Higher June 2023 Paper 2R Q16
16 Show that \(\;\dfrac{2\sqrt{3}}{\sqrt{3} - 1}\;\) can be written in the form \(\;a + \sqrt{a}\;\) where \(a\) is an integer.
Show your working clearly.
(3)
| Scheme | Marks |
|---|---|
\(\dfrac{2\sqrt{3}}{\sqrt{3} - 1} \times \dfrac{\sqrt{3} + 1}{\sqrt{3} + 1}\) or \(\dfrac{2\sqrt{3}}{\sqrt{3} - 1} \times \dfrac{-\sqrt{3} - 1}{-\sqrt{3} - 1}\) | M1 |
\(\dfrac{2 \times 3 + 2\sqrt{3}}{3 - 1}\) or \(\dfrac{6 + 2\sqrt{3}}{3 - 1}\) or \(\dfrac{6 + 2\sqrt{3}}{2}\) oe \(\dfrac{-2 \times 3 - 2\sqrt{3}}{-3 + 1}\) or \(\dfrac{-6 - 2\sqrt{3}}{-3 + 1}\) or \(\dfrac{-6 - 2\sqrt{3}}{-2}\) oe | M1 |
Working required Answer: \(3 + \sqrt{3}\) | A1 |
| (3) | |
| (3 marks) |
Notes
M1: for explicitly multiplying the numerator and the denominator by \(\sqrt{3} + 1\) or \(-\sqrt{3} - 1\)
M1: dep on M1 (numerator expanded for 2 terms which need to be all correct and denominator may be 4 terms which need to be all correct)
A1: allow \(\sqrt{3} + 3\) (dep on M2)