AQA FP1 2011 January — Question 5

Exam BoardAQA
ModuleFP1 (Further Pure Mathematics 1)
Year2011
SessionJanuary
TopicComplex Numbers Arithmetic
TypeVerifying roots satisfy equations
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.
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