Higher January 2023 Paper 1R Q18
18 Solve \(\;\sqrt{3}(x - 2\sqrt{3}) = x + 2\sqrt{3}\)
Give your answer in the form \(\;a + b\sqrt{3}\;\) where \(a\) and \(b\) are integers.
Show your working clearly.
(4)
| Scheme | Marks |
|---|---|
\(\sqrt{3}x - x = 6 + 2\sqrt{3}\) oe or \(x - x\sqrt{3} = -6 - 2\sqrt{3}\) (allow \(-2\sqrt{9}\) or \(-2(\sqrt{3})^2\) for −6 or \(2\sqrt{9}\) or \(2(\sqrt{3})^2\) for 6) | M1 |
| \((x =)\;\dfrac{6 + 2\sqrt{3}}{\sqrt{3} - 1}\) oe eg \(\dfrac{-6 - 2\sqrt{3}}{1 - \sqrt{3}}\) | A1 |
\((x =)\;\dfrac{(6 + 2\sqrt{3})}{(\sqrt{3} - 1)} \times \dfrac{(\sqrt{3} + 1)}{(\sqrt{3} + 1)}\) or \(\dfrac{(6 + 2\sqrt{3})(\sqrt{3} + 1)}{2}\) oe or \(\dfrac{(6 + 2\sqrt{3})}{(-1 + \sqrt{3})} \times \dfrac{(-1 - \sqrt{3})}{(-1 - \sqrt{3})}\) oe or \(\dfrac{(-6 - 2\sqrt{3})(1 + \sqrt{3})}{(1 - \sqrt{3})(1 + \sqrt{3})}\) oe | M1 |
Working required Answer: \(6 + 4\sqrt{3}\) | A1 |
| (4) | |
| (4 marks) |
Notes
M1: expanding bracket and collecting terms. Condone one error
A1: oe must be a correct fraction with irrational numerator and denominator
M1: (indep) Multiplying the numerator and denominator of their fraction by \(\sqrt{3} + 1\) oe or showing 2 or −2 as the denominator and multiplying the numerator by \(\sqrt{3} + 1\) oe
or rationalising their denominator, so long as it is of the form \(p + q\sqrt{3}\) where \(p\) and \(q\) are non zero integers
(condone missing brackets provided meaning is clear)
A1: dep on M1A1M1 with no errors seen