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7 marksComplex Numbers

CAPE Pure Mathematics Unit 2 · May/June 2008 (Rest of Region) · Paper 2 · Question 6(b)(i)

The complex number z is represented by the point T in an Argand diagram. Given that z = \frac{1}{3 + it}, where t is a variable and \bar{z} denotes the complex conjugate of z:

Show that z + \bar{z} = 6 z\bar{z}.

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Other parts of this question

  1. 6(a)(i)Express the complex number \frac{2 - 3i}{5 - i} in the form \lambda(1 - i).[4 marks]
  2. 6(a)(ii)State the value of \lambda.[1 mark]
  3. 6(a)(iii)Verify that \left(\frac{2 - 3i}{5 - i}\right)^4 is a real number and state its value.[5 marks]
  4. 6(b)(ii)Show that as t varies, T lies on a circle, and state the coordinates of the centre of this circle.[8 marks]

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