Roots of unity applications

A question is this type if and only if it involves finding or using nth roots of unity, including their geometric representation or algebraic properties.

9 questions · Standard +0.8

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CAIE FP1 2013 June Q11 OR
Standard +0.8
Show the cube roots of 1 on an Argand diagram. Show that the two non-real cube roots can be expressed in the form \(\omega\) and \(\omega ^ { 2 }\), and find these cube roots in exact cartesian form \(x + i y\). Evaluate the determinant $$\left| \begin{array} { c c c } 1 & 3 \omega & 2 \omega ^ { 2 } \\ 3 \omega ^ { 2 } & 2 & \omega \\ 2 \omega & \omega ^ { 2 } & 3 \end{array} \right|$$ It is given that \(z = ( 4 \sqrt { } 3 ) \left( \cos \frac { 4 } { 3 } \pi + i \sin \frac { 4 } { 3 } \pi \right) - 4 \left( \cos \frac { 11 } { 6 } \pi + i \sin \frac { 11 } { 6 } \pi \right)\). Express \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\), giving exact values for \(r\) and \(\theta\). Hence find the cube roots of \(z\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
CAIE FP1 2018 June Q11 EITHER
Challenging +1.2
  1. Show that if \(z = \mathrm { e } ^ { \mathrm { i } \theta }\) and \(z \neq - 1\) then $$\frac { z - 1 } { z + 1 } = \mathrm { i } \tan \frac { 1 } { 2 } \theta$$
  2. Hence, or otherwise, show that if \(z\) is a cube root of unity then $$\frac { z ^ { 3 } - 1 } { z ^ { 3 } + 1 } + \frac { z ^ { 2 } - 1 } { z ^ { 2 } + 1 } + \frac { z - 1 } { z + 1 } = 0$$
  3. Hence write down three roots of the equation $$\left( z ^ { 3 } - 1 \right) \left( z ^ { 2 } + 1 \right) ( z + 1 ) + \left( z ^ { 2 } - 1 \right) \left( z ^ { 3 } + 1 \right) ( z + 1 ) + ( z - 1 ) \left( z ^ { 3 } + 1 \right) \left( z ^ { 2 } + 1 \right) = 0$$ and find the other three roots. Give your answers in an exact form.
CAIE FP1 2005 November Q1
4 marks Standard +0.8
1 Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z ^ { 5 } = - 16 + ( 16 \sqrt { } 3 ) i$$ giving each root in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).
CAIE FP1 2007 November Q9
10 marks Challenging +1.2
9 Write down, in any form, all the roots of the equation \(z ^ { 5 } - 1 = 0\). Hence find all the roots of the equation $$( w - 1 ) ^ { 4 } + ( w - 1 ) ^ { 3 } + ( w - 1 ) ^ { 2 } + w = 0$$ and deduce that none of them is real. Find the arguments of the two roots which have the smaller modulus.
Edexcel CP2 2021 June Q8
11 marks Standard +0.8
  1. (i) The point \(P\) is one vertex of a regular pentagon in an Argand diagram.
The centre of the pentagon is at the origin.
Given that \(P\) represents the complex number \(6 + 6 \mathrm { i }\), determine the complex numbers that represent the other vertices of the pentagon, giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) (ii) (a) On a single Argand diagram, shade the region, \(R\), that satisfies both $$| z - 2 i | \leqslant 2 \quad \text { and } \quad \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi$$ (b) Determine the exact area of \(R\), giving your answer in simplest form.
Edexcel CP2 2023 June Q5
9 marks Challenging +1.2
  1. The points representing the complex numbers \(z _ { 1 } = 35 - 25 i\) and \(z _ { 2 } = - 29 + 39 i\) are opposite vertices of a regular hexagon, \(H\), in the complex plane.
The centre of \(H\) represents the complex number \(\alpha\)
  1. Show that \(\alpha = 3 + 7 \mathrm { i }\) Given that \(\beta = \frac { 1 + \mathrm { i } } { 64 }\)
  2. show that $$\beta \left( z _ { 1 } - \alpha \right) = 1$$ The vertices of \(H\) are given by the roots of the equation $$( \beta ( z - \alpha ) ) ^ { 6 } = 1$$
    1. Write down the roots of the equation \(w ^ { 6 } = 1\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\)
    2. Hence, or otherwise, determine the position of the other four vertices of \(H\), giving your answers as complex numbers in Cartesian form.
OCR MEI FP2 2009 January Q2
18 marks Standard +0.3
  1. Write down the modulus and argument of the complex number \(\mathrm { e } ^ { \mathrm { j } \pi / 3 }\).
  2. The triangle OAB in an Argand diagram is equilateral. O is the origin; A corresponds to the complex number \(a = \sqrt { 2 } ( 1 + \mathrm { j } ) ; \mathrm { B }\) corresponds to the complex number \(b\). Show A and the two possible positions for B in a sketch. Express \(a\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\). Find the two possibilities for \(b\) in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\).
  3. Given that \(z _ { 1 } = \sqrt { 2 } \mathrm { e } ^ { \mathrm { j } \pi / 3 }\), show that \(z _ { 1 } ^ { 6 } = 8\). Write down, in the form \(r \mathrm { e } ^ { \mathrm { j } \theta }\), the other five complex numbers \(z\) such that \(z ^ { 6 } = 8\). Sketch all six complex numbers in a new Argand diagram. Let \(w = z _ { 1 } \mathrm { e } ^ { - \mathrm { j } \pi / 12 }\).
  4. Find \(w\) in the form \(x + \mathrm { j } y\), and mark this complex number on your Argand diagram.
  5. Find \(w ^ { 6 }\), expressing your answer in as simple a form as possible.
OCR MEI Further Pure Core 2020 November Q11
8 marks Standard +0.8
11 In this question you must show detailed reasoning. In Fig. 11, the points \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E }\) and F represent the complex sixth roots of 64 on an Argand diagram. The midpoints of \(\mathrm { AB } , \mathrm { BC } , \mathrm { CD } , \mathrm { DE } , \mathrm { EF }\) and FA are \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L respectively. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c2be8838-50ec-4e82-b203-4608ab56c110-5_807_872_443_239} \captionsetup{labelformat=empty} \caption{Fig. 11}
\end{figure}
  1. Write down, in exponential ( \(r \mathrm { e } ^ { \mathrm { i } \theta }\) ) form, the complex numbers represented by the points \(\mathrm { A } , \mathrm { B }\), \(\mathrm { C } , \mathrm { D } , \mathrm { E }\) and F .
  2. When these complex numbers are multiplied by the complex number \(w\), the resulting complex numbers are represented by the points G, H, I, J, K and L. Find \(w\) in exponential form.
  3. You are given that \(\mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm { J } , \mathrm { K }\) and L represent roots of the equation \(z ^ { 6 } = p\). Find \(p\).
AQA Further Paper 2 2023 June Q6
5 marks Standard +0.3
6
  1. Express \(- 5 - 5 \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leq \pi\) 6
  2. The point on an Argand diagram that represents \(- 5 - 5 \mathrm { i }\) is one of the vertices of an equilateral triangle whose centre is at the origin. Find the complex numbers represented by the other two vertices of the triangle.
    Give your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leq \pi\)