FP2 June 2009 Q2
2. Solve the equation \[z^3 = 4\sqrt{2} - 4\sqrt{2}\mathrm{i},\] giving your answers in the form \(r(\cos\theta + \mathrm{i}\sin\theta)\), where \(-\pi < \theta \leqslant \pi\). (6)
\(z^3 = 4\sqrt{2} - 4\sqrt{2}\mathrm{i},\ -\pi < \theta \leqslant \pi\)
| Scheme | Marks |
|---|---|
| \(r = \sqrt{(4\sqrt{2})^2 + (-4\sqrt{2})^2} = \sqrt{32 + 32} = \sqrt{64} = 8\) \(\theta = -\tan^{-1}\left(\frac{4\sqrt{2}}{4\sqrt{2}}\right) = -\frac{\pi}{4}\) A valid attempt to find the modulus and argument of \(4\sqrt{2} - 4\sqrt{2}\mathrm{i}\). | M1 |
| \(z^3 = 8\left(\cos\left(-\frac{\pi}{4}\right) + \mathrm{i}\sin\left(-\frac{\pi}{4}\right)\right)\) | |
| So, \(z = (8)^{\frac{1}{3}}\left(\cos\left(\dfrac{-\frac{\pi}{4}}{3}\right) + \mathrm{i}\sin\left(\dfrac{-\frac{\pi}{4}}{3}\right)\right)\) Taking the cube root of the modulus and dividing the argument by 3. | M1 |
| \(\Rightarrow z = 2\left(\cos\left(-\frac{\pi}{12}\right) + \mathrm{i}\sin\left(-\frac{\pi}{12}\right)\right)\) \(2\left(\cos\left(-\frac{\pi}{12}\right) + \mathrm{i}\sin\left(-\frac{\pi}{12}\right)\right)\) | A1 |
| Also, \(z^3 = 8\left(\cos\left(\frac{7\pi}{4}\right) + \mathrm{i}\sin\left(\frac{7\pi}{4}\right)\right)\) or \(z^3 = 8\left(\cos\left(-\frac{9\pi}{4}\right) + \mathrm{i}\sin\left(-\frac{9\pi}{4}\right)\right)\) Adding or subtracting \(2\pi\) to the argument for \(z^3\) in order to find other roots. | M1 |
| \(\Rightarrow z = 2\left(\cos\frac{7\pi}{12} + \mathrm{i}\sin\frac{7\pi}{12}\right)\) Any one of the final two roots | A1 |
| and \(z = 2\left(\cos\left(\frac{-3\pi}{4}\right) + \mathrm{i}\sin\left(\frac{-3\pi}{4}\right)\right)\) Both of the final two roots. | A1 |
| (6 marks) |
Notes
Special Case 1: Award SC: M1M1A1M1A0A0 for ALL three of \(2\left(\cos\frac{\pi}{12} + \mathrm{i}\sin\frac{\pi}{12}\right)\), \(2\left(\cos\frac{3\pi}{4} + \mathrm{i}\sin\frac{3\pi}{4}\right)\) and \(2\left(\cos\left(\frac{-7\pi}{12}\right) + \mathrm{i}\sin\left(\frac{-7\pi}{12}\right)\right)\).
Special Case 2: If \(r\) is incorrect (and not equal to 8) and candidate states the brackets ( ) correctly then give the first accuracy mark ONLY where this is applicable.