Higher November 2019 Paper 1 Q18
18 The function f is given by
\[\mathrm{f}(x) = 2x^3 - 4\]
(a) Show that \(\mathrm{f}^{-1}(50) = 3\) (2)
The functions g and h are given by
\[\mathrm{g}(x) = x + 2 \quad \text{and} \quad \mathrm{h}(x) = x^2\]
(b) Find the values of \(x\) for which
\[\mathrm{hg}(x) = 3x^2 + x - 1\]
(4)| Answer | Mark | Mark scheme |
|---|---|---|
| Shown | C1 | for \(\mathrm{f}^{-1}(x) = \sqrt[3]{\dfrac{x+4}{2}}\) OR for \(2x^3 - 4 = 50\) OR for substituting \(x = 3\) to find f(3) |
| C1 | for substituting \(x = 50\) to show the result giving \(\mathrm{f}^{-1}(50) = 3\) OR solving for \(x\) to give \(x = 3\) OR for showing that \(\mathrm{f}(3) = 50\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(x = -1\) and \(x = 2.5\) | P1 | for \(\mathrm{hg}(x) = (x + 2)^2\) |
| P1 | (dep) for start to a process to derive a quadratic equation eg. \(x^2 + 4x + 4 = 3x^2 + x - 1\) | |
| P1 | for a process to solve the quadratic equation \(2x^2 - 3x - 5 = 0\) eg \((2x - 5)(x + 1)\ (= 0)\) or \(\dfrac{--3 \pm \sqrt{(-3)^2 - 4 \times 2 \times -5}}{2 \times 2}\) or \(2\left[\left(x - \dfrac{3}{4}\right)^2 - \dfrac{9}{16} - \dfrac{5}{2}\right]\ (= 0)\) | |
| A1 | for \(x = -1\) and \(x = 2.5\) |
Additional guidance
\((x + 2)^2\) must be correctly expanded
2.5 or \(2\frac{1}{2}\) or \(\frac{5}{2}\) acceptable