Question 3 - A-Level Maths

CAIE Further Paper 2 2020 November — Question 3 4 marks

Exam Board	CAIE
Module	Further Paper 2 (Further Paper 2)
Year	2020
Session	November
Marks	4
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Complex numbers 2
Type	Roots of unity with derived equations
Difficulty	Standard +0.3 This is a straightforward roots of unity problem requiring a standard substitution (z = w+1), finding the 6th roots of unity using de Moivre's theorem, then back-substituting. While it involves complex numbers and exact trigonometric values, it's a routine Further Maths technique with no novel insight required, making it slightly easier than average.
Spec	4.02r nth roots: of complex numbers

3 Find all the roots of the equation \(( w + 1 ) ^ { 6 } = 1\), giving your answers in the form \(\mathrm { x } + \mathrm { iy }\) where \(x\) and \(y\) are real and exact.

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Question 3:

Answer	Marks	Guidance
\(w+1 = \cos\frac{2n}{6}\pi + i\sin\frac{2n}{6}\pi,\quad n=0,\ldots,5\)	M1 A1	Uses the 6th roots of unity
\(w = 0,\ -\frac{1}{2}\pm\frac{1}{2}i\sqrt{3},\ -\frac{3}{2}\pm\frac{1}{2}i\sqrt{3},\ -2\)	A1 A1	A1 for 3 correct roots; A1 gives all six roots in exact form

## Question 3:

| $w+1 = \cos\frac{2n}{6}\pi + i\sin\frac{2n}{6}\pi,\quad n=0,\ldots,5$ | M1 A1 | Uses the 6th roots of unity |
|---|---|---|
| $w = 0,\ -\frac{1}{2}\pm\frac{1}{2}i\sqrt{3},\ -\frac{3}{2}\pm\frac{1}{2}i\sqrt{3},\ -2$ | A1 A1 | A1 for 3 correct roots; A1 gives all six roots in exact form |

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3 Find all the roots of the equation $( w + 1 ) ^ { 6 } = 1$, giving your answers in the form $\mathrm { x } + \mathrm { iy }$ where $x$ and $y$ are real and exact.\\

\hfill \mbox{\textit{CAIE Further Paper 2 2020 Q3 [4]}}

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