AS June 2025 Paper 1 Q3
3 The roots of the equation \(2x^2 + 3x + 5 = 0\) are denoted by \(\alpha\) and \(\beta\).
(a) Write down the value of \(\alpha + \beta\) and the value of \(\alpha\beta\). [2]
(b) Using the answers to part (a) determine the value of each of the following.
- \(\alpha^2 + \beta^2\)
- \(\dfrac{1}{\alpha} + \dfrac{1}{\beta}\)
| Scheme | Marks | AO |
|---|---|---|
| \(\alpha + \beta = -\dfrac{3}{2}\) | B1 | 1.1 |
| \(\alpha\beta = \dfrac{5}{2}\) | B1 | 1.1 |
| [2] |
| Scheme | Marks | AO |
|---|---|---|
| \(\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta\) | M1 | 3.1a |
| \(= \left(-\dfrac{3}{2}\right)^2 - 2\left(\dfrac{5}{2}\right) = \dfrac{9}{4} - \dfrac{20}{4} = -\dfrac{11}{4}\) oe | A1 | 1.1 |
| \(\dfrac{1}{\alpha} + \dfrac{1}{\beta} = \dfrac{\alpha + \beta}{\alpha\beta}\) | M1 | 1.1 |
| \(= \dfrac{\left(-\frac{3}{2}\right)}{\left(\frac{5}{2}\right)} = -\dfrac{3}{5}\) oe | A1 | 1.1 |
| [4] |
Notes
M1: Expressing \(\alpha^2 + \beta^2\) in terms of the symmetric functions
“Using answers to part a”, so some evidence of how these are used is needed
A1: \(-2.75\)
M1: Expressing \(1/\alpha + 1/\beta\) in terms of the symmetric functions
A1: Exact equivalent only \((-0.6)\)