Higher June 2017 Paper 1 Q16
16.
(a) Simplify each of the following.
(i) \(\dfrac{15}{\sqrt{5}}\) [2]
(ii) \(\sqrt{300} - \sqrt{27}\) [2]
(b) Tom says:
Give the reasons to support your decision. [3]
“Expanding and simplifying \(\left(a + \sqrt{2}\right)\left(b - \sqrt{2}\right)\), where \(a\) and \(b\) are integers, can never have an integer answer.”
Show that Tom’s statement is incorrect.Give the reasons to support your decision. [3]
| Scheme | Marks | Comments |
|---|---|---|
| (a)(i) \(\dfrac{15}{\sqrt5} \times \dfrac{\sqrt5}{\sqrt5}\) or \(\dfrac{15\sqrt5}{5}\) or \(\dfrac{3 \times 5}{\sqrt5}\) | M1 | |
| \(3\sqrt5\) | A1 | implies M1 |
| (a)(ii) \(10\sqrt3 - 3\sqrt3\) | M1 | |
| \(7\sqrt3\) | A1 |
| Scheme | Marks | Comments |
|---|---|---|
| Expands \(ab - a\sqrt2 + b\sqrt2 - \sqrt2 \times \sqrt2\) | M1 | Allow one error in expansion; may be in a grid |
| Identifies \(ab\) as an integer and \(\sqrt2 \times \sqrt2\) as an integer | M1 | May be implied eg \(ab - 2\) is an integer |
| (Not correct as ) an integer when \(a = b\) or equivalent | A1 | \(a = b\) must be seen or implied at some point |
Alternative method 1
| Scheme | Marks | Comments |
|---|---|---|
| Stating eg ‘difference of two squares if \(a = b\)’ | B1 | |
| Shows eg \(a^2 - 2\) | M1 | |
| Concludes integer | A1 |
Alternative method 2
| Scheme | Marks | Comments |
|---|---|---|
| Uses equal numerical values of \(a\) and \(b\) and expands with at most one error eg \(\left(7 + \sqrt2\right)\left(7 - \sqrt2\right) = 7 \times 7 - 7\sqrt2 + 7\sqrt2 - \sqrt2 \times \sqrt2\) | M2 | Or M1 for using numerical unequal values of \(a\) and \(b\) and expands with at most one error Or B1 for stating that \(a\) and \(b\) must be the same |
| Correct numerical working and integer answer | A1 |
| (7) |