Edexcel F2 2014 June — Question 3 5 marks

Exam BoardEdexcel
ModuleF2 (Further Pure Mathematics 2)
Year2014
SessionJune
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeDirect nth roots: general complex RHS
DifficultyStandard +0.3 This is a standard Further Maths question requiring conversion to polar form, application of De Moivre's theorem for nth roots, and systematic enumeration of all five roots. While it involves multiple steps (finding modulus/argument, dividing by 5, adding 2πk/5), these are routine procedures for FM students with no novel insight required.
Spec4.02d Exponential form: re^(i*theta)4.02r nth roots: of complex numbers
3. Solve the equation $$z ^ { 5 } = 16 - 16 \mathrm { i } \sqrt { 3 }$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(\theta\) is in terms of \(\pi\) and \(0 \leqslant \theta < 2 \pi\).
Question 3:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(r^5 = \sqrt{16^2+(16\sqrt{3})^2} = 32 \Rightarrow r = 32^{\frac{1}{5}} (=2)\)B1 Correct value for \(r\)
\(\arg(16-16i\sqrt{3}) = \frac{5\pi}{3}\)B1 Allow \(\frac{5\pi}{3}\) or \(-\frac{\pi}{3}\)
\(5\theta = \frac{11\pi}{3}, \frac{17\pi}{3}, \frac{23\pi}{3}, \frac{29\pi}{3}\)M1 \(\left(\frac{5\pi}{3}\right)+2n\pi,\ n=1,2,3,4\); at least 2 values which must be positive
\(z = 2e^{\frac{\pi}{5}i},\ 2e^{\frac{11\pi}{15}i},\ 2e^{\frac{17\pi}{15}i},\ 2e^{\frac{23\pi}{15}i},\ 2e^{\frac{29\pi}{15}i}\)B1 A1 2 or \(32^{\frac{1}{5}}\); \(e^{\frac{5\pi}{15}i}\) or \(e^{\frac{\pi}{3}i}\); all 4 remaining values 
# Question 3:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $r^5 = \sqrt{16^2+(16\sqrt{3})^2} = 32 \Rightarrow r = 32^{\frac{1}{5}} (=2)$ | B1 | Correct value for $r$ |
| $\arg(16-16i\sqrt{3}) = \frac{5\pi}{3}$ | B1 | Allow $\frac{5\pi}{3}$ or $-\frac{\pi}{3}$ |
| $5\theta = \frac{11\pi}{3}, \frac{17\pi}{3}, \frac{23\pi}{3}, \frac{29\pi}{3}$ | M1 | $\left(\frac{5\pi}{3}\right)+2n\pi,\ n=1,2,3,4$; at least 2 values which must be positive |
| $z = 2e^{\frac{\pi}{5}i},\ 2e^{\frac{11\pi}{15}i},\ 2e^{\frac{17\pi}{15}i},\ 2e^{\frac{23\pi}{15}i},\ 2e^{\frac{29\pi}{15}i}$ | B1 A1 | 2 or $32^{\frac{1}{5}}$; $e^{\frac{5\pi}{15}i}$ or $e^{\frac{\pi}{3}i}$; all 4 remaining values |

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3. Solve the equation

$$z ^ { 5 } = 16 - 16 \mathrm { i } \sqrt { 3 }$$

giving your answers in the form $r \mathrm { e } ^ { \mathrm { i } \theta }$ where $\theta$ is in terms of $\pi$ and $0 \leqslant \theta < 2 \pi$.\\

\hfill \mbox{\textit{Edexcel F2 2014 Q3 [5]}}
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