Edexcel FP2 2006 January — Question 5 5 marks

Exam BoardEdexcel
ModuleFP2 (Further Pure Mathematics 2)
Year2006
SessionJanuary
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicComplex numbers 2
TypeDirect nth roots: find and express roots
DifficultyStandard +0.3 This is a standard Further Maths FP2 question requiring routine application of de Moivre's theorem to find fifth roots. Students express i in polar form, divide the argument by 5, and add 2πk/5 for k=0,1,2,3,4. While it's Further Maths content (inherently harder), it's a textbook exercise with no problem-solving required, making it slightly easier than average overall.
Spec4.02r nth roots: of complex numbers
5. Solve the equation \(z ^ { 5 } = \mathrm { i }\) giving your answers in the form \(\cos \theta + \mathrm { i } \sin \theta\).
(Total 5 marks)
AnswerMarks Guidance
\(\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}\)B1
\(\cos\frac{\pi}{10} + i\sin\frac{\pi}{10}\)B1
\(\cos\left(\frac{(4k + 1)\pi}{10}\right) + i\sin\left(\frac{(4k + 1)\pi}{10}\right), k = 2,3,4\) (or equiv.)M1, A2, 1, 0 5
\(\left[\cos\left(\frac{9\pi}{10}\right) + i\sin\left(\frac{9\pi}{10}\right), \cos\left(\frac{13\pi}{10}\right) + i\sin\left(\frac{13\pi}{10}\right), \cos\left(\frac{17\pi}{10}\right) + i\sin\left(\frac{17\pi}{10}\right)\right]\)
[Degrees: 18, 90, 162, 234, 306] [5] 
| $\cos\frac{\pi}{2} + i\sin\frac{\pi}{2}$ | B1 | |
| $\cos\frac{\pi}{10} + i\sin\frac{\pi}{10}$ | B1 | |
| $\cos\left(\frac{(4k + 1)\pi}{10}\right) + i\sin\left(\frac{(4k + 1)\pi}{10}\right), k = 2,3,4$ (or equiv.) | M1, A2, 1, 0 | 5 |
| $\left[\cos\left(\frac{9\pi}{10}\right) + i\sin\left(\frac{9\pi}{10}\right), \cos\left(\frac{13\pi}{10}\right) + i\sin\left(\frac{13\pi}{10}\right), \cos\left(\frac{17\pi}{10}\right) + i\sin\left(\frac{17\pi}{10}\right)\right]$ | | |
| [Degrees: 18, 90, 162, 234, 306] | | [5] |
5. Solve the equation $z ^ { 5 } = \mathrm { i }$\\
giving your answers in the form $\cos \theta + \mathrm { i } \sin \theta$.\\
(Total 5 marks)\\

\hfill \mbox{\textit{Edexcel FP2 2006 Q5 [5]}}
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Q1 6 Q2 13 Q3 14 Q4 15 Q5 5 Q7 11 Q8 12