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3 marksComplex Numbers

CAPE Pure Mathematics Unit 2 · May/June 2018 · Paper 2 · Question 1(d)(i)

Write the complex number z = (1 - i) in the form r e^(iθ), where r = |z| and θ = arg(z).

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Other parts of this question

  1. 1(a)(i)Determine the gradient of the curve at the point (1/2, 1/2).[5 marks]
  2. 1(a)(ii)Hence, or otherwise, determine the x and y intercepts of the tangent to the curve at the point (1/2, 1/2).[4 marks]
  3. 1(b)Let the function f(x, y) = sin(kx) sin(aky). Determine d^2 f(x, y) / (dx dy).[3 marks]
  4. 1(c)Use De Moivre's theorem to show that sin(5θ) = sin^5(θ) - 10 sin^3(θ) cos^2(θ) + 5 cos^4(θ) sin(θ).[5 marks]
  5. 1(d)(ii)Hence, show that (1 - i)^9 = 16(1 - i).[5 marks]

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