Higher November 2019 Unit 1 Q15
15. Simplify
\[\left(2 + \sqrt{5}\right)^2 - \frac{\sqrt{500}}{\left(\sqrt{5}\right)^3}\]and indicate whether your answer is rational or irrational. [5]
The answer is: rational ☐ irrational ☐
| Scheme | Marks | Comments |
|---|---|---|
| \(9 + 4\sqrt{5}\) | B2 | If not B2, award B1 for 3 or 4 correct terms within \(4 + 2\sqrt{5} + 2\sqrt{5} + 5\) or \(4 + 2\sqrt{5} + 2\sqrt{5} + \sqrt{25}\) (without subsequent correct collection of terms) (\(4\sqrt{5}\) is equivalent to ‘two correct terms’) |
| (–) 2 | B2 | B1 for (numerator of) \(10\sqrt{5}\) or B1 for (denominator of) \(5\sqrt{5}\) or \(\sqrt{125}\) or B1 for appropriate factorisation of both numerator and denominator e.g. \(\dfrac{\sqrt{5} \times \sqrt{100}}{\sqrt{5} \times \sqrt{25}}\) or \(\dfrac{\sqrt{5} \times \sqrt{5} \times \sqrt{5} \times \sqrt{4}}{\sqrt{5} \times \sqrt{5} \times \sqrt{5}}\) |
| \(7 + 4\sqrt{5}\) AND irrational | B1 | Mark final answer. FT for equivalent difficulty (requiring collection of terms) AND either B2 awarded AND final answer is irrational. |