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6 marksComplex Numbers

CAPE Pure Mathematics Unit 2 · May/June 2015 · Paper 2 · Question 1(a)(ii)

Given three complex numbers z_1 = 1 + (7 - 4\sqrt{3})i, z_2 = \sqrt{3} + 3i, and z_3 = -2 + 2i.

Given that \arg w = \arg z_3 - [\arg z_1 + \arg z_2], |z_1| = 1 and \arg z_1 = \frac{\pi}{12}, rewrite w = \frac{z_3}{z_1 z_2} in the form r e^{i\theta} where r = |w| and \theta = \arg w.

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Other parts of this question

  1. 1(a)(i)Express the quotient \frac{z_3}{z_2} in the form x + iy where x, y \in \mathbb{R}.[3 marks]
  2. 1(b)A complex number v = x + iy is such that v^2 = 2 + i. Show that x^2 = \frac{2 + \sqrt{5}}{2}.[7 marks]
  3. 1(c)(i)Show that \frac{dy}{dx} = \frac{e^t (1 - t^2)}{t^2 + t - 1}.[6 marks]
  4. 1(c)(ii)Hence, show that f has no stationary value.[3 marks]

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