6. Solve the equation
$$z ^ { 5 } = - 16 \sqrt { } 3 + 16 i$$
giving your answers in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), where \(r > 0\) and \(- \pi < \theta < \pi\).
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Question 6:
Answer Marks
Guidance
Modulus \(= 32\) B1
Argument \(= \arctan\left(-\frac{1}{\sqrt{3}}\right) = \frac{5\pi}{6}\) M1 A1
\(z = 32^{\frac{1}{5}}\left(\cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right)\right)^{\frac{1}{5}} = 2\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right)\) M1 A1
OR \(2\left(\cos\left(\frac{\pi}{6} + \frac{2n\pi}{5}\right) + i\sin\left(\frac{\pi}{6} + \frac{2n\pi}{5}\right)\right)\) M1
\(2\left(\cos\frac{17\pi}{30} + i\sin\frac{17\pi}{30}\right),\ 2\left(\cos\frac{29\pi}{30} + i\sin\frac{29\pi}{30}\right)\) A1
\(2\left(\cos\frac{-7\pi}{30} + i\sin\frac{-7\pi}{30}\right),\ 2\left(\cos\frac{-19\pi}{30} + i\sin\frac{-19\pi}{30}\right)\) A1
(8 marks)
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## Question 6:
| Modulus $= 32$ | B1 | |
| Argument $= \arctan\left(-\frac{1}{\sqrt{3}}\right) = \frac{5\pi}{6}$ | M1 A1 | |
| $z = 32^{\frac{1}{5}}\left(\cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right)\right)^{\frac{1}{5}} = 2\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right)$ | M1 A1 | |
| OR $2\left(\cos\left(\frac{\pi}{6} + \frac{2n\pi}{5}\right) + i\sin\left(\frac{\pi}{6} + \frac{2n\pi}{5}\right)\right)$ | M1 | |
| $2\left(\cos\frac{17\pi}{30} + i\sin\frac{17\pi}{30}\right),\ 2\left(\cos\frac{29\pi}{30} + i\sin\frac{29\pi}{30}\right)$ | A1 | |
| $2\left(\cos\frac{-7\pi}{30} + i\sin\frac{-7\pi}{30}\right),\ 2\left(\cos\frac{-19\pi}{30} + i\sin\frac{-19\pi}{30}\right)$ | A1 | (8 marks) |
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6. Solve the equation
$$z ^ { 5 } = - 16 \sqrt { } 3 + 16 i$$
giving your answers in the form $r ( \cos \theta + \mathrm { i } \sin \theta )$, where $r > 0$ and $- \pi < \theta < \pi$.\\
\hfill \mbox{\textit{Edexcel FP2 2013 Q6 [8]}}