A2 June 2019 Paper 1 Q1
1 In this question you must show detailed reasoning.
The quadratic equation \(x^2 - 2x + 5 = 0\) has roots \(\alpha\) and \(\beta\).
(a) Write down the values of \(\alpha + \beta\) and \(\alpha\beta\). [1]
(b) Hence find a quadratic equation with roots \(\alpha + \dfrac{1}{\beta}\) and \(\beta + \dfrac{1}{\alpha}\). [3]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\alpha + \beta = 2,\ \alpha\beta = 5\) | B1 | 1.1 |
| [1] |
| Scheme | Marks | AO |
|---|---|---|
| DR \(\left(\alpha + \dfrac{1}{\beta}\right) + \left(\beta + \dfrac{1}{\alpha}\right) = \alpha + \beta + \left(\dfrac{\alpha + \beta}{\alpha\beta}\right) = 2 + \dfrac{2}{5} = \dfrac{12}{5}\) | M1 | 1.1a |
| \(\left(\alpha + \dfrac{1}{\beta}\right) \times \left(\beta + \dfrac{1}{\alpha}\right) = \alpha\beta + 2 + \dfrac{1}{\alpha\beta} = 7 + \dfrac{1}{5} = \dfrac{36}{5}\) | A1 | 1.1 |
| \(\Rightarrow 5x^2 - 12x + 36 = 0\) oe | A1 | 2.2a |
| [3] |
Notes
M1: Attempt both sum and product of new roots in terms of original roots.
DR so finding roots M0
A1: For one of 12/5 or 36/5
A1: For both, correctly interpreted as quadratic.