Higher January 2021 Paper 1 Q12
12
(a) Express \(\dfrac{4}{x - 2} - \dfrac{3}{x + 1}\) as a single fraction.
Give your answer in its simplest form. (3)
Give your answer in its simplest form. (3)
(b) Expand and simplify \(2x(x - 5)(x - 3)\) (3)
| Scheme | Marks |
|---|---|
\(\dfrac{4(x + 1) - 3(x - 2)}{(x - 2)(x + 1)}\) or \(\dfrac{4(x + 1)}{(x - 2)(x + 1)} - \dfrac{3(x - 2)}{(x - 2)(x + 1)}\) | M1 |
| \(\dfrac{4x + 4 - 3x + 6}{(x - 2)(x + 1)}\) or \(\dfrac{4x + 4 - 3x + 6}{x^2 - x - 2}\) | M1 |
Correct answer scores full marks (unless from obvious incorrect working) Answer: \(\dfrac{x + 10}{(x - 2)(x + 1)}\) | A1 |
| (3) |
Notes
M1: for expressing both fractions correctly with a common denominator.
M1: for removing brackets in a single fraction with a correct denominator. Allow denominator to be expanded. Allow one error in the expansion of the numerator.
A1: accept \(\dfrac{x + 10}{x^2 - x - 2}\) oe
| Scheme | Marks |
|---|---|
| \(2x(x - 5) = 2x^2 - 10x\) or \(2x(x - 3) = 2x^2 - 6x\) or \((x - 5)(x - 3) = x^2 - 5x - 3x + 15\;(= x^2 - 8x + 15)\) | M1 |
| \((2x^2 - 10x)(x - 3) = 2x^3 - 6x^2 - 10x^2 + 30x\) or \((2x^2 - 6x)(x - 5) = 2x^3 - 10x^2 - 6x^2 + 30x\) or \(2x(x^2 - 5x - 3x + 15) = 2x^3 - 10x^2 - 6x^2 + 30x\) or \(2x(x^2 - 8x + 15) = 2x^3 - 16x^2 + 30x\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \(2x^3 - 16x^2 + 30x\) | A1 |
| (3) | |
| (6 marks) |
Notes
M1: for multiplying \(2x\) by a bracket with both terms correct or the 2 brackets with at least 3 out of 4 terms correct or at least 2 out of 3 terms correct
M1: (dep) for multiplying the product of \(2x\) and the 1st bracket (ft from the 1st stage) by the 2nd bracket and getting at least 3 out of 4 terms correct
or multiplying the product of the 2 brackets (ft from the 1st stage) by the \(2x\), and getting at least 3 out of 4 or 2 out of 3 terms correct