C3 June 2006 Q1
1.
(a) Simplify \(\dfrac{3x^2 - x - 2}{x^2 - 1}\). (3)
(b) Hence, or otherwise, express \(\dfrac{3x^2 - x - 2}{x^2 - 1} - \dfrac{1}{x(x + 1)}\) as a single fraction in its simplest form. (3)
| Scheme | Marks |
|---|---|
| \(\dfrac{(3x + 2)(x - 1)}{(x + 1)(x - 1)}, \quad = \dfrac{3x + 2}{x + 1}\) | M1 B1, A1 |
| (3) |
Notes
M1 attempt to factorise numerator, usual rules
B1 factorising denominator seen anywhere in (a),
A1 given answer
If factorisation of denom. not seen, correct answer implies B1
| Scheme | Marks |
|---|---|
| Expressing over common denominator \(\dfrac{3x + 2}{x + 1} - \dfrac{1}{x(x + 1)} = \dfrac{x(3x + 2) - 1}{x(x + 1)}\) [Or “Otherwise”: \(\dfrac{(3x^2 - x - 2)x - (x - 1)}{x(x^2 - 1)}\)] | M1 |
| Multiplying out numerator and attempt to factorise \([\,3x^2 + 2x - 1 \equiv (3x - 1)(x + 1)\,]\) | M1 |
| Answer: \(\dfrac{3x - 1}{x}\) | A1 |
| (3) | |
| (6 marks) |