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6 marksCubic Functions and Equations

CAPE Pure Mathematics Unit 1 · 2022 · Paper 2 · Question 2(d)

The roots of the equation 2x^3 - x^2 + 3x - 1 = 0 are \alpha, \beta, and \gamma. Given that \alpha\beta + \alpha\gamma + \beta\gamma = \frac{3}{2} and \frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\alpha\gamma} = \frac{\alpha + \beta + \gamma}{\alpha\beta\gamma}, determine the equation whose roots are \frac{1}{\alpha}, \frac{1}{\beta} and \frac{1}{\gamma}.

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  1. 2(a)(i)Let f(x) = \frac{3x + 1}{x} and g(x) = e^{-2x} + 1. Show that f^{-1}(x) = \frac{1}{x - 3}.[3 marks]
  2. 2(a)(ii)Hence, or otherwise, write an expression for (f^{-1} \circ g)(x).[2 marks]
  3. 2(b)Solve the equation 6 - \frac{7}{2^{2x}} - \frac{3}{4^{2x}} = 0.[8 marks]
  4. 2(c)The function f(x) = 2x^3 - px^2 + qx - 10 is divisible by x - 1 and has a remainder of -6 when divided by x + 1. Calculate the values of p and q.[6 marks]

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