A2 June 2019 Paper 2 Q8
8 In this question you must show detailed reasoning.
| Scheme | Marks | AO |
|---|---|---|
| DR \(\sin\theta = \dfrac{\mathrm{e}^{\mathrm{i}\theta} - \mathrm{e}^{-\mathrm{i}\theta}}{2\mathrm{i}}\) | *B1 | 1.1a |
| \(\sin^6\theta = \left(\dfrac{\mathrm{e}^{\mathrm{i}\theta} - \mathrm{e}^{-\mathrm{i}\theta}}{2\mathrm{i}}\right)^6 = -\dfrac{1}{64}\left(\mathrm{e}^{\mathrm{i}\theta} - \mathrm{e}^{-\mathrm{i}\theta}\right)^6\) | M1 | 2.1 |
| \((\mathrm{e}^{\mathrm{i}\theta} - \mathrm{e}^{-\mathrm{i}\theta})^6 =\) \(\mathrm{e}^{6\mathrm{i}\theta} - 6\mathrm{e}^{4\mathrm{i}\theta} + 15\mathrm{e}^{2\mathrm{i}\theta} - 20 + 15\mathrm{e}^{-2\mathrm{i}\theta} - 6\mathrm{e}^{-4\mathrm{i}\theta} + \mathrm{e}^{-6\mathrm{i}\theta}\) | M1 | 1.1 |
| \(\mathrm{e}^{6\mathrm{i}\theta} + \mathrm{e}^{-6\mathrm{i}\theta} - 6\left(\mathrm{e}^{4\mathrm{i}\theta} + \mathrm{e}^{-4\mathrm{i}\theta}\right) + 15\left(\mathrm{e}^{2\mathrm{i}\theta} + \mathrm{e}^{-2\mathrm{i}\theta}\right) - 20\) \(= 2\cos 6\theta - 6 \times 2\cos 4\theta + 15 \times 2\cos 2\theta - 20\) | M1 | 2.1 |
| \(\therefore \sin^6\theta =\) \(-\dfrac{1}{64}(2\cos 6\theta - 12\cos 4\theta + 30\cos 2\theta - 20)\) \(= \dfrac{1}{32}(10 - 15\cos 2\theta + 6\cos 4\theta - \cos 6\theta)\) | dep*A1 | 1.1 |
| [5] |
Notes
*B1: Condone \(2\mathrm{i}\sin\theta = \mathrm{e}^{\mathrm{i}\theta} - \mathrm{e}^{-\mathrm{i}\theta}\)
M1: Raising expression for \(\sin\theta\) to the power 6 and \((2\mathrm{i})^6 = -64\).
Allow use of \(\sin\theta = \frac{\mathrm{e}^{\mathrm{i}\theta} + \mathrm{e}^{-\mathrm{i}\theta}}{2\mathrm{i}}\) for 1st two M marks only
M1: Genuine attempt to use binomial expansion with correct evaluated binomial coefficients. Condone sign errors.
If i omitted from denominator their expression for \(\sin\theta\) then only this M mark can still be awarded
M1: Collecting terms and using \(\mathrm{e}^{\mathrm{i}\phi} + \mathrm{e}^{-\mathrm{i}\phi} = 2\cos\phi\) at least once.
dep*A1: AG. Fully correct argument
| Scheme | Marks | AO |
|---|---|---|
| DR \(\theta = \dfrac{\pi}{8}\) and eg \(\cos 2\theta = \dfrac{\sqrt{2}}{2}\) | *M1 | 2.1 |
| \(\sin^6\dfrac{\pi}{8} = \dfrac{1}{32}\left(10 - 15 \times \dfrac{\sqrt{2}}{2} - \dfrac{-\sqrt{2}}{2}(+6(0))\right)\) | dep*M1 | 1.1 |
| \(\sin\dfrac{\pi}{8} = \sqrt[6]{\dfrac{1}{64}\left(20 - 15\sqrt{2} + \sqrt{2}\right)}\) \(= \dfrac{1}{2}\sqrt[6]{20 - 14\sqrt{2}}\) | A1 | 2.2a |
| [3] |
Notes
*M1: Choice of \(\theta\) soi and calculation of at least one cos term.
dep*M1: Substitution and calculation of all cos terms. Terms must be shown distinct either in this line or in the form of \(\cos n\frac{\pi}{8}\)
A1: AG Some intermediate working must be seen