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8 marksThe Real Number System – ℝ

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

This question involves evaluating expressions without calculators, proving logarithmic identities, and mathematical induction.

Without using calculators or tables, evaluate (27¹⁰ + 9¹⁰) / (27⁴ + 9¹¹).

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Other parts of this question

  1. 2(b)(i)Prove that logₙm = log₁₀m / log₁₀n, for m, n ∈ N.[4 marks]
  2. 2(b)(ii)Hence, given that y = (log₂ 3) (log₃ 4) (log₄ 5) ... (log₃₁ 32), calculate the exact value of y.[6 marks]
  3. 2(c)Prove, by the principle of mathematical induction, that f(n) = 7ⁿ - 1 is divisible by 6, for all n ∈ N.[7 marks]

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