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CAPE Pure Mathematics Unit 1 · May/June 2008 · Paper 2 · Question 3(a)(i)

This question involves vector properties and trigonometric identities and equations.

Let p = i - j. If q = λi + 2j, find values of λ such that q is parallel to p.

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Other parts of this question

  1. 3(a)(ii)q is perpendicular to p.[2 marks]
  2. 3(a)(iii)the angle between p and q is π/3.[5 marks]
  3. 3(b)Show that (1 - cos 2A + sin 2A) / (1 + cos 2A + sin 2A) = tan A.[6 marks]
  4. 3(c)(i)Using the formula for sin A + sin B, show that if t = 2 cos θ then sin (n + 1) θ = t sin nθ – sin (n − 1) θ.[2 marks]
  5. 3(c)(ii)Hence, show that sin 3θ = (t² − 1) sin θ.[2 marks]
  6. 3(c)(iii)Using (c) (ii) above, or otherwise, find ALL solutions of sin 3θ = sin θ, 0 ≤ θ ≤ π.[7 marks]

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