C3 January 2010 Q1
1. Express
\[\frac{x + 1}{3x^2 - 3} - \frac{1}{3x + 1}\]as a single fraction in its simplest form. (4)
| Scheme | Marks |
|---|---|
| \(\dfrac{x + 1}{3x^2 - 3} - \dfrac{1}{3x + 1}\) | |
| \(= \dfrac{x + 1}{3(x^2 - 1)} - \dfrac{1}{3x + 1}\) | |
| \(= \dfrac{x + 1}{3(x + 1)(x - 1)} - \dfrac{1}{3x + 1}\) | Award below |
| \(= \dfrac{1}{3(x - 1)} - \dfrac{1}{3x + 1}\) | |
| \(= \dfrac{3x + 1 - 3(x - 1)}{3(x - 1)(3x + 1)}\) | M1 |
| or \(\dfrac{3x + 1}{3(x - 1)(3x + 1)} - \dfrac{3(x - 1)}{3(x - 1)(3x + 1)}\) | A1 |
| Decide to award M1 here!! | M1 |
| \(= \dfrac{4}{3(x - 1)(3x + 1)}\) | A1 aef |
| (4 marks) |
Notes
M1 (awarded below): \(x^2 - 1 \to (x + 1)(x - 1)\) or \(3x^2 - 3 \to (x + 1)(3x - 3)\) or \(3x^2 - 3 \to (3x + 3)(x - 1)\) seen or implied anywhere in candidate’s working.
M1: Attempt to combine.
A1: Correct result.
A1 aef: Either \(\dfrac{4}{3(x - 1)(3x + 1)}\) or \(\dfrac{\frac{4}{3}}{(x - 1)(3x + 1)}\) or \(\dfrac{4}{(3x - 3)(3x + 1)}\) or \(\dfrac{4}{9x^2 - 6x - 3}\)