4MA0 Higher January 2014 Paper 3 Q20
20 f is the function \(\;\mathrm{f}(x) = 2x + 5\)
(a) Find \(\mathrm{f}(3)\) (1)
(b) Express the inverse function \(\mathrm{f}^{-1}\) in the form \(\mathrm{f}^{-1}(x) =\) (2)
g is the function \(\;\mathrm{g}(x) = x^2 - 25\)
(c) Find \(\mathrm{g}(-3)\) (1)
(d)
(i) Find \(\mathrm{gf}(x)\)
Give your answer as simply as possible.
Give your answer as simply as possible.
(ii) Solve \(\mathrm{gf}(x) = 0\)
(5)| Scheme | Marks |
|---|---|
| 11 | B1 |
| (1) |
Notes
B1: cao
| Scheme | Marks |
|---|---|
| \(y = 2x + 5\) \(y - 5 = 2x\) or \(x = 2y + 5\) \(x - 5 = 2y\) | M1 |
| \(\dfrac{x - 5}{2}\) oe | A1 |
| (2) |
Notes
M1: for a correct first stage – subtract 5 from both sides or divide all terms by 2
NB Accept f(\(x\)) in place of \(y\)
| Scheme | Marks |
|---|---|
| −16 | B1 |
| (1) |
Notes
B1: cao
| Scheme | Marks |
|---|---|
| (i) \((2x + 5)^2 - 25\) | M1 |
| \(4x^2 + 10x + 10x + 25\) oe | B1 |
| \(4x^2 + 20x\) | A1 |
| (ii) \(4x(x + 5)\;(= 0)\) or \(x(4x + 20)\;(=0)\) or \(2x(2x + 10)\;(=0)\) or \(x(x + 5)\;(=0)\) | M1 |
| \(x = 0,\; x = -5\) | A1 |
| (5) | |
| (9 marks) |
Notes
B1: for correct expansion of \((2x + 5)^2\)
A1: or a correct fully or partially factorised expression
M1: or for eg \(\dfrac{-20 \pm \sqrt{20^2 - 4 \times 4 \times 0}}{2 \times 4}\)
A1: for both solutions