Higher January 2020 Paper 1R Q13
13
(a) Use algebra to show that \(\;0.5\dot{7}\dot{2} = \dfrac{63}{110}\) (2)
Given that \(y\) is a prime number,
(b) express \(\;\dfrac{3}{2 - \sqrt{y}}\;\) in the form \(\;\dfrac{a + b\sqrt{y}}{c - y}\;\) where \(a\), \(b\) and \(c\) are integers. (2)
| Scheme | Marks |
|---|---|
| e.g. \(x = 0.57272\ldots\) and \(100x = 57.272\ldots\) OR e.g. \(10x = 5.7272\ldots\) and \(1000x = 572.72\ldots\) | M1 |
e.g. \(100x - x = 57.272\ldots - 0.57272\ldots = 56.7\) and \(\dfrac{56.7}{99} = \dfrac{63}{110}\) or \(1000x - 10x = 572.72\ldots - 5.7272\ldots = 567\) and \(\dfrac{567}{990} = \dfrac{63}{110}\) Working required Answer: Shown | A1 |
| (2) |
Notes
M1: For 2 recurring decimals with correct algebraic labels that when subtracted give a whole number or terminating decimal eg 56.7 or 567 etc e.g. \(100x = 57.272\ldots\) and \(x = 0.57272\ldots\) OR \(1000x = 572.72\ldots\) and \(10x = 5.7272\ldots\) with intention to subtract. (If recurring dots not shown then showing at least the digits 57272, ie 5sf)
A1: for completion to \(\dfrac{63}{110}\)
| Scheme | Marks |
|---|---|
| \(\dfrac{3}{2 - \sqrt{y}} \times \dfrac{2 + \sqrt{y}}{2 + \sqrt{y}}\) or \(6 + 3\sqrt{y}\) or \(4 - y\) | M1 |
| \(\dfrac{6 + 3\sqrt{y}}{4 - y}\) | A1 |
| (2) | |
| (4 marks) |
Notes
M1: for multiplying numerator and denominator by \((2 + \sqrt{y})\) or a correct expression for the numerator or denominator