Higher January 2019 Paper 1R Q23
23 The function f is defined as \(\mathrm{f}(x) = \dfrac{\sqrt{x^2 + k^2}}{x}\) for \(x \gt 0\) and where \(k\) is a positive number.
(a) Find the value of \(p\) for which \(\mathrm{f}^{-1}(p) = k\) (3)
The function g is defined as \(\mathrm{g}(x) = x^2\) for \(x \gt 0\)
(b) Given that \(\mathrm{gf}(a) = k\) for \(k \gt 1\)
find an expression for \(a\) in terms of \(k\). (3)
find an expression for \(a\) in terms of \(k\). (3)
| Scheme | Marks |
|---|---|
| \(y = \dfrac{\sqrt{x^2 + k^2}}{x}\), \(x^2y^2 = x^2 + k^2\) \(x^2(y^2 - 1) = k^2\) | M1 |
| \(\dfrac{k}{\sqrt{p^2 - 1}} = k\) | M1 |
| \(\sqrt{2}\) | A1 |
| (3) |
Notes
M1: for squaring and rearranging correctly to the form \(x^2(y^2 - 1) = k^2\)
M1: (dep) for “\(\mathrm{f}^{-1}(p)\)” = \(k\)
| Scheme | Marks |
|---|---|
| \(p = \mathrm{f}(k)\) | M1 |
| \(p = \dfrac{\sqrt{k^2 + k^2}}{k}\) | M1 |
| \(\sqrt{2}\) | A1 |
| Scheme | Marks |
|---|---|
| \((\mathrm{gf}(a) =)\left(\dfrac{\sqrt{a^2 + k^2}}{a}\right)^2\) or \((\mathrm{gf}(x) =)\left(\dfrac{\sqrt{x^2 + k^2}}{x}\right)^2\) | M1 |
| \(ka^2 - a^2 = k^2\) | M1 |
| \(\dfrac{k}{\sqrt{k - 1}}\) | A1 |
| (3) | |
| (6 marks) |
Notes
M1: (dep) for rearranging gf = \(k\) and isolating correctly the terms in \(a^2\)
A1: oe eg \(\sqrt{\dfrac{k^2}{k - 1}}\)