A2 June 2022 Paper 2 Q7
7 The function \(\mathrm{f}\) is defined by
\[\mathrm{f}(x) = \frac{ax - 5}{2x + b} \qquad x \in \mathbb{R},\ x \neq \frac{9}{2}\]where \(a\) and \(b\) are integers.
The graph of \(y = \mathrm{f}(x)\) has asymptotes \(x = \dfrac{9}{2}\) and \(y = 3\)
(a) Find the value of \(a\) and the value of \(b\) [2 marks]
(b) Solve the inequality\[\mathrm{f}(x) \leqslant x + 2\]
Fully justify your answer. [6 marks]
| Scheme | Marks | AO |
|---|---|---|
| Deduces correct value of \(b\) | B1 | 2.2a |
| Deduces correct value of \(a\) | B1 | 2.2a |
| (2) |
Typical solution
\[2 \times \frac{9}{2} + b = 0\]\[b = -9\]\[\frac{a}{2} = 3\]\[a = 6\]| Scheme | Marks | AO |
|---|---|---|
| Selects a suitable method to solve the inequality, for example Multiplies by square of denominator or sketches graphs of their \(y = \mathrm{f}(x)\) and \(y = x + 2\) | M1 | 3.1a |
| Simplifies their inequality or equation or Indicates points of intersection of the two graphs | M1 | 1.1a |
| Obtains at least two critical values of their inequality or equation | A1F | 1.1b |
| Excludes \(x = \dfrac{9}{2}\) (PI by final answer) | A1 | 2.2a |
| Deduces correct solution set for their inequality or graph. Condone inclusion of \(x = \dfrac{9}{2}\) Follow through their answers to part a) | A1F | 2.2a |
| Obtains completely correct solution OE with each step clearly shown | R1 | 2.1 |
| (6) | ||
| (8 marks) |
Typical solution
\[x + 2 \geqslant \frac{6x - 5}{2x - 9}\]\[(x + 2)(2x - 9)^2 \geqslant (6x - 5)(2x - 9)\]\[(2x - 9)\{(x + 2)(2x - 9) - (6x - 5)\} \geqslant 0\]\[(2x - 9)\{2x^2 - 5x - 18 - 6x + 5\} \geqslant 0\]\[(2x - 9)\{2x^2 - 11x - 13\} \geqslant 0\]\[(2x - 9)(x + 1)(2x - 13) \geqslant 0\]Change of sign occurs at
\[x = -1,\ x = \frac{9}{2},\ x = \frac{13}{2}\]Exclude \(x = \dfrac{9}{2}\) (f not defined)
\[-1 \leqslant x \lt \frac{9}{2} \text{ or } x \geqslant \frac{13}{2}\]