Roots of unity with derived equations

Questions that use roots of unity to derive or solve related polynomial equations, often involving transformations like (z+1)^n = z^n or finding roots of equations whose solutions are expressed in terms of roots of unity.

5 questions · Challenging +1.1

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CAIE Further Paper 2 2020 November Q3
4 marks Standard +0.3
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.
OCR FP3 2013 January Q5
7 marks Challenging +1.2
5
  1. Solve the equation \(z ^ { 5 } = 1\), giving your answers in polar form.
  2. Hence, by considering the equation \(( z + 1 ) ^ { 5 } = z ^ { 5 }\), show that the roots of $$5 z ^ { 4 } + 10 z ^ { 3 } + 10 z ^ { 2 } + 5 z + 1 = 0$$ can be expressed in the form \(\frac { 1 } { \mathrm { e } ^ { \mathrm { i } \theta } - 1 }\), stating the values of \(\theta\).
CAIE FP1 2012 June Q6
9 marks Challenging +1.2
6 Write down the values of \(\theta\), in the interval \(0 \leqslant \theta < 2 \pi\), for which \(\cos \theta + \mathrm { i } \sin \theta\) is a fifth root of unity. By writing the equation \(( z + 1 ) ^ { 5 } = z ^ { 5 }\) in the form $$\left( \frac { z + 1 } { z } \right) ^ { 5 } = 1$$ show that its roots are $$- \frac { 1 } { 2 } \left\{ 1 + \mathrm { i } \cot \left( \frac { k \pi } { 5 } \right) \right\} , \quad k = 1,2,3,4$$
OCR Further Pure Core 1 2022 June Q9
5 marks Challenging +1.8
9 The cube roots of unity are represented on the Argand diagram below by the points \(A , B\) and \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{23e58e5e-bbaa-4932-aad0-89b3de6647b2-8_760_800_303_244} The points \(L , M\) and \(N\) are the midpoints of the line segments \(A B , B C\) and \(C A\) respectively. Determine a degree 6 polynomial equation with integer coefficients whose roots are the complex numbers represented by the points \(A , B , C , L , M\) and \(N\). \section*{END OF QUESTION PAPER}
AQA FP2 2006 June Q7
17 marks Challenging +1.2
7
  1. Find the six roots of the equation \(z ^ { 6 } = 1\), giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \phi }\), where \(- \pi < \phi \leqslant \pi\).
  2. It is given that \(w = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta \neq n \pi\).
    1. Show that \(\frac { w ^ { 2 } - 1 } { w } = 2 \mathrm { i } \sin \theta\).
    2. Show that \(\frac { w } { w ^ { 2 } - 1 } = - \frac { \mathrm { i } } { 2 \sin \theta }\).
    3. Show that \(\frac { 2 \mathrm { i } } { w ^ { 2 } - 1 } = \cot \theta - \mathrm { i }\).
    4. Given that \(z = \cot \theta - \mathrm { i }\), show that \(z + 2 \mathrm { i } = z w ^ { 2 }\).
    1. Explain why the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ has five roots.
    2. Find the five roots of the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ giving your answers in the form \(a + \mathrm { i } b\).