AS June 2023 Paper 1 Q14
14 The inequality
\[(x^2 - 5x - 24)(x^2 + 7x + a) \lt 0\]has the solution set
\[\{x : -9 \lt x \lt -3\} \cup \{x : 2 \lt x \lt b\}\]Find the values of integers \(a\) and \(b\) [4 marks]
| Scheme | Marks | AO |
|---|---|---|
| Factorises \(x^2 - 5x - 24\) to \((x + m)(x + n)\) where \(m + n = -5\) or \(mn = -24\) and identifies their \(m, n \gt 2\) as \(b\) Or uses the coefficients of a quartic to form an equation in \(a\) and/or \(b\) eg \(-(-5 + 7) = -9 + (-3) + 2 + b\) eg \(-24a = -9 \times -3 \times 2 \times b\) Or multiplies two or more of the factors \((x + 9)\), \((x + 3)\), \((x - 2)\) and \((x - b)\) | M1 | 3.1a |
| Obtains \(a = -18\) or \(b = 8\) | A1 | 1.1b |
| Expands \((x + 9)(x - 2)\) and identifies the constant term as \(a\) Or correctly forms two equations in \(a\) and \(b\) Or divides the expanded quartic by a quadratic or a cubic formed by multiplying two or three of \((x + 9)\), \((x + 3)\), \((x - 2)\) and \((x - b)\) Or compares coefficients in the expansions of \((x^2 - 5x - 24)(x^2 + 7x + a)\) and \((x + 9)(x + 3)(x - 2)(x - b)\) | M1 | 3.1a |
| Obtains \(a = -18\) and \(b = 8\) | A1 | 1.1b |
| (4 marks) |
Typical solution
\[x^2 - 5x - 24 = (x - 8)(x + 3)\]\(\therefore\) the critical values include \(-3\) and 8
\[\therefore \ b = 8\]\[(x + 9)(x - 2) = (x^2 + 7x - 18)\]\[\therefore \ a = -18\]