\)or
\(q = \frac{7}{30}, \frac{19}{30}, \frac{31}{30}, \frac{43}{30}, \frac{55}{30}\) 
$(c + is)^2 = c^2 + 5c^4(is) + 10c^3(is)^2 + 10c^2(is)^3 + 5c(is)^4 + (is)^5$ | M1 |

$\theta = 5c^4s - 10c^2s^3 + s^5$ | M1 |

$= 5s(1 - s^2)^2 - 10s^3(1 - s^2) + s^5$ | M1 |

$= 16s^5 - 20s^3 + 5s$ (AG) | A1 OEW | [4]

$x = \sin \theta \Rightarrow \sin 5\theta = -1/2$ | M1 |

Roots are $\sin q\pi$ where $q = 7/30, 11/30, 31/30, 35/30, 43/30$ | A3 |

A1 for 1 root: + A1 for 2 further roots: + A1 for completion CWO | [4] |

**Alternative answers:**

$q = \frac{11}{30}, \frac{23}{30}, \frac{35}{30}, \frac{47}{30}, \frac{59}{30}$ | or |

$q = \frac{7}{30}, \frac{19}{30}, \frac{31}{30}, \frac{43}{30}, \frac{55}{30}$ | |

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5 Use de Moivre's theorem to show that

$$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$

Hence find all the roots of the equation

$$32 x ^ { 5 } - 40 x ^ { 3 } + 10 x + 1 = 0$$

in the form $\sin ( q \pi )$, where $q$ is a positive rational number.

\hfill \mbox{\textit{CAIE FP1 2010 Q5 [8]}}
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