Higher November 2024 Paper 1 Q16
16
(a) Rationalise the denominator of \(\dfrac{15}{\sqrt{5}}\)
Give your answer in its simplest form. (2)
Give your answer in its simplest form. (2)
(b) Write \(\dfrac{\sqrt{75} - 2}{1 + 2\sqrt{3}}\) in the form \(\dfrac{a - b\sqrt{3}}{c}\) where \(a\), \(b\) and \(c\) are integers. (4)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(3\sqrt{5}\) | M1 | for \(\dfrac{15}{\sqrt{5}} \times \dfrac{\sqrt{5}}{\sqrt{5}}\) or \(\dfrac{15}{\sqrt{5}} \times \dfrac{-\sqrt{5}}{-\sqrt{5}}\) |
| A1 | for \(3\sqrt{5}\) or \(\sqrt{45}\) |
| Answer | Mark | Mark scheme |
|---|---|---|
| \(\dfrac{32 - 9\sqrt{3}}{11}\) | M1 | (indep) for writing \(\sqrt{75}\) as \(5\sqrt{3}\) |
| M1 | for method to rationalise the denominator, eg \(\dfrac{\sqrt{75} - 2}{1 + 2\sqrt{3}} \times \dfrac{1 - 2\sqrt{3}}{1 - 2\sqrt{3}}\) or \(\dfrac{5\sqrt{3} - 2}{1 + 2\sqrt{3}} \times \dfrac{1 - 2\sqrt{3}}{1 - 2\sqrt{3}}\) | |
| M1 | (dep on previous M1) for expanding terms, condone one error in numerator or denominator eg \(\dfrac{\sqrt{75} - 2\sqrt{75}\sqrt{3} - 2 + 4\sqrt{3}}{1 - 2\sqrt{3} + 2\sqrt{3} - 4\sqrt{3}\sqrt{3}}\) or \(\dfrac{5\sqrt{3} - 10\sqrt{3}\sqrt{3} - 2 + 4\sqrt{3}}{1 - 2\sqrt{3} + 2\sqrt{3} - 4\sqrt{3}\sqrt{3}}\) | |
| A1 | for \(\dfrac{32 - 9\sqrt{3}}{11}\) oe eg \(\dfrac{-32 + 9\sqrt{3}}{-11}\) |
Additional guidance
This mark can be awarded whenever this is seen, which might be later in the process.
Accept \(a = 32\), \(b = 9\), \(c = 11\)