| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Simplify algebraic fractions by addition or subtraction |
| Difficulty | Moderate -0.8 This is a straightforward algebraic manipulation requiring factorization of a difference of squares (9x²-4 = (3x-2)(3x+2)), finding a common denominator, and simplifying. It's more routine than average A-level questions since it only tests basic algebraic skills without requiring problem-solving insight or multiple techniques. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| \(9x^2 - 4 = (3x-2)(3x+2)\) | B1 | For factorising using difference of two squares. Can be awarded at any stage of the answer but must be scored on first mark |
| Eliminating the common factor of \((3x+2)\) at any stage: \(\frac{2(3x+2)}{(3x-2)(3x+2)} = \frac{2}{3x-2}\) | B1 | For eliminating/cancelling out a factor of \((3x+2)\) at any stage |
| Use of a common denominator: \(\frac{2(3x+2)(3x+1)}{(9x^2-4)(3x+1)}\) or equivalent forms | 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. Amongst possible (incorrect) options scoring method marks: only one numerator adapted or separate fractions; invisible brackets, single fraction; \(\frac{2 \times 3x+1-2 \times 3x-2}{(3x-2)(3x+1)}\) |
| \(\frac{6}{(3x-2)(3x+1)}\) or \(\frac{6}{9x^2-3x-2}\) | A1 | This is not a given answer so allow recovery from 'invisible' brackets |
| Total | (4 marks) |
| Answer | Marks |
|---|---|
| \(\frac{2(3x+2)}{(9x^2-4)} - \frac{2}{(3x+1)} = \frac{2(3x+2)(3x+1)-2(9x^2-4)}{(9x^2-4)(3x+1)} = \frac{18x+12}{(9x^2-4)(3x+1)} = \frac{6(3x+2)}{(3x+2)(3x-2)(3x+1)} = \frac{6}{(3x-2)(3x+1)}\) | scores 0,0,1,0 so far → 1,1,1,0 → 1,1,1,1 |
| $9x^2 - 4 = (3x-2)(3x+2)$ | B1 | For factorising using difference of two squares. Can be awarded at any stage of the answer but must be scored on first mark |
|---|---|---|
| Eliminating the common factor of $(3x+2)$ at any stage: $\frac{2(3x+2)}{(3x-2)(3x+2)} = \frac{2}{3x-2}$ | B1 | For eliminating/cancelling out a factor of $(3x+2)$ at any stage |
| Use of a common denominator: $\frac{2(3x+2)(3x+1)}{(9x^2-4)(3x+1)}$ or equivalent forms | 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. Amongst possible (incorrect) options scoring method marks: only one numerator adapted or separate fractions; invisible brackets, single fraction; $\frac{2 \times 3x+1-2 \times 3x-2}{(3x-2)(3x+1)}$ |
| $\frac{6}{(3x-2)(3x+1)}$ or $\frac{6}{9x^2-3x-2}$ | A1 | This is not a given answer so allow recovery from 'invisible' brackets |
| **Total** | **(4 marks)** | |
**Alternative method:**
$\frac{2(3x+2)}{(9x^2-4)} - \frac{2}{(3x+1)} = \frac{2(3x+2)(3x+1)-2(9x^2-4)}{(9x^2-4)(3x+1)} = \frac{18x+12}{(9x^2-4)(3x+1)} = \frac{6(3x+2)}{(3x+2)(3x-2)(3x+1)} = \frac{6}{(3x-2)(3x+1)}$ | scores 0,0,1,0 so far → 1,1,1,0 → 1,1,1,1 |
---\begin{enumerate}
\item Express
\end{enumerate}
$$\frac { 2 ( 3 x + 2 ) } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$
as a single fraction in its simplest form.\\
\hfill \mbox{\textit{Edexcel C3 2012 Q1 [4]}}