4MA0 Higher June 2017 Paper 3 Q22
22 Simplify fully \(\dfrac{3}{2x + 12} - \dfrac{x - 15}{x^2 - 2x - 48}\)
Show clear algebraic working.
(5)
| Scheme | Marks |
|---|---|
E.g. \(\dfrac{3}{2(x + 6)} - \dfrac{x - 15}{(x - 8)(x + 6)}\) or \(\dfrac{3}{2x + 12} - \dfrac{x - 15}{(x - 8)(x + 6)}\) | M1 |
E.g. \(\dfrac{3(x - 8) - 2(x - 15)}{2(x - 8)(x + 6)}\) or \(\dfrac{3(x - 8)}{2(x - 8)(x + 6)} - \dfrac{2(x - 15)}{2(x - 8)(x + 6)}\) | M1 |
| \(\dfrac{3x - 24 - 2x + 30}{2(x - 8)(x + 6)}\) | M1 |
| \(\dfrac{x + 6}{2(x - 8)(x + 6)}\) | M1 |
| \(\dfrac{1}{2(x - 8)}\) | A1 |
| (5) | |
| (5 marks) |
Notes
M1: \(x^2 - 2x - 48\) correctly factorised
NB : May be seen at a later stage
M1: for a correct common denominator with numerators correct
This may be a single fraction or two fractions; denominators may be expanded – if so, must be correct.
M1: for a correct single fraction with brackets in numerator removed correctly; denominators may be expanded – if so, must be correct.
M1: for a correct single fraction with the numerator simplified; denominators may expanded – if so, must be correct.
Alternative scheme
| Scheme | Marks |
|---|---|
E.g. \(\dfrac{3}{2(x + 6)} - \dfrac{x - 15}{(x - 8)(x + 6)}\) or \(\dfrac{3}{2x + 12} - \dfrac{x - 15}{(x - 8)(x + 6)}\) | M1 |
| \(\dfrac{3(x^2 - 2x - 48) - (2x + 12)(x - 15)}{(2x + 12)(x^2 - 2x - 48)}\) | M1 |
E.g. \(\dfrac{3x^2 - 6x - 144 - 2x^2 + 30x - 12x + 180}{(2x + 12)(x^2 - 2x - 48)}\) or \(\dfrac{x^2 + 12x + 36}{(2x + 12)(x^2 - 2x - 48)}\) | M1 |
| E.g. \(\dfrac{(x + 6)^2}{(2x + 12)(x^2 - 2x - 48)}\) or \(\dfrac{x + 6}{2(x - 8)(x + 6)}\) | M1 |
| \(\dfrac{1}{2(x - 8)}\) | A1 |
Notes
M1: \(x^2 - 2x - 48\) correctly factorised
NB : May be seen at a later stage
M1: for a correct common denominator with numerators correct
This may be a single fraction or two fractions; denominators may be expanded – if so, must be correct.
M1: for a correct single fraction with brackets in numerator removed correctly; denominators may be expanded – if so, must be correct (\(2x^3 + 8x^2 - 120x - 576\))
M1: for a correct single fraction with the numerator factorised; denominators may expanded – if so, must be correct.