C3 January 2006 Q2
2. Express\[\frac{2x^2 + 3x}{(2x + 3)(x - 2)} - \frac{6}{x^2 - x - 2}\]as a single fraction in its simplest form. (7)
| Scheme | Marks |
|---|---|
| \(x^2 - x - 2 = (x - 2)(x + 1)\) At any stage | B1 |
| \(\dfrac{2x^2 + 3x}{(2x + 3)(x - 2)} = \dfrac{x(2x + 3)}{(2x + 3)(x - 2)} = \dfrac{x}{x - 2}\) | B1 |
| \(\dfrac{2x^2 + 3x}{(2x + 3)(x - 2)} - \dfrac{6}{x^2 - x - 2} = \dfrac{x(x + 1) - 6}{(x - 2)(x + 1)}\) | M1 |
| \(= \dfrac{x^2 + x - 6}{(x - 2)(x + 1)}\) | A1 |
| \(= \dfrac{(x + 3)(x - 2)}{(x - 2)(x + 1)}\) | M1 A1 |
| \(= \dfrac{x + 3}{x + 1}\) | A1 |
| (7) | |
| (7 marks) |
Alternative method
| \(x^2 - x - 2 = (x - 2)(x + 1)\) At any stage | B1 |
| \((2x + 3)\) appearing as a factor of the numerator at any stage | B1 |
| \(\dfrac{2x^2 + 3x}{(2x + 3)(x - 2)} - \dfrac{6}{(x - 2)(x + 1)} = \dfrac{(2x^2 + 3x)(x + 1) - 6(2x + 3)}{(2x + 3)(x - 2)(x + 1)}\) | M1 |
| \(= \dfrac{2x^3 + 5x^2 - 9x - 18}{(2x + 3)(x - 2)(x + 1)}\) can be implied | A1 |
| \(= \dfrac{(x - 2)(2x^2 + 9x + 9)}{(2x + 3)(x - 2)(x + 1)}\) or \(\dfrac{(2x + 3)(x^2 + x - 6)}{(2x + 3)(x - 2)(x + 1)}\) or \(\dfrac{(x + 3)(2x^2 - x - 6)}{(2x + 3)(x - 2)(x + 1)}\) Any one linear factor \(\times\) quadratic | M1 |
| \(= \dfrac{(2x + 3)(x - 2)(x + 3)}{(2x + 3)(x - 2)(x + 1)}\) Complete factors | A1 |
| \(= \dfrac{x + 3}{x + 1}\) | A1 (7) |