| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2021 |
| Session | May |
| Marks | 8 |
| Topic | Addition & Double Angle Formulae |
| Type | Product to sum using compound angles |
| Difficulty | Hard +2.3 This is a challenging Further Maths question requiring sophisticated use of binomial theorem with complex numbers (De Moivre's theorem), converting the sum to real parts of $(1+e^{i heta})^{20}$, then applying product-to-sum formulas and half-angle identities. The 8-mark allocation and non-standard manipulation of the result into the specific factored form requires substantial insight beyond routine application. |
| Spec | 1.04a Binomial expansion: (a+b)^n for positive integer n4.02q De Moivre's theorem: multiple angle formulae |
Let $C = \sum_{r=0}^{20} \binom{20}{r} \cos(r\theta)$. Show that $C = 2^{20} \cos^{20}\left(\frac{1}{2}\theta\right) \cos(10\theta)$. [8]
\hfill \mbox{\textit{SPS SPS FM Pure 2021 Q8 [8]}}