A2 October 2021 Paper 1 Q3
3 A function \(\mathrm{f}(z)\) is defined on all complex numbers \(z\) by \(\mathrm{f}(z) = z^3 - 3z^2 + kz - 5\) where \(k\) is a real constant. The roots of the equation \(\mathrm{f}(z) = 0\) are \(\alpha, \beta\) and \(\gamma\). You are given that \(\alpha^2 + \beta^2 + \gamma^2 = -5\).
(a) Explain why \(\mathrm{f}(z) = 0\) has only one real root. [3]
(b) Find the value of \(k\). [3]
(c) Find a cubic equation with integer coefficients that has roots \(\dfrac{1}{\alpha}, \dfrac{1}{\beta}\) and \(\dfrac{1}{\gamma}\). [2]
| Scheme | Marks | AO |
|---|---|---|
| e.g. \(\alpha^2 + \beta^2 + \gamma^2 = -5\) means that at least one root is complex | B1 | 2.4 |
| But complex roots come in complex pairs so there are 2 complex roots. | B1 | 2.4 |
| Given that there are 3 roots and 2 are complex one is real. | B1 | 2.4 |
| [3] |
| Scheme | Marks | AO |
|---|---|---|
| \(\alpha + \beta + \gamma = 3\) | B1 | 1.1 |
| \((\alpha + \beta + \gamma)^2 = \alpha^2 + \beta^2 + \gamma^2 + 2(\alpha\beta + \beta\gamma + \gamma\alpha)\) \(9 = -5 + 2(\alpha\beta + \beta\gamma + \gamma\alpha)\) | M1 | 3.1a |
| But \(k = \alpha\beta + \beta\gamma + \gamma\alpha\) \(\Rightarrow k = 7\) | A1 | 1.1 |
| [3] |
Notes
M1: Attempt to obtain identity and substitute
Condone missing 2 and sign errors
| Scheme | Marks | AO |
|---|---|---|
| \(\left(\dfrac{1}{u}\right)^3 - 3\left(\dfrac{1}{u}\right)^2 + 7\left(\dfrac{1}{u}\right) - 5 = 0\) | M1 | 1.1 |
| \(\Rightarrow -5u^3 + 7u^2 - 3u + 1 = 0\) oe | A1 | 1.1 |
| [2] |
Notes
M1: For the substitution
“=0” not necessary here but needed for A1
A1: Allow in terms of \(z\). Allow ft from their \(k\) in (b)
Alternate method
| Scheme | Marks |
|---|---|
| \(\dfrac{1}{\alpha} + \dfrac{1}{\beta} + \dfrac{1}{\gamma} = \dfrac{7}{5},\ \dfrac{1}{\alpha}\dfrac{1}{\beta} + \dfrac{1}{\beta}\dfrac{1}{\gamma} + \dfrac{1}{\gamma}\dfrac{1}{\alpha} = \dfrac{3}{5},\ \dfrac{1}{\alpha}\dfrac{1}{\beta}\dfrac{1}{\gamma} = \dfrac{1}{5}\) | M1 |
| Answer as above | A1 |
| [2] |
M1: For calculating the sum, product and sum of product of pairs of reciprocals of \(\alpha, \beta, \gamma\)