Higher November 2024 Paper 1 Q17
17
(a) Express \(\sqrt{675}\) in the form \(n\sqrt{27}\) where \(n\) is a positive integer. (1)
(b) Show that \(\dfrac{5 - \sqrt{2}}{\sqrt{2} - 1}\) can be written in the form \(a + b\sqrt{2}\) where \(a\) and \(b\) are integers. (3)
| Scheme | Marks |
|---|---|
| \(5\sqrt{27}\) | B1 |
| (1) |
Notes
B1: Allow \(n = 5\)
Do not accept 5 by itself
| Scheme | Marks |
|---|---|
| \(\dfrac{5 - \sqrt{2}}{\sqrt{2} - 1} \times \dfrac{\sqrt{2} + 1}{\sqrt{2} + 1}\) or \(\dfrac{5 - \sqrt{2}}{\sqrt{2} - 1} \times \dfrac{-\sqrt{2} - 1}{-\sqrt{2} - 1}\) | M1 |
eg \(\dfrac{5\sqrt{2} + 5 - 2 - \sqrt{2}}{2 - 1}\) oe or \(\dfrac{5\sqrt{2} + 5 - 2 - \sqrt{2}}{\sqrt{4} + \sqrt{2} - \sqrt{2} - 1}\) oe or \(5\sqrt{2} + 5 - 2 - \sqrt{2}\) | M1 |
Working required Answer: \(3 + 4\sqrt{2}\) | A1 |
| (3) | |
| (4 marks) |
Notes
M1: for rationalising the denominator by multiplying numerator and denominator by \(\sqrt{2} + 1\) or \(-\sqrt{2} - 1\)
M1: (numerator must be expanded to 4 terms, denominator may be 4 terms which need to be all correct)
Accept 1 in the denominator without working
A1: or for stating \(a = 3\) and \(b = 4\)
dep on M2