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.

7 questions · Standard +0.9

4.02r nth roots: of complex numbers
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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 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]
[diagram]
\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 \(p\).