A2 June 2023 Paper 1 Q1
1. The cubic equation
\[x^3 - 7x^2 - 12x + 6 = 0\]has roots \(\alpha\), \(\beta\) and \(\gamma\).
Without solving the equation, determine a cubic equation whose roots are \((\alpha + 2)\), \((\beta + 2)\) and \((\gamma + 2)\), giving your answer in the form \(w^3 + pw^2 + qw + r = 0\), where \(p\), \(q\) and \(r\) are integers to be found.
(5)
| Scheme | Marks | AO |
|---|---|---|
| \(\{w = x + 2 \Rightarrow\}\ x = w - 2\) | B1 | 3.1a |
| \((w - 2)^3 - 7(w - 2)^2 - 12(w - 2) + 6\ (= 0)\) | M1 | 1.1b |
| \(\left(w^3 - 6w^2 + 12w - 8\right) - 7\left(w^2 - 4w + 4\right) - 12(w - 2) + 6\) \(w^3 - 6w^2 + 12w - 8 - 7w^2 + 28w - 28 - 12w + 24 + 6\) \(= w^3 + \ldots w^2 + \ldots w + \ldots\) | M1 | 3.1a |
| \(w^3 - 13w^2 + 28w - 6 = 0\) | A1 A1 | 1.1b 1.1b |
| (5) | ||
| (5 marks) |
Notes
Alternative using sum, pair sum and product of roots:
| Scheme | Marks | AO |
|---|---|---|
| \(\alpha + \beta + \gamma = 7,\ \alpha\beta + \beta\gamma + \alpha\gamma = -12,\ \alpha\beta\gamma = -6\) | B1 | 3.1a |
| New sum: \(\alpha + 2 + \beta + 2 + \gamma + 2 = (\alpha + \beta + \gamma) + 6 = 7 + 6 = 13\) New pair sum: \((\alpha + 2)(\beta + 2) + (\alpha + 2)(\gamma + 2) + (\beta + 2)(\gamma + 2)\) \(= (\alpha\beta + \alpha\gamma + \beta\gamma) + 4(\alpha + \beta + \gamma) + 12 = -12 + 4 \times 7 + 12 = 28\) New product: \((\alpha + 2)(\beta + 2)(\gamma + 2)\) \(= \alpha\beta\gamma + 2(\alpha\beta + \alpha\gamma + \beta\gamma) + 4(\alpha + \beta + \gamma) + 8\) \(= -6 + 2 \times -12 + 4 \times 7 + 8 = 6\) | M1 | 3.1a |
| \(p = -\text{"}13\text{"},\ q = 28,\ r = -\text{"}6\text{"}\) or \(w^3 - \text{"}13\text{"}w^2 + \text{"}28\text{"}w - \text{"}6\text{"}\ (= 0)\) | M1 | 1.1b |
| \(w^3 - 13w^2 + 28w - 6 = 0\) | A1 A1 | 1.1b 1.1b |
Notes
Allow a variable other than \(w\) to be used for the first 4 marks.
The “= 0” is not required until the final mark.
B1: Selects the method of making a connection between \(x\) and \(w\) by writing \(x = w - 2\)
M1: Applies the process of substituting their \(x = \text{"}w - 2\text{"}\) into the equation for all occurrences of \(x\).
M1: Depends on having attempted substituting either \(x = w - 2\) or \(x = w + 2\) into the equation. This mark is for manipulating their resulting equation into the required form so must have gathered terms. Condone poor squaring/cubing of brackets as long as a cubic expression is obtained.
A1: At least two of \(p\), \(q\) and \(r\) correct.
A1: Correct final equation (including “= 0”). Must be an equation in \(w\).
Note if they say e.g. \(x = w - 2\) and then substitute \(w + 2\), it is possible to score B1 M0 M1
Note if they say e.g. \(x = w + 2\) and then substitute \(w - 2\), allow recovery
Alternative:
B1: Selects the method of giving three correct equations for the sum, pair sum and product in terms of \(\alpha\), \(\beta\) and \(\gamma\). Note that the correct values may be seen embedded when they attempt the new sum, pair sum and product e.g. \((\alpha + 2)(\beta + 2)(\gamma + 2) = \alpha\beta\gamma + 2(\alpha\beta + \alpha\gamma + \beta\gamma) + 4(\alpha + \beta + \gamma) + 8 = \underline{-6} + 2(\underline{-12}) + 4(\underline{7}) + 8\)
M1: Applies the process of finding the new sum, pair sum and product. Mark positively here and allow slips provided they are attempting \(\alpha + 2 + \beta + 2 + \gamma + 2\), \((\alpha + 2)(\beta + 2) + (\alpha + 2)(\gamma + 2) + (\beta + 2)(\gamma + 2)\) and \((\alpha + 2)(\beta + 2)(\gamma + 2)\)
M1: In this method, this mark is for choosing \(p = -\)(their new sum), \(q =\) their new pair sum, \(r = -\)(their new product) or forming \(w^3 - (\text{new sum})w^2 + (\text{new pair sum})w - (\text{new product})\)
(corrected from the printed mark scheme: the first term is printed as \(w^2\))
A1: At least two of \(p\), \(q\) and \(r\) correct. As values or seen in their equation.
A1: Correct final equation (including “= 0”). Must be an equation in \(w\).
In all methods, the final A mark depends on all the previous marks.