Higher June 2019 Paper 2 Q22
22 Show that \(\quad \left(5\sqrt{3} - \sqrt{12}\right)^2 \quad\) simplifies to an integer. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| \((\sqrt{12} =)\ 2\sqrt{3}\) | M1 | |
| \(5\sqrt{3} - 2\sqrt{3} = 3\sqrt{3}\) | A1 | implies M1A1 |
| 27 with M1A1 seen | A1 | |
| Alternative method 2 | ||
| \(5\sqrt{3}\,5\sqrt{3} - 5\sqrt{3}\,\sqrt{12} - 5\sqrt{3}\,\sqrt{12} + \sqrt{12}\,\sqrt{12}\) or \(25\sqrt{3}\,\sqrt{3} - 10\sqrt{3}\,\sqrt{12} + \sqrt{12}\,\sqrt{12}\) or \((5\sqrt{3}\,5\sqrt{3} =)\ 75\) or \((5\sqrt{3}\,\sqrt{12} =)\ 30\) or \((10\sqrt{3}\,\sqrt{12} =)\ 60\) or \((\sqrt{12}\,\sqrt{12} =)\ 12\) | M1 | oe expansion eg1 \(\sqrt{75}\,\sqrt{75} - \sqrt{75}\,\sqrt{12} - \sqrt{75}\,\sqrt{12} + \sqrt{12}\,\sqrt{12}\) eg2 \(\sqrt{75}\,\sqrt{75} - \sqrt{900} - \sqrt{900} + \sqrt{12}\,\sqrt{12}\) |
| \(75 - 30 - 30 + 12\) or \(75 - 60 + 12\) | A1 | implies M1A1 |
| 27 with M1A1 seen | A1 | |
Additional guidance
| 27 with no working (\(2\sqrt{3}\) not seen) | M0A0A0 |
| Alt 1 \(5\sqrt{3} - \sqrt{12} = 3\sqrt{3}\) (\(2\sqrt{3}\) not seen) | M0A0A0 |
| Alt 2 \(75 - 30 - 30 - 12\) | M1A0A0 |
| Alt 1 \(5\sqrt{3} - 2\sqrt{3} = 3\sqrt{3}\) followed by \(3\sqrt{3}^2 = 27\) (condone missing brackets) | M1A1A1 |
| Only converting to decimals | M0A0A0 |