| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2010 |
| Session | January |
| 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 fraction problem requiring finding a common denominator and simplifying. It's easier than average because it's purely mechanical manipulation with no conceptual depth—factorising the quadratic, finding the LCD, combining numerators, and simplifying. No problem-solving insight required, just routine algebraic technique. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(\frac{x+1}{3x^2-3} - \frac{1}{3x+1} = \frac{x+1}{3(x^2-1)} - \frac{1}{3x+1}\) | ||
| \(= \frac{x+1}{3(x+1)(x-1)} - \frac{1}{3x+1}\) | \(x^2-1 \to (x+1)(x-1)\) or \(3x^2-3 \to (x+1)(3x-3)\) or \(3x^2-3 \to (3x+3)(x-1)\) seen or implied anywhere | |
| \(= \frac{1}{3(x-1)} - \frac{1}{3x+1}\) | ||
| \(= \frac{3x+1-3(x-1)}{3(x-1)(3x+1)}\) | M1 | Attempt to combine |
| or \(\frac{3x+1}{3(x-1)(3x+1)} - \frac{3(x-1)}{3(x-1)(3x+1)}\) | A1 | Correct result |
| *Decide to award M1 here!!* | M1 | |
| \(= \frac{4}{3(x-1)(3x+1)}\) | A1 aef | Either \(\frac{4}{3(x-1)(3x+1)}\) or \(\frac{\frac{4}{3}}{(x-1)(3x+1)}\) or \(\frac{4}{(3x-3)(3x+1)}\) or \(\frac{4}{9x^2-6x-3}\) |
## Question 1:
| Working | Mark | Guidance |
|---------|------|----------|
| $\frac{x+1}{3x^2-3} - \frac{1}{3x+1} = \frac{x+1}{3(x^2-1)} - \frac{1}{3x+1}$ | | |
| $= \frac{x+1}{3(x+1)(x-1)} - \frac{1}{3x+1}$ | | $x^2-1 \to (x+1)(x-1)$ or $3x^2-3 \to (x+1)(3x-3)$ or $3x^2-3 \to (3x+3)(x-1)$ seen or implied anywhere | Award below |
| $= \frac{1}{3(x-1)} - \frac{1}{3x+1}$ | | |
| $= \frac{3x+1-3(x-1)}{3(x-1)(3x+1)}$ | M1 | Attempt to combine |
| or $\frac{3x+1}{3(x-1)(3x+1)} - \frac{3(x-1)}{3(x-1)(3x+1)}$ | A1 | Correct result |
| *Decide to award M1 here!!* | M1 | |
| $= \frac{4}{3(x-1)(3x+1)}$ | A1 **aef** | Either $\frac{4}{3(x-1)(3x+1)}$ or $\frac{\frac{4}{3}}{(x-1)(3x+1)}$ or $\frac{4}{(3x-3)(3x+1)}$ or $\frac{4}{9x^2-6x-3}$ |
**Total: [4]**
---\begin{enumerate}
\item Express
\end{enumerate}
$$\frac { x + 1 } { 3 x ^ { 2 } - 3 } - \frac { 1 } { 3 x + 1 }$$
as a single fraction in its simplest form.\\
\hfill \mbox{\textit{Edexcel C3 2010 Q1 [4]}}