AS June 2019 Paper 1 Q14
14 The graph of \(y = x^3 - 3x\) is shown below.
The two stationary points have \(x\)-coordinates of \(-1\) and \(1\)
The cubic equation
\[x^3 - 3x + p = 0\]where \(p\) is a real constant, has the roots \(\alpha\), \(\beta\) and \(\gamma\).
The roots \(\alpha\) and \(\beta\) are not real.
(a) Explain why \(\alpha + \beta = -\gamma\) [1 mark]
(b) Find the set of possible values for the real constant \(p\). [2 marks]
(c) \(\mathrm{f}(x) = 0\) is a cubic equation with roots \(\alpha + 1\), \(\beta + 1\) and \(\gamma + 1\)
(i) Show that the constant term of \(\mathrm{f}(x)\) is \(p + 2\) [3 marks]
(ii) Write down the \(x\)-coordinates of the stationary points of \(y = \mathrm{f}(x)\) [1 mark]
| Scheme | Marks | AO |
|---|---|---|
| Gives a full and correct explanation. Accept \(\alpha + \beta + \gamma = 0\) followed by \(\alpha + \beta = -\gamma\) without further justification. | E1 | 2.4 |
Typical solution
The coefficient of \(x^2\) is zero \(\therefore \alpha + \beta + \gamma = 0\)
\[\alpha + \beta = -\gamma\]| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(-1\) or \(1\) into \(x^3 - 3x\) Possibly implied by \(-2\) or \(2\) | M1 | 3.1a |
| Finds the correct set of values for \(p\). Condone ‘and’. NMS scores 2/2 | A1 | 2.2a |
Typical solution
max point \(= (-1, 2)\) and min point \(= (1, -2)\)
\[p \lt -2, \quad p \gt 2\]| Scheme | Marks | AO |
|---|---|---|
| (i) Correctly expands \((\alpha + 1)(\beta + 1)(\gamma + 1)\) Or recognises that \(y = \mathrm{f}(x)\) is a horizontal translation of \(y = x^3 - 3x + p\) (possibly implied by the sight of \(x + 1\) or \(x - 1\)) Accept any sensible alternative for \(x\). | B1 | 3.1a |
| Substitutes 0 for \(\alpha + \beta + \gamma\), and \(\pm 3\) for \(\alpha\beta + \beta\gamma + \gamma\alpha\), and \(\pm p\) for \(\alpha\beta\gamma\) Or substitutes \(x - 1\) for \(x\) in \(x^3 - 3x + p\) Accept any sensible alternative for \(x\). | M1 | 1.1a |
| Shows the required result. NMS scores 0/3 | R1 | 2.1 |
| (ii) Gives the correct \(x\)-intercepts and no others. | B1 | 2.2a |
| (7 marks) |
Typical solution
(i)
Let \(w = x + 1\)
\[x = w - 1\]\[(w - 1)^3 - 3(w - 1) + p = 0\]\[w^3 - 3w^2 + 3w - 1 - 3w + 3 + p = 0\]\[w^3 - 3w^2 + p + 2 = 0\]constant term \(= p + 2\)
(ii)
0 and 2