FP2 June 2018 Q3
3.
(a) By writing \(\dfrac{\pi}{12} = \dfrac{\pi}{3} - \dfrac{\pi}{4}\), show that
(i) \(\sin\left(\dfrac{\pi}{12}\right) = \dfrac{1}{4}\left(\sqrt{6} - \sqrt{2}\right)\)
(ii) \(\cos\left(\dfrac{\pi}{12}\right) = \dfrac{1}{4}\left(\sqrt{6} + \sqrt{2}\right)\) (4)
(b) Hence find the exact values of \(z\) for which \[z^4 = 4\left(\cos\frac{\pi}{3} + \mathrm{i}\sin\frac{\pi}{3}\right)\] Give your answers in the form \(z = a + \mathrm{i}b\) where \(a, b \in \mathbb{R}\) (5)
| Scheme | Marks |
|---|---|
| \(\sin\left(\dfrac{\pi}{3} - \dfrac{\pi}{4}\right) = \sin\dfrac{\pi}{3}\cos\dfrac{\pi}{4} - \cos\dfrac{\pi}{3}\sin\dfrac{\pi}{4}\) \(\sin\dfrac{\pi}{12} = \dfrac{\sqrt{3}}{2}\dfrac{\sqrt{2}}{2} - \dfrac{1}{2}\dfrac{\sqrt{2}}{2}\) Correct expansion for sine, including surd values for all 4 trig functions. \(\dfrac{\sqrt{2}}{2}\text{ or }\dfrac{1}{\sqrt{2}}\) accepted | M1 |
| (i) \(\sin\dfrac{\pi}{12} = \dfrac{\sqrt{6}}{4} - \dfrac{\sqrt{2}}{4} = \dfrac{1}{4}\left(\sqrt{6} - \sqrt{2}\right)\) ** Completion to given answer: No errors seen, cso | A1cso |
| \(\cos\left(\dfrac{\pi}{3} - \dfrac{\pi}{4}\right) = \cos\dfrac{\pi}{3}\cos\dfrac{\pi}{4} + \sin\dfrac{\pi}{3}\sin\dfrac{\pi}{4}\) \(\cos\dfrac{\pi}{12} = \dfrac{1}{2}\dfrac{\sqrt{2}}{2} + \dfrac{\sqrt{3}}{2}\dfrac{\sqrt{2}}{2}\) Correct expansion for cosine, including surd values for all 4 trig functions. \(\dfrac{\sqrt{2}}{2}\text{ or }\dfrac{1}{\sqrt{2}}\) accepted OR other complete method eg using \(\sin^2\theta + \cos^2\theta = 1\) | M1 NB A1 on e-PEN |
| (ii) \(\cos\left(\dfrac{\pi}{12}\right) = \dfrac{\sqrt{2}}{4} + \dfrac{\sqrt{6}}{4} = \dfrac{1}{4}\left(\sqrt{6} + \sqrt{2}\right)\) ** Completion to given answer: No errors seen, cso | A1cso |
| (4) |
| Scheme | Marks |
|---|---|
| Allow all marks using EXACT calculator values for the trig functions. Decimal answers qualify for M marks only. | |
| \(z^4 = 4\left(\cos\left(2k\pi + \dfrac{\pi}{3}\right) + \mathrm{i}\sin\left(2k\pi + \dfrac{\pi}{3}\right)\right)\) OR \(z^4 = 4\mathrm{e}^{\mathrm{i}\left(2k\pi + \frac{\pi}{3}\right)}\) Use a valid method to generate at least 2 roots (eg use of \(2k\pi\) or rotate through \(\dfrac{\pi}{2}\), multiply by I, symmetry) | M1 |
| \(z = 4^{\frac{1}{4}}\left(\cos\left(\dfrac{k\pi}{2} + \dfrac{\pi}{12}\right) + \mathrm{i}\sin\left(\dfrac{k\pi}{2} + \dfrac{\pi}{12}\right)\right)\) OR \(z = 4^{\frac{1}{4}}\mathrm{e}^{\mathrm{i}\left(\frac{k\pi}{2} + \frac{\pi}{12}\right)}\) Application of de Moivre’s theorem resulting in at least 1 root being found. (\(4 \to \sqrt{2}\) and arg divided by 4) \(4^{\frac{1}{4}}\) or \(\sqrt{2}\) accepted | M1 |
| \((k = 0 \to)\) \(z = \sqrt{2}\left(\cos\dfrac{\pi}{12} + \mathrm{i}\sin\dfrac{\pi}{12}\right)\) or \(\sqrt{2}\mathrm{e}^{\mathrm{i}\left(\frac{\pi}{12}\right)}\) or \(\dfrac{\sqrt{2}}{4}\left(\sqrt{6} + \sqrt{2}\right) + \dfrac{\mathrm{i}\sqrt{2}}{4}\left(\sqrt{6} - \sqrt{2}\right)\) or \(\dfrac{1 + \sqrt{3}}{2} + \mathrm{i}\dfrac{-1 + \sqrt{3}}{2}\) oe Any correct root (this is the most likely one if only one found) Can be in any exact form (\(4^{\frac{1}{4}}\) or \(\sqrt{2}\) oe) Can be unsimplified using results from (a) ie \(\dfrac{\sqrt{2}}{4}\left(\sqrt{6} + \sqrt{2}\right) + \dfrac{\mathrm{i}\sqrt{2}}{4}\left(\sqrt{6} - \sqrt{2}\right)\) with \(\dfrac{\sqrt{2}}{4}\) or \(\dfrac{1}{2\sqrt{2}}\) or \(4^{-\frac{3}{4}}\) oe Or simplified/calculator values ie \(\dfrac{1 + \sqrt{3}}{2} + \mathrm{i}\dfrac{-1 + \sqrt{3}}{2}\) | B1 |
| \(\left\{(k = 1 \to)\ z = \sqrt{2}\left(\cos\dfrac{7\pi}{12} + \mathrm{i}\sin\dfrac{7\pi}{12}\right) \text{ or } \sqrt{2}\mathrm{e}^{\mathrm{i}\left(\frac{7\pi}{12}\right)}\right\}\) \(= \dfrac{-\sqrt{2}}{4}\left(\sqrt{6} - \sqrt{2}\right) + \dfrac{\mathrm{i}\sqrt{2}}{4}\left(\sqrt{6} + \sqrt{2}\right)\) or \(\dfrac{1}{2}\left(1 - \sqrt{3}\right) + \dfrac{\mathrm{i}}{2}\left(1 + \sqrt{3}\right)\) | |
| \(\left\{(k = 2 \to)\ z = \sqrt{2}\left(\cos\dfrac{13\pi}{12} + \mathrm{i}\sin\dfrac{13\pi}{12}\right) \text{ or } \sqrt{2}\mathrm{e}^{\mathrm{i}\left(\frac{13\pi}{12}\right)}\right\}\) \(= \dfrac{-\sqrt{2}}{4}\left(\sqrt{6} + \sqrt{2}\right) - \dfrac{\mathrm{i}\sqrt{2}}{4}\left(\sqrt{6} - \sqrt{2}\right)\) or \(-\dfrac{1}{2}\left(1 + \sqrt{3}\right) + \dfrac{\mathrm{i}}{2}\left(1 - \sqrt{3}\right)\) | |
| \(\left\{(k = 3 \to)\ z = \sqrt{2}\left(\cos\dfrac{19\pi}{12} + \mathrm{i}\sin\dfrac{19\pi}{12}\right) \text{ or } \sqrt{2}\mathrm{e}^{\mathrm{i}\left(\frac{19\pi}{12}\right)}\right\}\) \(= \dfrac{\sqrt{2}}{4}\left(\sqrt{6} - \sqrt{2}\right) - \dfrac{\mathrm{i}\sqrt{2}}{4}\left(\sqrt{6} + \sqrt{2}\right)\) or \(\dfrac{1}{2}\left(-1 + \sqrt{3}\right) - \dfrac{\mathrm{i}}{2}\left(1 + \sqrt{3}\right)\) | |
| Two correct roots in form \(a + \mathrm{i}b\) unsimplified or calculator values, must be exact surd form | A1 |
| All 4 correct roots in form \(a + \mathrm{i}b\) unsimplified or calculator values must be exact surd form. | A1 |
| (5) | |
| (9 marks) |