FP2 June 2017 Q3
3. Solve the equation \[z^3 + 32 + 32\mathrm{i}\sqrt{3} = 0\] giving your answers in the form \(r\mathrm{e}^{\mathrm{i}\theta}\) where \(r \gt 0\) and \(-\pi \lt \theta \leqslant \pi\) (6)
| Scheme | Marks |
|---|---|
| \(z^3 + 32 + 32\mathrm{i}\sqrt{3} = 0\) | |
| \(\arg\left(z^3\right) = \dfrac{4\pi}{3}\) or \(-\dfrac{2\pi}{3}\) M1: Uses tan to find arg \(z^3\) \(\arctan\sqrt{3},\ \arctan\dfrac{1}{\sqrt{3}},\ \dfrac{\pi}{3}\) or \(\dfrac{\pi}{6}\) seen. Allow equivalent angles A1: Either of values shown | M1A1 |
| \(|z| = r = 4\) Correct \(r\) seen anywhere (eg only in answers) | B1 |
| \(3\theta = \dfrac{4\pi}{3},\ -\dfrac{2\pi}{3},\ -\dfrac{8\pi}{3}\) | |
| \(\theta = \dfrac{4\pi}{9},\ -\dfrac{2\pi}{9},\ -\dfrac{8\pi}{9}\) Divides by 3 to obtain at least 2 values of \(\theta\) which differ by \(\dfrac{2\pi}{3}\) or \(\dfrac{4\pi}{3}\). | M1 |
| \(\theta = \dfrac{4\pi}{9},\ -\dfrac{2\pi}{9}\) or \(\dfrac{16\pi}{9},\ -\dfrac{8\pi}{9}\) or \(\dfrac{10\pi}{9}\) At least 2 correct (and distinct) values from list shown | A1 |
| \(z = 4\mathrm{e}^{\frac{4\pi}{9}\mathrm{i}},\ 4\mathrm{e}^{-\frac{2\pi}{9}\mathrm{i}},\ 4\mathrm{e}^{-\frac{8\pi}{9}\mathrm{i}}\) or \(4\mathrm{e}^{\mathrm{i}\theta}\) where \(\theta = \ldots\) A1: All correct and in either of the forms shown Ignore extra answers outside the range | A1 |
| (6 marks) |