Higher November 2017 Paper 1 Q17
17.
Diagram not drawn to scale
The area of the rectangle is 22 cm2.
The width of the rectangle is \((1 + 2\sqrt{3})\) cm.
(a) Find the length of the rectangle. Express your answer in the form \(a + b\sqrt{3}\), where \(a\) and \(b\) are integers. [4]
(b) The length of the diagonal of the rectangle is \(\sqrt{x}\) cm.
Show that \(x = (65 - 12\sqrt{3})\) cm. [2]
Show that \(x = (65 - 12\sqrt{3})\) cm. [2]
| Scheme | Marks | Comments |
|---|---|---|
| \((\text{length} =)\ \dfrac{22}{1 + 2\sqrt{3}}\) | M1 | |
| \(\dfrac{22}{1 + 2\sqrt{3}} \times \dfrac{1 - 2\sqrt{3}}{1 - 2\sqrt{3}}\) | M1 | |
| \(\dfrac{22 - 44\sqrt{3}}{1 - 4(3)}\) | M1 | |
| \(-2 + 4\sqrt{3}\) | A1 |
| Scheme | Marks | Comments |
|---|---|---|
| \(x = \left(-2 + 4\sqrt{3}\right)^2 + \left(1 + 2\sqrt{3}\right)^2\) | M1 | FT ‘their \(-2 + 4\sqrt{3}\)’ for M1 only |
| \(4 - 8\sqrt{3} - 8\sqrt{3} + 16(3) +\) \(1 + 2\sqrt{3} + 2\sqrt{3} + 4(3) = 65 - 12\sqrt{3}\) | A1 | NB Answer is given |
| (6) |