October 2020 Paper 1 Q2
2 Express \(\dfrac{a+\sqrt{2}}{3-\sqrt{2}}\) in the form \(p + q\sqrt{2}\), giving \(p\) and \(q\) in terms of \(a\). [3]
| Scheme | Marks | AO |
|---|---|---|
| \(\dfrac{\left(a+\sqrt{2}\right)\left(3+\sqrt{2}\right)}{\left(3-\sqrt{2}\right)\left(3+\sqrt{2}\right)}\) | M1 | 1.1a |
| \(= \dfrac{3a + a\sqrt{2} + 3\sqrt{2} + 2}{9-2}\) | M1 | 1.1a |
| \(= \dfrac{3a+2}{7} + \dfrac{a+3}{7}\sqrt{2}\) | A1 | 1.1 |
| [3] |
Notes
M1: Attempt to rationalize the denominator
M1: Attempt to expand the brackets
A1: must be in the form \(p + q\sqrt{2}\)
Mark final answer
Allow for instead stating \(p = \dfrac{3a+2}{7},\ q = \dfrac{a+3}{7}\)