Solve polynomial equations with complex roots

A question is this type if and only if it asks to solve a polynomial equation (typically quadratic or cubic) with complex coefficients or to find complex roots, giving answers in Cartesian form x + iy.

8 questions · Standard +0.5

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CAIE P3 2023 March Q4
5 marks Standard +0.8
4 Solve the equation $$\frac { 5 z } { 1 + 2 \mathrm { i } } - z z ^ { * } + 30 + 10 \mathrm { i } = 0$$ giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
CAIE P3 2022 November Q6
6 marks Standard +0.8
6 Solve the quadratic equation \(( 1 - 3 \mathrm { i } ) z ^ { 2 } - ( 2 + \mathrm { i } ) z + \mathrm { i } = 0\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
OCR FP3 2007 January Q3
7 marks Standard +0.3
3
  1. Solve the equation \(z ^ { 2 } - 6 z + 36 = 0\), and give your answers in the form \(r ( \cos \theta \pm \mathrm { i } \sin \theta )\), where \(r > 0\) and \(0 \leqslant \theta \leqslant \pi\).
  2. Given that \(Z\) is either of the roots found in part (i), deduce the exact value of \(Z ^ { - 3 }\).
AQA FP3 Q6
17 marks Challenging +1.2
6 It is given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\).
    1. Show that $$z + \frac { 1 } { z } = 2 \cos \theta$$
    2. Find a similar expression for $$z ^ { 2 } + \frac { 1 } { z ^ { 2 } }$$ (2 marks)
    3. Hence show that $$z ^ { 2 } - z + 2 - \frac { 1 } { z } + \frac { 1 } { z ^ { 2 } } = 4 \cos ^ { 2 } \theta - 2 \cos \theta$$ (3 marks)
  2. Hence solve the quartic equation $$z ^ { 4 } - z ^ { 3 } + 2 z ^ { 2 } - z + 1 = 0$$ giving the roots in the form \(a + \mathrm { i } b\).
Pre-U Pre-U 9795/1 2013 June Q11
13 marks Standard +0.3
11
  1. Determine \(p\) and \(q\) given that \(( p + \mathrm { i } q ) ^ { 2 } = 63 - 16 \mathrm { i }\) and that \(p\) and \(q\) are real.
  2. Let \(\mathrm { f } ( z ) = z ^ { 3 } - A z ^ { 2 } + B z - C\) for complex numbers \(A , B\) and \(C\).
    1. Given that the cubic equation \(\mathrm { f } ( z ) = 0\) has roots \(\alpha = - 7 \mathrm { i } , \beta = 3 \mathrm { i }\) and \(\gamma = 4\), determine each of \(A , B\) and \(C\).
    2. Find the roots of the equation \(\mathrm { f } ^ { \prime } ( z ) = 0\).
Pre-U Pre-U 9795/1 2014 June Q13
8 marks Challenging +1.8
13 The complex number \(w\) has modulus 1. It is given that $$w ^ { 2 } - \frac { 2 } { w } + k \mathrm { i } = 0$$ where \(k\) is a positive real constant.
  1. Show that \(k = ( 3 - \sqrt { 3 } ) \sqrt { \frac { 1 } { 2 } \sqrt { 3 } }\).
  2. Prove that at least one of the remaining two roots of the equation \(z ^ { 2 } - \frac { 2 } { z } + k i = 0\) has modulus greater than 1 .
OCR Further Pure Core 2 2024 June Q2
6 marks Moderate -0.8
In this question you must show detailed reasoning.
  1. Solve the equation \(x^2 - 6x + 58 = 0\). Give your solutions in the form \(a + bi\) where \(a\) and \(b\) are real numbers. [3]
  2. Determine, in exact form, \(\arg(-10 + (5\sqrt{12})i)^3\). [3]
OCR Further Pure Core 2 2021 June Q1
5 marks Moderate -0.5
In this question you must show detailed reasoning. Solve the equation \(4z^2 - 20z + 169 = 0\). Give your answers in modulus-argument form. [5]