4MA0 Higher June 2014 Paper 3R Q16
16
(a) Expand \(\;(5 + 3\sqrt{2})^2\)
Give your answer in the form \((a + b\sqrt{2})\), where \(a\) and \(b\) are integers.
Show your working clearly. (2)
Give your answer in the form \((a + b\sqrt{2})\), where \(a\) and \(b\) are integers.
Show your working clearly. (2)
(b) \((5 + 3\sqrt{2})^2 = p + \dfrac{q}{\sqrt{8}}\), where \(p\) and \(q\) are integers.
Find the value of \(q\). (3)
Find the value of \(q\). (3)
| Scheme | Marks |
|---|---|
\(25 + 15\sqrt{2} + 15\sqrt{2} + 9 \times 2\) or \(25 + 15\sqrt{2} + 15\sqrt{2} + 18\) or \(25 + 30\sqrt{2} + 9 \times 2\) or \(25 + 30\sqrt{2} + 18\) | M1 |
| \(43 + 30\sqrt{2}\) | A1 |
| (2) |
Notes
M1: Expand to give four terms – (must have surds not decimals), at least three correct,
or three terms with irrational term and one other correct.
Accept \(\left(\sqrt{2}\right)^2\) for 2
A1: dep on M1 awarded
Working required in part (a): the correct answer alone does not imply a correct method.
| Scheme | Marks |
|---|---|
| \(\sqrt{8} = 2\sqrt{2}\) or \((q =)\sqrt{8} \times \text{“}30\sqrt{2}\text{”}\) | M1 |
\(\dfrac{1}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}\) or \(\dfrac{1}{\sqrt{8}} = \dfrac{\sqrt{8}}{8}\) or \((q =)30\sqrt{16}\) or \((q =)60\sqrt{4}\) or \((q =)30\sqrt{2} \times 2\sqrt{2}\) | M1 |
| 120 | A1 |
| (3) | |
| (5 marks) |
Notes
A1: ft 4 × “30” from (a)
M1 M1: Award M marks independently for simplifying \(\sqrt{8}\) and rationalising the denominator, seen at any points in the solution. ft from (a) for \(30\sqrt{16}\)