Roots of unity and special equations

A question is this sub-type if and only if it involves finding roots of equations of the form z^n = c (including roots of unity) or other special polynomial forms that can be solved using exponential/polar form.

5 questions · Moderate -0.7

4.02r nth roots: of complex numbers
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Edexcel CP2 2024 June Q7
9 marks Moderate -0.8
  1. (a) Determine the roots of the equation
$$z ^ { 6 } = 1$$ giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\) (b) Show the roots of the equation in part (a) on a single Argand diagram.
(c) Show that $$( \sqrt { 3 } + i ) ^ { 6 } = - 64$$ (d) Hence, or otherwise, solve the equation $$z ^ { 6 } + 64 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(0 \leqslant \theta < 2 \pi\)
Pre-U Pre-U 9794/1 2018 June Q3
4 marks Easy -1.2
3 Given that \(z = 1\) is the real root of the equation \(z ^ { 3 } - 1 = 0\), find the two complex roots.
CAIE Further Paper 2 2023 November Q2
5 marks Standard +0.3
Find the roots of the equation \((z + 5i)^3 = 4 + 4\sqrt{3}i\), giving your answers in the form \(r\cos\theta + ir\sin\theta - 5)\), where \(r > 0\) and \(0 \leq \theta < 2\pi\). [5]
Edexcel FP1 Q11
7 marks Moderate -0.8
  1. Using that 3 is the real root of the cubic equation \(x^3 - 27 = 0\), show that the complex roots of the cubic satisfy the quadratic equation \(x^2 + 3x + 9 = 0\). [2]
  2. Hence, or otherwise, find the three cube roots of 27, giving your answers in the form \(a + ib\), where \(a, b \in \mathbb{R}\). [3]
  3. Show these roots on an Argand diagram. [2]
Edexcel FP1 Q45
7 marks Moderate -0.8
  1. Write down the value of the real root of the equation \(x^3 - 64 = 0\). [1]
  2. Find the complex roots of \(x^3 - 64 = 0\) , giving your answers in the form \(a + ib\), where \(a\) and \(b\) are real. [4]
  3. Show the three roots of \(x^3 - 64 = 0\) on an Argand diagram. [2]