| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Simplify algebraic fractions by addition or subtraction |
| Difficulty | Moderate -0.5 This question requires factorising a quadratic, finding a common denominator, and simplifying - all standard algebraic techniques for C3. While it involves multiple steps, it's a routine exercise with no conceptual difficulty or problem-solving insight required, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x^2 + x - 12 = (x+4)(x-3)\) | B1 | For correctly factorising \(x^2 + x - 12 = (x+4)(x-3)\). Can appear anywhere in solution |
| Attempt as single fraction \(\frac{(3x+5)(x-3) - 2(x^2+x-12)}{(x^2+x-12)(x-3)}\) or \(\frac{3x+5-2(x+4)}{(x+4)(x-3)}\) | M1 | Attempt to combine two fractions. Denominator must be correct for 'their' fractions. Terms could be separate but one term must have been modified. Condone invisible brackets. |
| \(= \frac{x-3}{(x+4)(x-3)}\) | A1 | Correct unsimplified answer \(\frac{x-3}{(x+4)(x-3)}\). If \(\frac{x^2-6x-9}{(x^2+x-12)(x-3)}\) scored M1, fraction must be subsequently reduced to \(\frac{x-3}{x^2+x-12}\) or \(\frac{(x-3)(x-3)}{(x+4)(x-3)(x-3)}\) to score this mark |
| \(= \frac{1}{(x+4)}\) | A1 | cao \(\frac{1}{(x+4)}\). Do not isw in this question. |
## Question 1(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x^2 + x - 12 = (x+4)(x-3)$ | B1 | For correctly factorising $x^2 + x - 12 = (x+4)(x-3)$. Can appear anywhere in solution |
| Attempt as single fraction $\frac{(3x+5)(x-3) - 2(x^2+x-12)}{(x^2+x-12)(x-3)}$ or $\frac{3x+5-2(x+4)}{(x+4)(x-3)}$ | M1 | Attempt to combine two fractions. Denominator must be correct for 'their' fractions. Terms could be separate but one term must have been modified. Condone invisible brackets. |
| $= \frac{x-3}{(x+4)(x-3)}$ | A1 | Correct unsimplified answer $\frac{x-3}{(x+4)(x-3)}$. If $\frac{x^2-6x-9}{(x^2+x-12)(x-3)}$ scored M1, fraction must be subsequently reduced to $\frac{x-3}{x^2+x-12}$ or $\frac{(x-3)(x-3)}{(x+4)(x-3)(x-3)}$ to score this mark |
| $= \frac{1}{(x+4)}$ | A1 | cao $\frac{1}{(x+4)}$. **Do not isw in this question.** |
**Note:** Method of partial fractions is acceptable for full marks:
$$\frac{3x+5}{(x+4)(x-3)} - \frac{2}{x-3} = \frac{1}{x+4} + \frac{2}{x-3} - \frac{2}{x-3} = \frac{1}{x+4}$$
**(4 marks total)**\begin{enumerate}
\item Express
\end{enumerate}
$$\frac { 3 x + 5 } { x ^ { 2 } + x - 12 } - \frac { 2 } { x - 3 }$$
as a single fraction in its simplest form.\\
\hfill \mbox{\textit{Edexcel C3 2013 Q1 [4]}}