4MA0 Higher January 2013 Paper 3 Q19
19 The diagram shows a rectangular playground of width \(x\) metres and length \(3x\) metres.
Diagram NOT accurately drawn
The playground is extended, by adding 10 metres to its width and 20 metres to its length, to form a larger rectangular playground.
The area of the larger rectangular playground is double the area of the original playground.
(a) Show that \(3x^2 - 50x - 200 = 0\) (3)
(b) Calculate the area of the original playground. (5)
| Scheme | Marks |
|---|---|
| \(2 \times 3x \times x = (x + 10)(3x + 20)\) or \(6x^2 = (x + 10)(3x + 20)\) | M2 |
| \(6x^2 = 3x^2 + 50x + 200\) | A1 |
| (3) |
Notes
M2: If not M2 then M1 for \(2 \times 3x \times x\) or \(2 \times 3x^2\) or \(6x^2\) or \((x + 10)(3x + 20)\)
A1: Dependent on at least M1
| Scheme | Marks |
|---|---|
| \((3x + 10)(x - 20)\ (=0)\) Marks can be awarded in b) if seen in a) | M2 |
| \(x = 20\) | A1 |
| 20 x 3 x 20 | M1 |
| 1200 | A1 |
| (5) |
Notes
M2: or \(x = \dfrac{50 \pm \sqrt{2500 + 2400}}{6}\)
If not M2 then M1 for \((3x \pm 10)(x \pm 20)\) or \(x = \dfrac{--50 \pm \sqrt{(-50)^2 - 4 \times 3 \times -200}}{2 \times 3}\) condone 1 sign error
A1: dep on M1 in b). Ignore negative root (−3.3 rec)
A1: dep on 1st M1 in b)