AS June 2018 Paper 1 Q2
2. The cubic equation
\[z^3 - 3z^2 + z + 5 = 0\]has roots \(\alpha\), \(\beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are \((2\alpha + 1)\), \((2\beta + 1)\) and \((2\gamma + 1)\), 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 |
|---|---|---|
| \(w = 2z + 1 \Rightarrow z = \dfrac{w - 1}{2}\) | B1 | 3.1a |
| \[\left(\frac{w - 1}{2}\right)^3 - 3\left(\frac{w - 1}{2}\right)^2 + \left(\frac{w - 1}{2}\right) + 5 = 0\] | M1 | 3.1a |
| \[\frac{1}{8}\left(w^3 - 3w^2 + 3w - 1\right) - \frac{3}{4}\left(w^2 - 2w + 1\right) + \frac{w - 1}{2} + 5 = 0\] | ||
| \[w^3 - 9w^2 + 19w + 29 = 0\] | M1 A1 A1 | 1.1b 1.1b 1.1b |
| (5) | ||
| (5 marks) |
Notes
B1: Selects the method of making a connection between \(z\) and \(w\) by writing \(z = \dfrac{w - 1}{2}\)
M1: Applies the process of substituting their \(z = \dfrac{w - 1}{2}\) into \(z^3 - 3z^2 + z + 5 = 0\)
(Allow \(z = 2w + 1\))
M1: Manipulates their equation into the form \(w^3 + pw^2 + qw + r\,(= 0)\) having substituted their \(z\) in terms of \(w\). Note that the “= 0” can be missing for this mark.
A1: At least two of \(p\), \(q\), \(r\) correct. Note that the “= 0” can be missing for this mark.
A1: Fully correct equation including “= 0”
The first 4 marks are available if another letter is used instead of \(w\) but the final answer must be in terms of \(w\).
Alternative (ALT 1)
| Scheme | Marks | AO |
|---|---|---|
| \(\alpha + \beta + \gamma = 3,\ \alpha\beta + \beta\gamma + \alpha\gamma = 1,\ \alpha\beta\gamma = -5\) | B1 | 3.1a |
| New sum \(= 2(\alpha + \beta + \gamma) + 3 = 9\) New pair sum \(= 4(\alpha\beta + \beta\gamma + \gamma\alpha) + 4(\alpha + \beta + \gamma) + 3 = 19\) New product \(= 8\alpha\beta\gamma + 4(\alpha\beta + \beta\gamma + \gamma\alpha) + 2(\alpha + \beta + \gamma) + 1 = -29\) | M1 | 3.1a |
| \[w^3 - 9w^2 + 19w + 29 = 0\] | M1 A1 A1 | 1.1b 1.1b 1.1b |
| (5) |
B1: Selects the method of giving three correct equations containing \(\alpha\), \(\beta\) and \(\gamma\)
M1: Applies the process of finding the new sum, new pair sum, new product
M1: Applies \(w^3 - (\text{new sum})w^2 + (\text{new pair sum})w - (\text{new product})\,(= 0)\)
or identifies \(p\) as –(new sum) \(q\) as (new pair sum) and \(r\) as –(new product)
A1: At least two of \(p\), \(q\), \(r\) correct.
A1: Fully correct equation including “= 0”
The first 4 marks are available if another letter is used instead of \(w\) but the final answer must be in terms of \(w\).