C3 June 2014 (R) Q1
1. Express\[\frac{3}{2x+3}-\frac{1}{2x-3}+\frac{6}{4x^2-9}\]as a single fraction in its simplest form. (4)
| Scheme | Marks |
|---|---|
| Factorise \(4x^2-9=(2x-3)(2x+3)\) | B1 |
| Use of common denominator \(\dfrac{3}{2x+3}-\dfrac{1}{2x-3}+\dfrac{6}{4x^2-9}=\dfrac{3(2x-3)-1(2x+3)+6}{(2x+3)(2x-3)}\) | M1 |
| \(=\dfrac{4x-6}{(2x+3)(2x-3)}\) | A1 |
| \(=\dfrac{2\cancel{(2x-3)}}{(2x+3)\cancel{(2x-3)}}=\dfrac{2}{2x+3}\) | A1 |
| (4 marks) |
Alternative where \(4x^2-9\) is not factorised
| Scheme | Marks |
|---|---|
| \(\dfrac{3}{2x+3}-\dfrac{1}{2x-3}+\dfrac{6}{4x^2-9}=\dfrac{3(2x-3)(4x^2-9)-1(2x+3)(4x^2-9)+6(2x+3)(2x-3)}{(2x+3)(2x-3)(4x^2-9)}\) | M1 |
| \(=\dfrac{2(2x-3)(4x^2-9)}{(2x+3)(2x-3)(4x^2-9)}\) or \(\dfrac{(4x-6)(4x^2-9)}{(2x+3)(2x-3)(4x^2-9)}\) or \(\dfrac{(2x-3)(8x^2-18)}{(2x+3)(2x-3)(4x^2-9)}\) | B1 |
| \(=\dfrac{(4x-6)\cancel{(4x^2-9)}}{(2x+3)(2x-3)\cancel{(4x^2-9)}}\) or \(\dfrac{\overset{2}{\cancel{(4x-6)}}(4x^2-9)}{(2x+3)\cancel{(2x-3)}(4x^2-9)}\) | A1 |
| \(=\dfrac{2}{2x+3}\) | A1 |
Notes
B1 For factorising \(4x^2-9\) to \((2x-3)(2x+3)\) at any point. Note that this is not scored for combining the terms \((2x-3)(2x+3)\) and writing the product as \(4x^2-9\)
M1 Use of common denominator – combines three fractions to form one. The denominator must be correct for their fractions and at least one numerator must have been adapted. Condone missing brackets.
\(\dfrac{16x^3-24x^2-36x+54}{(4x^2-9)^2}\) is a correct intermediate stage but needs to be factorised and cancelled before A1
Examples of incorrect fractions scoring this mark are: \(\dfrac{3(2x-3)-2x+3+6}{(2x+3)(2x-3)}\) missing bracket
\(\dfrac{3(4x^2-9)-4x^2-9+6(2x+3)(2x-3)}{(2x+3)(2x-3)(4x^2-9)}\) denominator correct and at least one numerator has been adapted.
A1 Correct simplified intermediate answer. It must be a CORRECT \(\dfrac{\text{Linear}}{\text{Quadratic}}\) or \(\dfrac{\text{Quadratic}}{\text{Cubic}}\)
Accept versions of \(\dfrac{4x-6}{(2x+3)(2x-3)}\) or \(\dfrac{8x^2-18}{(2x+3)(4x^2-9)}\)
A1 cao \(=\dfrac{2}{2x+3}\)
Allow recovery from invisible brackets for all 4 marks as the answer is not given.