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3 marksFunctions

CAPE Pure Mathematics Unit 1 · May/June 2014 · Paper 2 · Question 2(a)(i)a)

The functions f and g are defined as follows: f(x) = 2x² + 1 where 1 ≤ x < ∞, x ∈ R. g(x) = √(x-1)/2

Determine, in terms of x, f²(x)

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  1. 2(a)(i)b)Determine, in terms of x, f[g(x)].[3 marks]
  2. 2(a)(ii)Hence, or otherwise, state the relationship between f and g.[1 mark]
  3. 2(b)Given that a² + b³ + 3a²b = 5ab², show that 3 log((a+b)/2) = log a + 2 log b.[5 marks]
  4. 2(c)(i)Solve EACH of the following equations: eˣ + 1/eˣ - 2 = 0[4 marks]
  5. 2(c)(ii)Solve EACH of the following equations: log₂(x + 1) – log₂(3x + 1) = 2[4 marks]
  6. 2(d)Without the use of a calculator, show that (√3-1)/(√3+1) + (√3+1)/(√3-1) + (√2-1)/(√2+1) + (√2+1)/(√2-1) = 10.[5 marks]

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