| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | Specimen |
| 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 factorisation of a difference of squares (9x²-4 = (3x-2)(3x+1)), 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 conceptual steps. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | \(9x^2 – 4 = (3x − 2)(3x + 2)\) at any stage | B1 |
| Answer | Marks |
|---|---|
| \[\frac{2}{(3x-2)} = \frac{2(3x+2)}{(3x-2)(3x+2)} = \frac{2}{3x-2}\] | M1 |
| Answer | Marks |
|---|---|
| \[\frac{2(3x+2)(3x+1)}{(9x^2-4)(3x+1)} - \frac{2(9x^2-4)}{(9x^2-4)(3x+1)}\] or \[\frac{2(3x+1)}{(3x-2)(3x+1)} - \frac{2(3x-2)}{(3x+1)(3x-2)}\] | M1 |
| \[\frac{6}{(3x-2)(3x+1)}\] or \[\frac{6}{9x^2 - 3x - 2}\] | A1 |
# Question 1
**1** | $9x^2 – 4 = (3x − 2)(3x + 2)$ at any stage | B1
Eliminating the common factor of $(3x + 2)$ at any stage
$$\frac{2}{(3x-2)} = \frac{2(3x+2)}{(3x-2)(3x+2)} = \frac{2}{3x-2}$$ | M1
Use of a common denominator
$$\frac{2(3x+2)(3x+1)}{(9x^2-4)(3x+1)} - \frac{2(9x^2-4)}{(9x^2-4)(3x+1)}$$ or $$\frac{2(3x+1)}{(3x-2)(3x+1)} - \frac{2(3x-2)}{(3x+1)(3x-2)}$$ | M1
$$\frac{6}{(3x-2)(3x+1)}$$ or $$\frac{6}{9x^2 - 3x - 2}$$ | 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: $\frac{2(3x+2)}{(9x^2-4)(3x+1)} - \frac{2(9x^2-4)}{(9x^2-4)(3x+1)}$ (only one numerator adapted, separate fractions) or $\frac{2 \cdot 3x+1 - 2 \cdot 3x-2}{(3x-2)(3x+1)}$ (invisible brackets, single fraction).
**A1:** $$\frac{6}{(3x-2)(3x+1)}$$ This is not a given answer so you can allow recovery from 'invisible' brackets.
**Alternative:** $\frac{2(3x+2)}{(9x^2-4)} - \frac{2}{(3x+1)} = \frac{2(3x+2)(3x+1)-2(9x^2-4)}{(9x^2-4)(3x+1)} = \frac{18x+12}{(9x^2-4)(3x+1)}$ has scored 0,0,1,0 so far. Then $\frac{6(3x+2)}{(3x+2)(3x-2)(3x+1)}$ is now 1,1,1,0. Then $\frac{6}{(3x-2)(3x+1)}$ scores 1,1,1,1.\begin{enumerate}
\item Express
\end{enumerate}
$$\frac { 6 x + 4 } { 9 x ^ { 2 } - 4 } - \frac { 2 } { 3 x + 1 }$$
as a single fraction in its simplest form.\\
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\hfill \mbox{\textit{Edexcel P3 2018 Q1 [4]}}