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5 marksComplex Numbers

CAPE Pure Mathematics Unit 2 · May/June 2012 · Paper 2 · Question 6(a)(ii)

Let A = \frac{1 + i}{1 - i} and B = \frac{\sqrt{2}}{1 - i}.

Hence, or otherwise, show that the argument of \frac{(1 + \sqrt{2} + i)}{1 - i} is EXACTLY \frac{3\pi}{8}.

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Other parts of this question

  1. 6(a)(i)Draw the points A and B on an Argand diagram.[6 marks]
  2. 6(b)(i)Find ALL complex numbers, z, such that z^2 = i.[3 marks]
  3. 6(b)(ii)Hence, find ALL complex roots of the equation z^2 - (3 + 5i)z - (4 - 7i) = 0.[5 marks]
  4. 6(c)Use de Moivre's theorem to show that \cos 6\theta = \cos^6 \theta - 15\cos^4 \theta \sin^2 \theta + 15\cos^2 \theta \sin^4 \theta - \sin^6 \theta.[6 marks]

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