C3 June 2013 (R) Q1
1. Express\[\frac{3x+5}{x^2+x-12}-\frac{2}{x-3}\]as a single fraction in its simplest form. (4)
| Scheme | Marks |
|---|---|
| \(x^2+x-12=(x+4)(x-3)\) | B1 |
| Attempt as a single fraction \(\dfrac{(3x+5)(x-3)-2(x^2+x-12)}{(x^2+x-12)(x-3)}\) or \(\dfrac{3x+5-2(x+4)}{(x+4)(x-3)}\) | M1 |
| \(=\dfrac{x-3}{(x+4)(x-3)}\ ,=\dfrac{1}{(x+4)}\) cao | A1, A1 |
| (4 marks) |
Notes
B1 For correctly factorising \(x^2+x-12=(x+4)(x-3)\). It could appear anywhere in their solution
M1 For an attempt to combine two fractions. The denominator must be correct for ‘their’ fractions.
The terms could be separate but one term must have been modified.
Condone invisible brackets.
Examples of work scoring this mark are;
\(\dfrac{(3x+5)(x-3)}{(x^2+x-12)(x-3)}-\dfrac{2(x^2+x-12)}{(x^2+x-12)(x-3)}\) Two separate terms
\(\dfrac{3x+5-2x+4}{(x+4)(x-3)}\) Single term, invisible bracket
\(\dfrac{(3x+5)}{(x^2+x-12)(x-3)}-\dfrac{2(x^2+x-12)}{(x^2+x-12)(x-3)}\) Separate terms, only one numerator modified
A1 Correct un simplified answer \(\dfrac{x-3}{(x+4)(x-3)}\)
If \(\dfrac{x^2-6x-9}{(x^2+x-12)(x-3)}\) scored M1 the fraction must be subsequently be reduced to a correct \(\dfrac{x-3}{x^2+x-12}\) or \(\dfrac{(x-3)(x-3)}{(x+4)(x-3)(x-3)}\) to score this mark.
A1 cao \(\dfrac{1}{(x+4)}\)
Do Not isw in this question.
The method of partial fractions is perfectly acceptable and can score full marks
\(\underbrace{\dfrac{3x+5}{(x+4)(x-3)}}_{B1}-\dfrac{2}{x-3}=\underbrace{\dfrac{1}{x+4}+\dfrac{2}{x-3}}_{M1A1}-\dfrac{2}{x-3}=\underbrace{\dfrac{1}{x+4}}_{A1}\)