4400 Higher November 2007 Paper 3 Q20
20
(a) Expand \((1 + \sqrt{3})^2\)
Give your answer in the form \(a + b\sqrt{3}\) where \(a\) and \(b\) are integers. (2)
Give your answer in the form \(a + b\sqrt{3}\) where \(a\) and \(b\) are integers. (2)
(b) 
Give your answer as a surd. (4)
Diagram NOT accurately drawn
Calculate the exact length of \(PR\).Give your answer as a surd. (4)
| Scheme | Marks |
|---|---|
| \(1 + \sqrt{3} + \sqrt{3} + 3\) | M1 |
| \(4 + 2\sqrt{3}\) | A1 |
| (2) |
Notes
M1: oe
| Scheme | Marks |
|---|---|
| \(2^2 + (1 + \sqrt{3})^2 - 2 \times 2 \times (1 + \sqrt{3})\cos 60\) | M1 |
| \(= 4 + \text{“}4 + 2\sqrt{3}\text{”} - 2(1 + \sqrt{3})\) | M1 |
| \(= 6\) | A1 |
| \(\sqrt{6}\) | A1 |
| (4) | |
| (6 marks) |
Notes
M1: oe allow \(2^2 + 2.73^2 - 2 \times 2 \times 2.73\cos 60\) oe
M1: oe ft (a), as long as in form \(a + \sqrt{b}\)
must have exp’d bracket & subst’d cos60
A1: not ISW decimals can score only 1st M1