AS June 2024 Paper 1 Q13
13 The cubic equation \(x^3 - x - 7 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\)
The cubic equation \(\mathrm{p}(x) = 0\) has roots \(\alpha - 1\), \(\beta - 1\) and \(\gamma - 1\)
The coefficient of \(x^3\) in \(\mathrm{p}(x)\) is 1
(a) Describe fully the transformation that maps the graph of \(y = x^3 - x - 7\) onto the graph of \(y = \mathrm{p}(x)\) [2 marks]
(b) Find \(\mathrm{p}(x)\) [3 marks]
| Scheme | Marks | AO |
|---|---|---|
| Identifies the transformation as a translation. Do not accept alternatives such as ‘move’. | M1 | 3.1a |
| States Translation \(\begin{bmatrix} -1 \\ 0 \end{bmatrix}\) Accept 1 to the left instead of \(\begin{bmatrix} -1 \\ 0 \end{bmatrix}\) Do not accept any extra erroneous description. | A1 | 3.2a |
| (2) |
Typical solution
Translation \(\begin{bmatrix} -1 \\ 0 \end{bmatrix}\)
| Scheme | Marks | AO |
|---|---|---|
| Substitutes \(y + 1\) for \(x\) Accept any letter, including \(x\) Or Writes one of: \(\alpha + \beta + \gamma = 0\) \(\alpha\beta + \beta\gamma + \gamma\alpha = -1\) \(\alpha\beta\gamma = 7\) | B1 | 1.1b |
| Expands and simplifies their \((my + c)^3 - (my + c) - 7 = 0\) where \(m\) is non-zero Allow one incorrect simplified coefficient. Or Correctly calculates two of \(\sum(\alpha - 1)\) \(\sum(\alpha - 1)(\beta - 1)\) \((\alpha - 1)(\beta - 1)(\gamma - 1)\) for their values of \(\alpha + \beta + \gamma\) and/or \(\alpha\beta + \beta\gamma + \gamma\alpha\) and/or \(\alpha\beta\gamma\) PI by two correct terms in a cubic equation other than \(x^3\) | M1 | 1.1a |
| Obtains \(x^3 + 3x^2 + 2x - 7\) | R1 | 2.2a |
| (3) | ||
| (5 marks) |
Typical solution
Let \(y = x - 1\)
\[\Rightarrow x = y + 1\]So the new equation is
\[(y + 1)^3 - (y + 1) - 7 = 0\]\[\Rightarrow y^3 + 3y^2 + 3y + 1 - y - 1 - 7 = 0\]\[\Rightarrow y^3 + 3y^2 + 2y - 7 = 0\]So \(\mathrm{p}(x) = x^3 + 3x^2 + 2x - 7\)