Finding specific root values

A question is this type if and only if it gives enough information about relationships between roots to determine their actual numerical or exact values.

2 questions · Standard +0.3

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AQA FP2 2013 June Q5
9 marks Standard +0.3
5 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + q z + 37 - 36 \mathrm { i } = 0$$ where \(p\) and \(q\) are constants, has three complex roots, \(\alpha , \beta\) and \(\gamma\). It is given that \(\beta = - 2 + 3 \mathrm { i }\) and \(\gamma = 1 + 2 \mathrm { i }\).
    1. Write down the value of \(\alpha \beta \gamma\).
    2. Hence show that \(( 8 + \mathrm { i } ) \alpha = 37 - 36 \mathrm { i }\).
    3. Hence find \(\alpha\), giving your answer in the form \(m + n \mathrm { i }\), where \(m\) and \(n\) are integers.
  2. Find the value of \(p\).
  3. Find the value of the complex number \(q\).
WJEC Further Unit 1 2018 June Q2
6 marks Standard +0.3
  1. Find the values of \(\alpha , \beta\), and \(\gamma\).
  2. Find the cubic equation with roots \(3 \alpha , 3 \beta , 3 \gamma\). Give your answer in the form \(a x ^ { 3 } + b x ^ { 2 } + c x + d = 0\), where \(a , b , c , d\) are constants to be determined.