A2 June 2023 Paper 1 Q11
11 The function \(\mathrm{f}\) is defined by
\[\mathrm{f}(x) = 4x^3 - 8x^2 - 51x - 45 \qquad (x \in \mathbb{R})\](a)
(i) Fully factorise \(\mathrm{f}(x)\) [2 marks]
(ii) Hence, solve the inequality \(\mathrm{f}(x) \lt 0\) [2 marks]
(b) The graph of \(y = \mathrm{f}(x)\) is translated by the vector \(\begin{bmatrix} 7 \\ 0 \end{bmatrix}\)
The new graph is then reflected in the \(x\)-axis, to give the graph of \(y = \mathrm{g}(x)\)
Solve the inequality \(\mathrm{g}(x) \leqslant 0\) [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| (i) Obtains one factor of \(\mathrm{f}(x)\) | M1 | 1.1a |
| (i) Deduces the correct factorisation | A1 | 2.2a |
| (2) | ||
| (ii) Deduces that \(x \lt 5\) | M1 | 2.2a |
| (ii) Obtains a completely correct solution to the inequality | A1 | 1.1b |
| (2) |
Typical solution
(i)
\[\mathrm{f}(x) = (x - 5)(2x + 3)^2\](ii)
If \(\mathrm{f}(x) \lt 0\),
\[x \lt -\tfrac{3}{2},\ -\tfrac{3}{2} \lt x \lt 5\]| Scheme | Marks | AO |
|---|---|---|
| Obtains \(\pm 5.5\) or \(\pm 12\) ft 7 + their critical values from (a)(ii) | M1 | 2.2a |
| Obtains 5.5 and 12 ft 7 + their critical values from (a)(ii) | A1F | 2.2a |
| Obtains a completely correct solution to the inequality | A1 | 1.1b |
| (3) | ||
| (7 marks) |
Typical solution
If \(\mathrm{g}(x) \leqslant 0\),
\[x = \tfrac{11}{2} \text{ or } x \geqslant 12\]