Functions, modulus and partial fractions – Year 13 A level maths test to download as PDF

The functions \(f\) and \(g\) are defined by \(f(x)=2x+1\) and \(g(x)=x^2-3\) for all real \(x\). (a) Find \(fg(x)\) and \(gf(x)\). [2 marks] (b) Solve \(gf(x)=6\). [2 marks]

(a) Solve \(|3x-2|=x+4\). [3 marks] (b) Solve \(|2x+1|\le5\). [1 mark]

(a) Express \(\dfrac{5x+3}{(x-1)(x+3)}\) in partial fractions. [2 marks] (b) Express \(\dfrac{x^2}{(x-1)(x+1)}\) in partial fractions. [2 marks]

The function \(h\) is defined by \(h(x)=|x-2|+1\), \(x\in\mathbb{R}\). (a) State the range of \(h\). [1 mark] (b) Solve \(h(x)=4\). [1 mark] (c) The function \(k\) is \(h(x)\) for \(x\ge2\). Find \(k^{-1}(x)\) and state its domain. [2 marks]

After switching off, an oven cools according to \(T=25+175e^{-0.05t}\), where \(T\) is in °C and \(t\) in minutes. (a) Find the temperature when it is switched off and the long-term temperature. [1 mark] (b) Find the time for the oven to reach 100 °C, to 3 significant figures. [2 marks] (c) State the domain of the inverse function. [1 mark]