4MA0 Higher June 2014 Paper 4 Q25
25 Simplify fully \(\dfrac{5}{2x - 6} - \dfrac{x + 2}{x^2 - 4x + 3}\)
(5)
| Scheme | Marks |
|---|---|
\(\dfrac{5}{2(x - 3)} - \dfrac{x + 2}{(x - 3)(x - 1)}\) or \(\dfrac{5}{2x - 6} - \dfrac{x + 2}{(x - 3)(x - 1)}\) | M1 |
| \(\dfrac{5(x - 1)}{2(x - 3)(x - 1)} - \dfrac{2(x + 2)}{2(x - 3)(x - 1)}\) | M1 |
| \(\dfrac{5x - 5 - 2x - 4}{2(x - 3)(x - 1)}\) | M1 |
| \(\dfrac{3(x - 3)}{2(x - 3)(x - 1)}\) | M1 |
| \(\dfrac{3}{2(x - 1)}\) | A1 |
| (5 marks) |
Notes
M1: \(x^2 - 4x + 3\) factorised correctly
M1: a correct common denominator – may be a single fraction or two fractions with correct numerators; denominator may be expanded correctly
M1: correct single fraction with numerator expanded correctly; denominator may be expanded correctly
M1: correct factorisation of numerator ; denominator may be expanded correctly
A1: Accept \(\dfrac{3}{2x - 2}\)
(corrected from the printed mark scheme: the total for this question is printed as “Total 3 marks”; the question is worth 5 marks)
Alternative
| Scheme | Marks |
|---|---|
| \(\dfrac{5(x^2 - 4x + 3)}{(2x - 6)(x^2 - 4x + 3)} - \dfrac{(2x - 6)(x + 2)}{(2x - 6)(x^2 - 4x + 3)}\) | M1 |
| \(\dfrac{5x^2 - 20x + 15 - 2x^2 - 4x + 6x + 12}{(2x - 6)(x^2 - 4x + 3)}\) | M1 |
| \(\dfrac{3x^2 - 18x + 27}{(2x - 6)(x - 3)(x - 1)}\) | M1 |
| \(\dfrac{3(x - 3)^2}{2(x - 3)(x - 3)(x - 1)}\) | M1 |
| \(\dfrac{3}{2(x - 1)}\) | A1 |
Notes
M1: a correct common denominator – may be a single fraction or two fractions with correct numerators; denominator may be expanded correctly
M1: correct single fraction with numerator expanded correctly; denominator may be expanded correctly;
M1: \(x^2 - 4x + 3\) factorised correctly – could occur earlier
M1: correct fully factorised numerator and denominator
A1: Accept \(\dfrac{3}{2x - 2}\)