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2 marksThe Real Number System – ℝ

CAPE Pure Mathematics Unit 1 · May/June 2010 · Paper 2 · Question 2(a)(i)

Use the fact that Sₙ = ∑r = 1 to n of r = (1/2) n (n + 1) to express S₂ₙ = ∑r = 1 to 2n of r in terms of n.

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Other parts of this question

  1. 2(a)(ii)Find constants p and q such that S₂ₙ - Sₙ = pn² + qn.[5 marks]
  2. 2(a)(iii)Hence, or otherwise, find n such that S₂ₙ - Sₙ = 260.[5 marks]
  3. 2(b)(i)Write down the solution set of the inequality x² (3 - x) ≤ 0.[4 marks]
  4. 2(b)(ii)Given that the equation x² (3 - x) = k has three real solutions for x, write down the set of possible values for k.[3 marks]
  5. 2(b)(iii)a)By using (b) (ii) above, or otherwise, show that f has an inverse.[6 marks]
  6. 2(b)(iii)b)By using (b) (ii) above, or otherwise, show that g does NOT have an inverse.[1 mark]

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