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6 marksComplex Numbers

CAPE Pure Mathematics Unit 2 · May/June 2021 · Paper 2 · Question 1(a)(i)

Two complex numbers are given as z₁ = √(5/3) + √5i and z₂ = 2 – 2i.

Show that the complex number z₁/z₂ = (√5)/6 e^(i7π/12).

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Other parts of this question

  1. 1(a)(ii)Hence, without using a calculator, determine the value of (z₁/z₂)². [3 marks]
  2. 1(b)One root of a quadratic equation is given as 4 – 7i. Determine the quadratic equation with real coefficients which has the root 4 – 7i.[4 marks]
  3. 1(c)Show that the derivative of sin⁻¹(cos x / (1 + sin x)) with respect to x is -1 / √( (1+sin x)² - cos² x ).[6 marks]
  4. 1(d)A curve is defined parametrically by x = (3 – 2t)², y = t³ – 2t. Determine the equation of the tangent to the curve at the point where t = 2.[6 marks]

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