Higher June 2023 Unit 1 Q17
17.
(a) Expand and simplify \(\left(4 - \sqrt{6}\right)\left(1 + \sqrt{6}\right)\). [2]
(b)
(i) Write down an integer value of \(x\) that is greater than 5, for which \(x^{\frac{3}{2}}\) is rational. [1]
\(x =\) ……………………
(ii) Write down an integer value of \(x\) that is greater than 5, for which \(x^{\frac{2}{3}}\) is rational. [1]
\(x =\) ……………………
(iii) Write down an integer value of \(x\) that is greater than 5, for which \(x^{\frac{3}{2}}\) and \(x^{\frac{2}{3}}\) are both rational. [1]
\(x =\) ……………………
| Scheme | Marks | Comments |
|---|---|---|
| \(4 + 4 \times \sqrt{6} - (1\ \times)\ \sqrt{6} - 6\) or equivalent | M1 | \((\sqrt{6})^2\) or \(\sqrt{6}\sqrt{6}\) is insufficient for 6. |
| \(= 3\sqrt{6} - 2\) or \(-2 + 3\sqrt{6}\) | A1 | Mark final answer. If no marks awarded, SC1 for 3 of the 4 terms correct. |
| Scheme | Marks | Comments |
|---|---|---|
| (i) Any square number greater than 5 e.g. 9, 16, 25, 36, … | B1 | |
| (ii) Any cube number greater than 5 e.g. 8, 27, 64, 125, 216, ...1000, … | B1 | |
| (iii) Any value of n6 where n is an integer > 1. e.g 64 , 729, … 1 000 000, … | B1 | Allow 26 or 36 or … |