Higher June 2017 Paper 1 Q21
21. A manufacturer wants to test the performance of a new fuel.
In the test, a total of 20 litres of the fuel is shared between a car and a motorbike.
(a) The car uses \(x\) litres of fuel in the test.
Write an expression in terms of \(x\) for the number of litres used by the motorbike. [1]
Write an expression in terms of \(x\) for the number of litres used by the motorbike. [1]
(b) The vehicles are driven and the distance travelled is recorded.
The table shows some of the test results.
The table shows some of the test results.
| Distance travelled (miles) | Fuel used (litres) | Fuel consumption (Miles per litre) | |
|---|---|---|---|
| Car | 160 | \(x\) | \(\dfrac{160}{x}\) |
| Motorbike | 48 |
The motorbike averaged 2 miles per litre more than the car in the test.
(i) Show that \(x\) satisfies the equation [5]\[x^2 + 84x - 1600 = 0.\]
(ii) Using an algebraic method, find the number of litres of fuel used by the motorbike. [3]
| Scheme | Marks | Comments |
|---|---|---|
| \(20 - x\) | B1 | Accept \(\dfrac{48x}{160 + 2x}\) |
| Scheme | Marks | Comments |
|---|---|---|
| (b)(i) \(\dfrac{160}{x} + 2 = \dfrac{48}{20 - x}\) or \(\dfrac{48x}{160 + 2x} = 20 - x\) or equivalent | M2 | FT ‘their two-term \(20 - x\)’ for M2 M1 for \(\dfrac{48}{20 - x}\) or M1 for \(\dfrac{160}{x} + 2 =\) ‘\(\mathit{their}\ \dfrac{48}{20 - x}\)’ or equivalent if ‘their \(20 - x\)’ is not two-term ‘\(\mathit{their}\ \dfrac{48}{20 - x}\)’ must be in terms of \(x\) |
| Attempts to clear both of the fractions or attempts to use a common denominator on left hand side and cross multiply \(160(20 - x) + 2x(20 - x) = 48x\) or equivalent | M1 | FT ‘their two-term \(20 - x\)’ |
| Expanding the brackets \(3200 - 160x + 40x - 2x^2 = 48x\) or equivalent | M1 | FT ‘their two-term \(20 - x\)’ |
| Collects terms and correctly simplifies to \(x^2 + 84x - 1600 = 0\) | A1 | Convincingly shown as answer given |
| (b)(ii) \((x + 100)(x - 16)\) | M1 | Correct factorisation or correct application of the quadratic formula as far as \(b^2 - 4ac\) simplified; Solution of equation may be seen in part (b)(i), allow the marks if this is the case |
| \(x = 16\) or \(x = -100\) | A1 | |
| [motorbike used] 4 [litres] | A1 | No FT on part (a) here |
| (9) |