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3 marksComplex Numbers

CAPE Pure Mathematics Unit 2 · May/June 2019 · Paper 2 · Question 1(c)(i)

Write the complex number z = (-1 + √3 i)⁷ in the form re^(iθ), where r = |z| and θ = arg z.

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Other parts of this question

  1. 1(a)(i)Use implicit differentiation to show that dy/dx = -(8x + 3y² + 7) / (3(1 + 2xy)).[5 marks]
  2. 1(a)(ii)Show that 6(∂f(x,y)/∂y) - 10 = (∂²f(x,y)/∂y²) + (∂²f(x,y)/∂y∂x) + (∂²f(x,y)/∂x²).[5 marks]
  3. 1(b)Use de Moivre's theorem to prove that sin 5x = 16 sin⁵ x - 20 sin³ x + 5 sin x.[6 marks]
  4. 1(c)(ii)Hence, prove that (-1 + √3 i)⁷ = 64 (-1 + √3 i).[6 marks]

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