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6 marksThe Real Number System – ℝ

CAPE Pure Mathematics Unit 1 · May/June 2006 · Paper 2 · Question 6(a)(i)

Show that the area S is approximately 1/n² + 2/n² + 3/n² + .... + (n-1)/n².

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Other parts of this question

  1. 6(a)(ii)Given that Σ (from r=1 to n-1) r = (1/2)n(n - 1), show that S = (1/2)(1 - 1/n).[2 marks]
  2. 6(b)(i)Show that for f(x) = 2x/(x²+4), f'(x) = (8-2x²)/(x²+4)².[4 marks]
  3. 6(b)(ii)Hence, evaluate ∫ (from 0 to 1) (24-6x²)/(x²+4)² dx.[3 marks]
  4. 6(c)Find the value of u > 0 if ∫ (from 1 to u) (2/x⁴) dx = 7/192.[5 marks]

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