4MA0 Higher January 2014 Paper 4 Q18
18 A rectangular lawn has a length of \(3x\) metres and a width of \(2x\) metres.
The lawn has a path of width 1 metre on three of its sides.
Diagram NOT accurately drawn
The total area of the lawn and the path is 100 m2
(a) Show that \(\;6x^2 + 7x - 98 = 0\) (2)
(b) Calculate the area of the lawn.
Show clear algebraic working. (5)
Show clear algebraic working. (5)
| Scheme | Marks |
|---|---|
| \((3x + 2)(2x + 1) = 100\) | M1 |
| \(6x^2 + 4x + 3x + 2 = 100\) Answer: \(6x^2 + 7x - 98 = 0\) * | A1 |
| (2) |
Notes
M1: or \((2x \times 3x) + 2(2x + 1) + 3x = 100\) oe
or \((2x \times 3x) + (2 \times 2x(\times 1)) + 1) + 3x + 1 + 1 = 100\) oe
other partitions are acceptable but partitioning must go on to form a correct equation.
A1: Accept \(6x^2 + 7x + 2 = 100\) if M1 awarded
* Answer given
The mark scheme states that for this part the correct answer alone does not imply a correct method.
| Scheme | Marks |
|---|---|
| \((3x + 14)(2x - 7)\;(= 0)\) | M2 |
| \(x = 3.5\) | A1 |
| (Area =) \(6 \times \text{“}3.5\text{”}^2\) or \((3 \times \text{“}3.5\text{”}) \times (2 \times \text{“}3.5\text{”})\) | M1ft |
| 73.5 | A1 |
| (5) | |
| (7 marks) |
Notes
M2: or \((x =)\dfrac{-7 \pm \sqrt{49 + 2352}}{12}\) or \((x =)\dfrac{-7 \pm \sqrt{2401}}{12}\)
If not M2 then M1 for \((3x \pm 14)(2x \pm 7)\)
or \((x =)\dfrac{-7 \pm \sqrt{7^2 - 4 \times 6 \times -98}}{2 \times 6}\)
condone + in place of \(\pm\) and 1 sign error.
A1: Dependent on at least M1 Ignore negative root.
M1ft: Dependent on at least M1 and \(x \gt 0\)
A1: cao Dependent on first M1