FP2 June 2010 Q4
4. \[z = -8 + (8\sqrt{3})\mathrm{i}\]
(a) Find the modulus of \(z\) and the argument of \(z\). (3)
Using de Moivre’s theorem,
(b) find \(z^3\), (2)
(c) find the values of \(w\) such that \(w^4 = z\), giving your answers in the form \(a + \mathrm{i}b\), where \(a, b \in \mathbb{R}\). (5)
| Scheme | Marks |
|---|---|
| Modulus \(= 16\) | B1 |
| Argument \(= \arctan(-\sqrt{3}) = \dfrac{2\pi}{3}\) | M1A1 |
| (3) |
| Scheme | Marks |
|---|---|
| \(z^3 = 16^3\left(\cos\left(\frac{2\pi}{3}\right) + \mathrm{i}\sin\left(\frac{2\pi}{3}\right)\right)^3 = 16^3(\cos 2\pi + \mathrm{i}\sin 2\pi) = 4096\) or \(16^3\) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(w = 16^{\frac{1}{4}}\left(\cos\left(\frac{2\pi}{3}\right) + \mathrm{i}\sin\left(\frac{2\pi}{3}\right)\right)^{\frac{1}{4}} = 2\left(\cos\left(\frac{\pi}{6}\right) + \mathrm{i}\sin\left(\frac{\pi}{6}\right)\right)\ \left(= \sqrt{3} + \mathrm{i}\right)\) | M1 A1ft |
| OR \(-1 + \sqrt{3}\mathrm{i}\) OR \(-\sqrt{3} - \mathrm{i}\) OR \(1 - \sqrt{3}\mathrm{i}\) | M1A2(1,0) |
| (5) | |
| (10 marks) |