A2 June 2019 Paper 1 Q16
16
Series \(C\) and \(S\) are defined by
\[\begin{aligned} C &= \frac{1}{2}\cos\theta + \frac{1}{4}\cos 2\theta + \frac{1}{8}\cos 3\theta + \ldots + \frac{1}{2^n}\cos n\theta, \\ S &= \frac{1}{2}\sin\theta + \frac{1}{4}\sin 2\theta + \frac{1}{8}\sin 3\theta + \ldots + \frac{1}{2^n}\sin n\theta. \end{aligned}\]| Scheme | Marks | AO |
|---|---|---|
| \((2 - \mathrm{e}^{\mathrm{i}\theta})(2 - \mathrm{e}^{-\mathrm{i}\theta}) = 4 - 2(\mathrm{e}^{\mathrm{i}\theta} + \mathrm{e}^{-\mathrm{i}\theta}) + 1\) | M1 | 1.1b |
| \(= 5 - 2.2\cos\theta\) | M1 | 1.1b |
| \(= 5 - 4\cos\theta\,*\) | A1 | 2.2a |
| [3] |
Notes
M1: \(\mathrm{e}^{\mathrm{i}\theta}.\mathrm{e}^{-\mathrm{i}\theta} = 1\)
M1: \(\mathrm{e}^{\mathrm{i}\theta} + \mathrm{e}^{-\mathrm{i}\theta} = 2\cos\theta\) used
A1: NB AG
| Scheme | Marks | AO |
|---|---|---|
| \(C + \mathrm{i}S = \dfrac{1}{2}\mathrm{e}^{\mathrm{i}\theta} + \dfrac{1}{4}\mathrm{e}^{2\mathrm{i}\theta} + \dfrac{1}{8}\mathrm{e}^{3\mathrm{i}\theta} + \ldots + \dfrac{1}{2^n}\mathrm{e}^{n\mathrm{i}\theta}\) | M1 A1 | 3.1a 2.1 |
| \(= \dfrac{\frac{1}{2}\mathrm{e}^{\mathrm{i}\theta}\left(1 - (\frac{1}{2}\mathrm{e}^{\mathrm{i}\theta})^n\right)}{1 - \frac{1}{2}\mathrm{e}^{\mathrm{i}\theta}}\) | M1 A1 | 2.1 1.1b |
| (4) | ||
| \(= \dfrac{\mathrm{e}^{\mathrm{i}\theta}\left(1 - (\frac{1}{2}\mathrm{e}^{\mathrm{i}\theta})^n\right)(2 - \mathrm{e}^{-\mathrm{i}\theta})}{(2 - \mathrm{e}^{\mathrm{i}\theta})(2 - \mathrm{e}^{-\mathrm{i}\theta})}\) | M1 | 2.1 |
| \(= \dfrac{\mathrm{e}^{\mathrm{i}\theta}\left(2 - \frac{1}{2^{n-1}}\mathrm{e}^{n\mathrm{i}\theta} - \mathrm{e}^{-\mathrm{i}\theta} + \frac{1}{2^n}\mathrm{e}^{(n-1)\mathrm{i}\theta}\right)}{5 - 4\cos\theta}\) | M1 A1 | 2.1 |
| (3) | ||
| \(= \dfrac{2^{n+1}\mathrm{e}^{\mathrm{i}\theta} - 2\mathrm{e}^{(n+1)\mathrm{i}\theta} - 2^n + \mathrm{e}^{n\mathrm{i}\theta}}{2^n(5 - 4\cos\theta)}\) \(C = \mathrm{Re}\left(\dfrac{2^{n+1}\mathrm{e}^{\mathrm{i}\theta} - 2\mathrm{e}^{(n+1)\mathrm{i}\theta} - 2^n + \mathrm{e}^{n\mathrm{i}\theta}}{2^n(5 - 4\cos\theta)}\right)\) | M1 | 2.1 |
| \(= \dfrac{2^n(2\cos\theta - 1) - 2\cos(n + 1)\theta + \cos n\theta}{2^n(5 - 4\cos\theta)}\,*\) | A1cao | 2.2a |
| (2) | ||
| [9] |
Notes
M1: at least 2 terms
A1: correct (\(n\)th term soi)
M1: sum of GP (condone \(S_\infty\))
A1: correct expression
M1: \(\times\) top and bottom by complex conjugate
M1: expand brackets; allow 1 slip, not on \(S_\infty\)
M1: taking real part; not on \(S_\infty\)
A1cao: NB AG; need evidence of clearing subsidiary denominators
Alternative solution
| Scheme | Marks |
|---|---|
| \(C = \dfrac{1}{2}\left(\dfrac{\mathrm{e}^{\mathrm{i}\theta} + \mathrm{e}^{-\mathrm{i}\theta}}{2}\right) + \dfrac{1}{4}\left(\dfrac{\mathrm{e}^{2\mathrm{i}\theta} + \mathrm{e}^{-2\mathrm{i}\theta}}{2}\right) + \ldots + \dfrac{1}{2^n}\left(\dfrac{\mathrm{e}^{n\mathrm{i}\theta} + \mathrm{e}^{-n\mathrm{i}\theta}}{2}\right)\) | M1 |
| \(= \dfrac{1}{2}\left[\dfrac{\frac{1}{2}\mathrm{e}^{\mathrm{i}\theta}\left(1 - (\frac{1}{2}\mathrm{e}^{\mathrm{i}\theta})^n\right)}{1 - \frac{1}{2}\mathrm{e}^{\mathrm{i}\theta}} + \dfrac{\frac{1}{2}\mathrm{e}^{-\mathrm{i}\theta}\left(1 - (\frac{1}{2}\mathrm{e}^{-\mathrm{i}\theta})^n\right)}{1 - \frac{1}{2}\mathrm{e}^{-\mathrm{i}\theta}}\right]\) | M1 A1 |
| \(= \dfrac{1}{2} \times \dfrac{\mathrm{e}^{\mathrm{i}\theta}\left(1 - (\frac{1}{2}\mathrm{e}^{\mathrm{i}\theta})^n\right)(2 - \mathrm{e}^{-\mathrm{i}\theta}) + \mathrm{e}^{-\mathrm{i}\theta}\left(1 - (\frac{1}{2}\mathrm{e}^{-\mathrm{i}\theta})^n\right)(2 - \mathrm{e}^{\mathrm{i}\theta})}{(2 - \mathrm{e}^{\mathrm{i}\theta})(2 - \mathrm{e}^{-\mathrm{i}\theta})}\) | M1 A1 |
| \(= \dfrac{\mathrm{e}^{\mathrm{i}\theta} + \mathrm{e}^{-\mathrm{i}\theta} - \frac{1}{2^n}(\mathrm{e}^{(n+1)\mathrm{i}\theta} + \mathrm{e}^{-(n+1)\mathrm{i}\theta}) - 1 + \frac{1}{2^{n+1}}(\mathrm{e}^{n\mathrm{i}\theta} + \mathrm{e}^{-n\mathrm{i}\theta})}{(2 - \mathrm{e}^{\mathrm{i}\theta})(2 - \mathrm{e}^{-\mathrm{i}\theta})}\) | M1 A1 |
| \(= \dfrac{2\cos\theta - \frac{1}{2^{n-1}}\cos(n + 1)\theta - 1 + \frac{1}{2^n}\cos n\theta}{5 - 4\cos\theta}\) | M1 |
| \(= \dfrac{2^n(2\cos\theta - 1) - 2\cos(n + 1)\theta + \cos n\theta}{2^n(5 - 4\cos\theta)}\,*\) | A1 |
| [9] |
M1: substituting for cosines in terms of \(\mathrm{e}^{\mathrm{i}\theta}\)s; M1 A1: sum of GP; M1 A1: combining fractions; M1 A1: expanding; M1: expressing in cosines; A1: NB AG
(corrected from the printed mark scheme: in the printed alternative the first line has a stray factor \(\mathrm{e}^{\mathrm{i}\theta}\) after the first term and is cut off after \(\frac{1}{2^n}\); the second and third lines omit the overall factor \(\frac{1}{2}\); and the constant term in the fourth and fifth lines is printed as \(-2\) instead of \(-1\).)