Verifying roots satisfy equations

A question is this type if and only if it asks to verify or show that a given complex number satisfies a particular equation by direct substitution and simplification.

5 questions

AQA Further AS Paper 1 2023 June Q12
12
  1. Show that \(( 1 + i ) ^ { 4 } = - 4\)
    12
  2. The function f is defined by $$f ( z ) = z ^ { 4 } + 3 z ^ { 2 } - 6 z + 10 \quad z \in \mathbb { C }$$ 12
    1. Show that (1+i) is a root of \(\mathrm { f } ( \mathrm { z } ) = 0\)
      12
  4. (ii) Hence write down another root of \(\mathrm { f } ( \mathrm { z } ) = 0\)
    12
  5. (iii) One of the linear factors of \(\mathrm { f } ( \mathrm { z } )\) is $$( z - ( 1 + i ) )$$ Write down another linear factor and hence, or otherwise, find a quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
    12
  6. (iv) Find another quadratic factor of \(\mathrm { f } ( \mathrm { z } )\) with real coefficients.
    12
  7. (v) Hence explain why the graph of \(y = \mathrm { f } ( x )\) does not intersect the \(x\)-axis.
OCR Further Pure Core AS 2018 June Q5
5 In this question you must show detailed reasoning.
  1. Express \(( 2 + 3 \mathrm { i } ) ^ { 3 }\) in the form \(a + \mathrm { i } b\).
  2. Hence verify that \(2 + 3\) i is a root of the equation \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52 = 0\).
  3. Express \(3 z ^ { 3 } - 8 z ^ { 2 } + 23 z + 52\) as the product of a linear factor and a quadratic factor with real coefficients.
AQA FP1 2006 January Q5
5
    1. Calculate \(( 2 + \mathrm { i } \sqrt { 5 } ) ( \sqrt { 5 } - \mathrm { i } )\).
    2. Hence verify that \(\sqrt { 5 } - \mathrm { i }\) is a root of the equation $$( 2 + \mathrm { i } \sqrt { 5 } ) z = 3 z ^ { * }$$ where \(z ^ { * }\) is the conjugate of \(z\).
  2. The quadratic equation $$x ^ { 2 } + p x + q = 0$$ in which the coefficients \(p\) and \(q\) are real, has a complex root \(\sqrt { 5 } - \mathrm { i }\).
    1. Write down the other root of the equation.
    2. Find the sum and product of the two roots of the equation.
    3. Hence state the values of \(p\) and \(q\).
AQA FP1 2007 January Q1
1
  1. Solve the following equations, giving each root in the form \(a + b \mathrm { i }\) :
    1. \(x ^ { 2 } + 16 = 0\);
    2. \(x ^ { 2 } - 2 x + 17 = 0\).
    1. Expand \(( 1 + x ) ^ { 3 }\).
    2. Express \(( 1 + \mathrm { i } ) ^ { 3 }\) in the form \(a + b \mathrm { i }\).
    3. Hence, or otherwise, verify that \(x = 1 + \mathrm { i }\) satisfies the equation $$x ^ { 3 } + 2 x - 4 \mathrm { i } = 0$$
AQA FP1 2011 January Q5
5
  1. It is given that \(z _ { 1 } = \frac { 1 } { 2 } - \mathrm { i }\).
    1. Calculate the value of \(z _ { 1 } ^ { 2 }\), giving your answer in the form \(a + b \mathrm { i }\).
    2. Hence verify that \(z _ { 1 }\) is a root of the equation $$z ^ { 2 } + z ^ { * } + \frac { 1 } { 4 } = 0$$
  2. Show that \(z _ { 2 } = \frac { 1 } { 2 } + \mathrm { i }\) also satisfies the equation in part (a)(ii).
  3. Show that the equation in part (a)(ii) has two equal real roots.