June 2022 Paper 1 Q10
10 Rationalise and simplify fully \(\qquad \dfrac{\sqrt{3}}{3 + \sqrt{3}}\) [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \(\dfrac{\sqrt{3}}{3 + \sqrt{3}} \times \dfrac{3 - \sqrt{3}}{3 - \sqrt{3}}\) or \(\dfrac{\sqrt{3}}{3 + \sqrt{3}} \times \dfrac{\sqrt{3} - 3}{\sqrt{3} - 3}\) | M1 | \(\times\,(3 - \sqrt{3})\) can still gain full marks if recovered but doesn’t gain M1 if the second M mark isn’t awarded |
| \(\dfrac{3\sqrt{3} - 3}{9 - 3}\) | M1dep | oe eg \(\dfrac{3\sqrt{3} - \sqrt{3}\sqrt{3}}{9 + 3\sqrt{3} - 3\sqrt{3} - \sqrt{3}\sqrt{3}}\) or \(\dfrac{3\sqrt{3} - \sqrt{9}}{9 + 3\sqrt{3} - 3\sqrt{3} - \sqrt{9}}\) or \(\dfrac{3\sqrt{3}}{6} - \dfrac{3}{6}\) |
| \(\dfrac{\sqrt{3} - 1}{2}\) | A1 | oe to something fully simplified eg \(\dfrac{\sqrt{3}}{2} - \dfrac{1}{2}\) or \(\dfrac{1 - \sqrt{3}}{-2}\) |
| Alternative method 2 | ||
| \(\dfrac{\sqrt{3}}{3 + \sqrt{3}} = \dfrac{1}{\sqrt{3} + 1}\) | M1 | |
| \(\dfrac{1}{\sqrt{3} + 1} \times \dfrac{\sqrt{3} - 1}{\sqrt{3} - 1}\) or \(\dfrac{1}{\sqrt{3} + 1} \times \dfrac{1 - \sqrt{3}}{1 - \sqrt{3}}\) | M1dep | oe \(\times\,(\sqrt{3} - 1)\) can still gain full marks if recovered but doesn’t gain M1 if the A mark isn’t awarded |
| \(\dfrac{\sqrt{3} - 1}{2}\) | A1 | oe eg \(\dfrac{\sqrt{3}}{2} - \dfrac{1}{2}\) or \(\dfrac{1 - \sqrt{3}}{-2}\) |
| Alternative method 3 | ||
| \(\dfrac{\sqrt{3}}{3 + \sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{3}{3\sqrt{3} + 3}\) | M1 | |
| \(\dfrac{3}{3\sqrt{3} + 3} \times \dfrac{3\sqrt{3} - 3}{3\sqrt{3} - 3}\) or \(\dfrac{3}{3\sqrt{3} + 3} \times \dfrac{3 - 3\sqrt{3}}{3 - 3\sqrt{3}}\) | M1dep | oe \(\times\,(3\sqrt{3} - 3)\) can still gain full marks if recovered but doesn’t gain M1 if the A mark isn’t awarded |
| \(\dfrac{\sqrt{3} - 1}{2}\) | A1 | oe eg \(\dfrac{\sqrt{3}}{2} - \dfrac{1}{2}\) or \(\dfrac{1 - \sqrt{3}}{-2}\) |
Additional guidance
| Penalise further incorrect working |