A2 June 2025 Paper 1 Q15
15
The infinite series \(C\) and \(S\) are defined as follows.
\(C = \cos\theta + \dfrac{1}{3}\cos 5\theta + \dfrac{1}{9}\cos 9\theta + \dfrac{1}{27}\cos 13\theta + \ldots\)
\(S = \sin\theta + \dfrac{1}{3}\sin 5\theta + \dfrac{1}{9}\sin 9\theta + \dfrac{1}{27}\sin 13\theta + \ldots\)
| Scheme | Marks | AO |
|---|---|---|
| \(\left(3 - \mathrm{e}^{4\mathrm{i}\theta}\right)\left(3 - \mathrm{e}^{-4\mathrm{i}\theta}\right) = 9 - 3\mathrm{e}^{4\mathrm{i}\theta} - 3\mathrm{e}^{-4\mathrm{i}\theta} + 1\) | M1 | 2.1 |
| \(= 10 - 6\cos 4\theta\) | A1 | 1.1 |
| [2] |
Notes
M1: expanding correctly and fully to give at least three terms, allow \(10 - 3\left(\mathrm{e}^{4\mathrm{i}\theta} + \mathrm{e}^{-4\mathrm{i}\theta}\right)\). Must be seen. Condone \(\mathrm{e}^0 = 1\).
A1: www. Condone only incorrect values quoted for \(a\) and \(b\). No intermediate step required.
| Scheme | Marks | AO |
|---|---|---|
| \(C + \mathrm{i}S = \mathrm{e}^{\mathrm{i}\theta} + \frac{1}{3}\mathrm{e}^{5\mathrm{i}\theta} + \frac{1}{9}\mathrm{e}^{9\mathrm{i}\theta} + \frac{1}{27}\mathrm{e}^{13\mathrm{i}\theta} + \ldots\) | M1 | 2.1 |
| \(= \mathrm{e}^{\mathrm{i}\theta} + \frac{1}{3}\left(\mathrm{e}^{\mathrm{i}\theta}\right)^5 + \frac{1}{9}\left(\mathrm{e}^{\mathrm{i}\theta}\right)^9 + \frac{1}{27}\left(\mathrm{e}^{\mathrm{i}\theta}\right)^{13} + \ldots\) This is a GP with \(a = \mathrm{e}^{\mathrm{i}\theta}, r = \frac{1}{3}\left(\mathrm{e}^{\mathrm{i}\theta}\right)^4\) | A1 | 1.1 |
| so \(C + \mathrm{i}S = \dfrac{\mathrm{e}^{\mathrm{i}\theta}}{1 - \frac{1}{3}\left(\mathrm{e}^{\mathrm{i}\theta}\right)^4}\) | M1 | 3.1a |
| \(= \dfrac{3\mathrm{e}^{\mathrm{i}\theta}}{3 - \mathrm{e}^{4\mathrm{i}\theta}}\) | A1 | 2.1 |
| [4] |
Notes
M1: at least two terms of series in exponential form soi by correct GP formula
A1: writing series as powers of \(\mathrm{e}^{\mathrm{i}\theta}\) or identifying geometric series with correct first term and common ratio soi by correct GP formula
M1: correct use of sum to infinity formula. Must be seen.
A1: AG www
| Scheme | Marks | AO |
|---|---|---|
| \(C + \mathrm{i}S = \dfrac{3\mathrm{e}^{\mathrm{i}\theta}\left(3 - \mathrm{e}^{-4\mathrm{i}\theta}\right)}{\left(3 - \mathrm{e}^{4\mathrm{i}\theta}\right)\left(3 - \mathrm{e}^{-4\mathrm{i}\theta}\right)}\) | M1 | 3.1a |
| \(C + \mathrm{i}S = \dfrac{9(\cos\theta + \mathrm{i}\sin\theta) - 3(\cos 3\theta - \mathrm{i}\sin 3\theta)}{10 - 6\cos 4\theta}\) | M1 | 2.1 |
| \(C = \dfrac{9\cos\theta - 3\cos 3\theta}{10 - 6\cos 4\theta}\) | A1 | 2.2a |
| [3] |
Notes
M1: multiplying numerator and denominator by a multiple of \(3 - \mathrm{e}^{-4\mathrm{i}\theta}\); multiplication of numerator must be shown but denominator could be given immediately as \(10 - 6\cos 4\theta\).
M1: converting to sine and cosine form after denominator has been simplified to a real expression. Must be seen. No errors allowed FT their expression. Condone only missing brackets around \((-3\theta)\).
A1: AG www. Condone only missing brackets around \((-3\theta)\).