A2 June 2024 Paper 1 Q4
4 In this question you must show detailed reasoning.
The equation \(2x^3 + 3x^2 + 6x - 3 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
Determine a cubic equation with integer coefficients that has roots \(\alpha^2\beta\gamma, \alpha\beta^2\gamma\) and \(\alpha\beta\gamma^2\). [3]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\alpha\beta\gamma = -\left(-\tfrac{3}{2}\right) \;\left(= \tfrac{3}{2}\right)\) | M1 | 1.1 |
| \(w = \tfrac{3}{2}x\) | M1 | 3.1a |
| \(2\left(\tfrac{2}{3}w\right)^3 + 3\left(\tfrac{2}{3}w\right)^2 + 6\left(\tfrac{2}{3}w\right) - 3 (= 0)\) \(16w^3 + 36w^2 + 108w - 81 = 0\) | A1 | 2.2a |
| [3] |
Notes
M1: For \(\alpha\beta\gamma = -\left(-\tfrac{3}{2}\right)\).
M1: Appropriate substitution of the form \(w = (\alpha\beta\gamma)x\) with their value of \(\alpha\beta\gamma\) (need not be substituted into given cubic for this mark). Condone reciprocal e.g. \(x = \tfrac{3}{2}w\).
A1: Allow any integer multiple but must have integer coefficients. Must be an equation (so must = 0 or terms on both sides of an equation) with all terms simplified. Condone if in terms of \(x\).
Alternative method
| Scheme | Marks |
|---|---|
| \(\sum\alpha = -\tfrac{3}{2}, \quad \sum\alpha\beta = 3, \quad \alpha\beta\gamma = -\left(-\tfrac{3}{2}\right)\) | M1 |
| \(\alpha\beta\gamma(\alpha + \beta + \gamma) = -\dfrac{9}{4}\) \((\alpha\beta\gamma)^2(\alpha\beta + \beta\gamma + \gamma\alpha) = \dfrac{27}{4}\) \((\alpha\beta\gamma)^4 = \dfrac{81}{16}\) | M1 |
| \(w^3 + \dfrac{9}{4}w^2 + \dfrac{27}{4}w - \dfrac{81}{16} = 0\) \(16w^3 + 36w^2 + 108w - 81 = 0\) | A1 |
| [3] |
M1: For at least one of \(\alpha\beta\gamma, \alpha + \beta + \gamma, \alpha\beta + \beta\gamma + \gamma\alpha\) correct. Can be implied by one correct coefficient in new cubic equation.
M1: For at least two of \((\alpha\beta\gamma)^4\), \((\alpha\beta\gamma)^2(\alpha\beta + \beta\gamma + \gamma\alpha), \alpha\beta\gamma(\alpha + \beta + \gamma)\) correct – not from incorrect values of \(\alpha\beta\gamma, \alpha + \beta + \gamma, \alpha\beta + \beta\gamma + \gamma\alpha\).
A1: Allow any integer multiple but must have integer coefficients. Must be an equation (so must = 0 or terms on both sides of an equation) with all terms simplified. Condone if in terms of \(x\).