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6 marksComplex Numbers

CAPE Pure Mathematics Unit 2 · May/June 2013 · Paper 2 · Question 1(c)

Use de Moivre's theorem to prove that \cos 5\theta = 16\cos^5 \theta - 20\cos^3 \theta + 5\cos \theta.

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  1. 1(a)Calculate the gradient of the curve \ln(x^2 y) - \sin y = 3x - 2y at the point (1, 0).[5 marks]
  2. 1(b)Let f(x, y, z) = 3yz^2 - e^{4x}\cos 4z - 3y^2 - 4 = 0. Given that \frac{\partial z}{\partial y} = -\frac{\partial f / \partial y}{\partial f / \partial z},…[5 marks]
  3. 1(d)(i)Write the complex number z = (-1 + i)^7 in the form r e^{i\theta}, where r = |z| and \theta = \arg z.[3 marks]
  4. 1(d)(ii)Hence, prove that (-1 + i)^7 = -8(1 + i).[6 marks]

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