Higher November 2019 Paper 1 Q20
20.
(a) Write \(\sqrt{44} + \sqrt{275}\) in the form \(k\sqrt{11}\), where \(k\) is an integer. [2]
(b) Show that \(\dfrac{\left(\sqrt{3} - 1\right)^2}{\sqrt{3}}\) can be written as \(c\sqrt{3} + d\), where \(c\) and \(d\) are values to be found. [3]
(c) \[2\sqrt{x} - \sqrt{y} = 0\]Find a value for \(x\) and a value for \(y\) so that \(\sqrt{x}\) and \(\sqrt{y}\) are surds. [2]
| Scheme | Marks | Comments |
|---|---|---|
| \(7\sqrt{11}\) | B2 | Allow \(k = 7\); B1 for \(2\sqrt{11}\) or \(5\sqrt{11}\) seen |
| Scheme | Marks | Comments |
|---|---|---|
| M marks may be awarded in reverse order | ||
| \(\dfrac{3 - \sqrt{3} - \sqrt{3} + 1}{\sqrt{3}}\) or better | M1 | For squaring the numerator with at least 3 terms out of 4 correct; may be implied by e.g. \(\dfrac{3 - 2\sqrt{3} - 1}{\sqrt{3}}\) |
| \(\dfrac{3 - \sqrt{3} - \sqrt{3} + 1}{\sqrt{3}} \times \dfrac{\sqrt{3}}{\sqrt{3}}\) or \(\dfrac{4}{\sqrt{3}} - 2\) | M1 | For rationalising the denominator or writing as a simplified binomial |
| \(\dfrac{4}{3}\sqrt{3} - 2\) | A1 | For correct completion to given form |
| Alternative method: | ||
| \(\dfrac{3 - \sqrt{3} - \sqrt{3} + 1}{\sqrt{3}} \quad (= c\sqrt{3} + d)\) | M1 | |
| \(4 - 2\sqrt{3} = 3c + d\sqrt{3}\) and e.g. compares appropriate terms | M1 | |
| \(c = \dfrac{4}{3}\) \(d = -2\) | A1 | |
| Scheme | Marks | Comments |
|---|---|---|
| Any values such that \(x\) and \(y\) are not square and \(y = 4x\) | B2 | B1 for values \(x\) and \(y\) such that \(y = 4x\) oe, si |
| (7) |