C3 June 2012 Q1
1. Express
\[\frac{2(3x + 2)}{9x^2 - 4} - \frac{2}{3x + 1}\]
as a single fraction in its simplest form. (4)
| Scheme | Marks |
|---|---|
| \(9x^2 - 4 = (3x - 2)(3x + 2)\) At any stage | B1 |
| Eliminating the common factor of \((3x + 2)\) at any stage \(\dfrac{2\cancel{(3x + 2)}}{(3x - 2)\cancel{(3x + 2)}} = \dfrac{2}{3x - 2}\) | B1 |
| Use of a common denominator \(\dfrac{2(3x + 2)(3x + 1)}{(9x^2 - 4)(3x + 1)} - \dfrac{2(9x^2 - 4)}{(9x^2 - 4)(3x + 1)}\) or \(\dfrac{2(3x + 1)}{(3x - 2)(3x + 1)} - \dfrac{2(3x - 2)}{(3x + 1)(3x - 2)}\) | M1 |
| \(\dfrac{6}{(3x - 2)(3x + 1)}\) or \(\dfrac{6}{9x^2 - 3x - 2}\) | A1 |
| (4 marks) |
Notes
B1 For factorising \(9x^2 - 4 = (3x - 2)(3x + 2)\) using difference of two squares. It can be awarded at any stage of the answer but it must be scored on E pen as the first mark
B1 For eliminating/cancelling out a factor of \((3x + 2)\) at any stage of the answer.
M1 For combining two fractions to form a single fraction with a common denominator. Allow slips on the numerator but at least one must have been adapted. Condone invisible brackets. Accept two separate fractions with the same denominator as shown in the mark scheme. Amongst possible (incorrect) options scoring method marks are
\(\dfrac{2(3x + 2)}{(9x^2 - 4)(3x + 1)} - \dfrac{2(9x^2 - 4)}{(9x^2 - 4)(3x + 1)}\) Only one numerator adapted, separate fractions
\(\dfrac{2 \times 3x + 1 - 2 \times 3x - 2}{(3x - 2)(3x + 1)}\) Invisible brackets, single fraction
A1 \(\dfrac{6}{(3x - 2)(3x + 1)}\)
This is not a given answer so you can allow recovery from ‘invisible’ brackets.
Alternative method
\(\dfrac{2(3x + 2)}{(9x^2 - 4)} - \dfrac{2}{(3x + 1)} = \dfrac{2(3x + 2)(3x + 1) - 2(9x^2 - 4)}{(9x^2 - 4)(3x + 1)} = \dfrac{18x + 12}{(9x^2 - 4)(3x + 1)}\) has scored 0,0,1,0 so far
\(= \dfrac{6\cancel{(3x + 2)}}{\cancel{(3x + 2)}(3x - 2)(3x + 1)}\) is now 1,1,1,0
\(= \dfrac{6}{(3x - 2)(3x + 1)}\) and now 1,1,1,1