Higher June 2023 Paper 1 Q23
23 Write \(\dfrac{3\sqrt{3}}{4 - \sqrt{3}} - \dfrac{2}{\sqrt{3}}\) in the form \(\dfrac{a\sqrt{3} + b}{c}\) where \(a\), \(b\) and \(c\) are integers. (4)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(\dfrac{10\sqrt{3} + 27}{39}\) | M1 | for method to rationalise one of the fractions, eg \(\dfrac{3\sqrt{3}}{4 - \sqrt{3}} \times \dfrac{4 + \sqrt{3}}{4 + \sqrt{3}}\ \left(= \dfrac{12\sqrt{3} + 9}{16 - 3}\right)\) or \(\dfrac{2}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}}\ \left(= \dfrac{2\sqrt{3}}{3}\right)\) oe |
| M1 | for method to rationalise both of the fractions eg \(\dfrac{3\sqrt{3}}{4 - \sqrt{3}} \times \dfrac{4 + \sqrt{3}}{4 + \sqrt{3}}\ \left(= \dfrac{12\sqrt{3} + 9}{16 - 3}\right)\) and \(\dfrac{2}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}}\ \left(= \dfrac{2\sqrt{3}}{3}\right)\) oe | |
| M1 | for method to write correct fractions with a common denominator eg \(\dfrac{3(12\sqrt{3} + 9)}{13 \times 3} - \dfrac{13 \times 2\sqrt{3}}{13 \times 3}\) or \(\dfrac{36\sqrt{3} + 27}{39} - \dfrac{26\sqrt{3}}{39}\) oe | |
| A1 | for \(\dfrac{10\sqrt{3} + 27}{39}\) oe | |
| Alternative | ||
| M1 | for method to write fractions with a common denominator, eg \(\dfrac{3\sqrt{3} \times \sqrt{3}}{\sqrt{3}(4 - \sqrt{3})} - \dfrac{2 \times (4 - \sqrt{3})}{\sqrt{3}(4 - \sqrt{3})}\) oe | |
| M1 | for writing as a correct single fraction without brackets, eg \(\dfrac{9 - 8 + 2\sqrt{3}}{4\sqrt{3} - 3}\) oe | |
| M1 | for method to rationalise a fraction of the form \(\dfrac{a + b\sqrt{3}}{c\sqrt{3} - d}\) where \(a\), \(b\), \(c\) and \(d\) are integers, eg \(\dfrac{1 + 2\sqrt{3}}{4\sqrt{3} - 3} \times \dfrac{4\sqrt{3} + 3}{4\sqrt{3} + 3}\) or \(\dfrac{4\sqrt{3} + 3 + 24 + 6\sqrt{3}}{48 + 12\sqrt{3} - 12\sqrt{3} - 9}\) oe | |
| A1 | for \(\dfrac{10\sqrt{3} + 27}{39}\) oe |