Challenging +1.2 This is a structured multi-part Further Maths question following a standard template: prove de Moivre's theorem by induction (routine for FP1), derive a multiple angle formula using binomial expansion and equating real parts (mechanical but lengthy), then solve a polynomial by substituting x=cos θ and using the derived formula. While it requires several techniques and careful algebra across multiple steps, each component follows well-established procedures taught explicitly in FP1 with no novel insight required. The length and coordination of parts elevates it above average difficulty, but it remains a textbook-style examination question.
11 Prove de Moivre's theorem for a positive integral exponent:
$$\text { for all positive integers } n , \quad ( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta \text {. }$$
Use de Moivre's theorem to show that
$$\cos 7 \theta = 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$
Hence obtain the roots of the equation
$$128 x ^ { 7 } - 224 x ^ { 5 } + 112 x ^ { 3 } - 14 x + 1 = 0$$
in the form \(\cos q \pi\), where \(q\) is a rational number.
11 Prove de Moivre's theorem for a positive integral exponent:
$$\text { for all positive integers } n , \quad ( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta \text {. }$$
Use de Moivre's theorem to show that
$$\cos 7 \theta = 64 \cos ^ { 7 } \theta - 112 \cos ^ { 5 } \theta + 56 \cos ^ { 3 } \theta - 7 \cos \theta$$
Hence obtain the roots of the equation
$$128 x ^ { 7 } - 224 x ^ { 5 } + 112 x ^ { 3 } - 14 x + 1 = 0$$
in the form $\cos q \pi$, where $q$ is a rational number.
\hfill \mbox{\textit{CAIE FP1 2006 Q11 [13]}}