Higher November 2024 Paper 1 Q25
25 The function f is such that \(\mathrm{f}(x) = 2x^2 - 24x + 7\) where \(x \geqslant 6\)
Find the inverse function \(\mathrm{f}^{-1}(x)\)
(4)
| Scheme | Marks |
|---|---|
\((y =)\;2(x^2 - 12x) + 7\) or \((y =)\;2\left(x^2 - 12x + \dfrac{7}{2}\right)\) or \(\dfrac{y - 7}{2} = x^2 - 12x\) or \((x =)\;2(y^2 - 12y) + 7\) or \((x =)\;2\left(y^2 - 12y + \dfrac{7}{2}\right)\) or \(\dfrac{x - 7}{2} = y^2 - 12y\) | M1 |
eg \((y =)\;2\left((x - 6)^2 - 6^2\right) + 7\) or \((y =)\;2\left((x - 6)^2 - 6^2 + \dfrac{7}{2}\right)\) or \((y =)\;2(x - 6)^2 - 65\) oe or \(\dfrac{y - 7}{2} = (x - 6)^2 - 6^2\) oe or eg \((x =)\;2\left((y - 6)^2 - 6^2\right) + 7\) or \((x =)\;2\left((y - 6)^2 - 6^2 + \dfrac{7}{2}\right)\) or \((x =)\;2(y - 6)^2 - 65\) oe or \(\dfrac{x - 7}{2} = (y - 6)^2 - 6^2\) oe | M1 |
\((x - 6)^2 = \dfrac{y + 65}{2}\) oe or \((x - 6)^2 = \dfrac{y - 7}{2} + 6^2\) oe or \((y - 6)^2 = \dfrac{x + 65}{2}\) oe or \((y - 6)^2 = \dfrac{x - 7}{2} + 6^2\) oe | M1 |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(6 + \sqrt{\dfrac{x + 65}{2}}\) | A1 |
| (4) | |
| (4 marks) |
Notes
M1: for a correct first step in order to complete the square
For each method mark the function must be correct
M1: dep on M1
A1: oe eg \(6 + \sqrt{\dfrac{x - 7}{2} + 36}\)
Must be in terms of \(x\)
M3A0 for \(6 \pm \sqrt{\dfrac{x + 65}{2}}\) or \(6 \pm \sqrt{\dfrac{y + 65}{2}}\)
or \(6 + \sqrt{\dfrac{y + 65}{2}}\)
Note: Allow candidates to swap \(x\) and \(y\) when finding the inverse
| Scheme | Marks |
|---|---|
| \(2x^2 - 24x + 7 - y\;(= 0)\) | M1 |
\((x =)\;\dfrac{24 \pm \sqrt{576 - 8(7 - y)}}{4}\) or \((x =)\;\dfrac{24 + \sqrt{576 - 8(7 - y)}}{4}\) or \(2\left((x - 6)^2 - 6^2\right) + 7 - y\;(= 0)\) or \(2\left((x - 6)^2 - 6^2 + \dfrac{7}{2}\right) - y\;(= 0)\) | M1 |
\((x =)\;6 \pm \sqrt{\dfrac{y + 65}{2}}\) or \((x - 6)^2 = \dfrac{y + 65}{2}\) | M1 |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(6 + \sqrt{\dfrac{x + 65}{2}}\) | A1 |
Notes
M1: for a correct first step
M1: dep on M1
A1: oe eg \(6 + \sqrt{\dfrac{x - 7}{2} + 36}\)
Must be in terms of \(x\)
M3A0 for \(6 \pm \sqrt{\dfrac{x + 65}{2}}\) or \(6 \pm \sqrt{\dfrac{y + 65}{2}}\)
or \(6 + \sqrt{\dfrac{y + 65}{2}}\)
Note: Allow candidates to swap \(x\) and \(y\) when finding the inverse