AS June 2023 Paper 1 Q5
5 In this question you must show detailed reasoning.
The roots of the equation \(5x^2 - 3x + 12 = 0\) are \(\alpha\) and \(\beta\).
By considering the symmetric functions of the roots, \(\alpha + \beta\) and \(\alpha\beta\), determine the exact value of \(\dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2}\). [4]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\alpha + \beta = \dfrac{3}{5}\) | B1 | 1.1 |
| \(\alpha\beta = \dfrac{12}{5}\) | B1 | 1.1 |
| \(\dfrac{1}{\alpha^2} + \dfrac{1}{\beta^2} = \dfrac{\beta^2 + \alpha^2}{\alpha^2\beta^2} = \dfrac{(\alpha + \beta)^2 - 2\alpha\beta}{(\alpha\beta)^2}\) | M1 | 3.1a |
| \(= \dfrac{\left(\frac{3}{5}\right)^2 - 2 \times \frac{12}{5}}{\left(\frac{12}{5}\right)^2} = -\dfrac{37}{48}\) | A1 | 1.1 |
| [4] |
Notes
M1: Rewrite the expression in terms of the standard symmetric functions
Need to see \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\) oe
No need to see \(\alpha^2\beta^2 = (\alpha\beta)^2\)
A1: cao
Accept eg \(-0.7708\dot{3}\) but not rounded, incorrect or incomplete decimal form.
SC B1 for correct answer if B0B0M0
Accept any equivalent fraction