| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Repeated linear factor with distinct linear factor – decompose and integrate |
| Difficulty | Moderate -0.3 This is a standard partial fractions decomposition with a repeated linear factor. The technique is routine for C4: set up A/(2x+1) + B/(x+1) + C/(x+1)², multiply through, and solve by substitution or comparing coefficients. It requires methodical application of a well-practiced algorithm with no conceptual challenges or novel problem-solving. |
| Spec | 1.02y Partial fractions: decompose rational functions |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| At least one of \(A\) or \(C\) correct | B1 | At least one of "\(A\)" or "\(C\)" are correct |
| \(A = 2,\ C = 2\) | B1 cso | Breaks up partial fraction correctly into three terms and both \(A=2\) and \(C=2\) |
| \(5x+3 \equiv A(x+1)^2 + B(2x+1)(x+1) + C(2x+1)\); attempts to find value of \(A\), \(B\), or \(C\) | M1 | Writes down a correct identity and attempts to find the value of either one of "\(A\)" or "\(B\)" or "\(C\)" (by substitution or comparing coefficients) |
| \(B = -1\) | A1 cso | Correct value for "\(B\)" found using a correct identity, following from their partial fraction decomposition |
## Question 1:
$$\frac{5x+3}{(2x+1)(x+1)^2} \equiv \frac{A}{(2x+1)} + \frac{B}{(x+1)} + \frac{C}{(x+1)^2}$$
| Working/Answer | Mark | Guidance |
|---|---|---|
| At least one of $A$ or $C$ correct | B1 | At least one of "$A$" or "$C$" are correct |
| $A = 2,\ C = 2$ | B1 **cso** | Breaks up partial fraction correctly into three terms **and** both $A=2$ and $C=2$ |
| $5x+3 \equiv A(x+1)^2 + B(2x+1)(x+1) + C(2x+1)$; attempts to find value of $A$, $B$, or $C$ | M1 | Writes down a **correct identity** and attempts to find the value of either one of "$A$" or "$B$" or "$C$" (by substitution or comparing coefficients) |
| $B = -1$ | A1 **cso** | Correct value for "$B$" found using a correct identity, following from their partial fraction decomposition |
**Final answer:** $\dfrac{5x+3}{(2x+1)(x+1)^2} \equiv \dfrac{2}{(2x+1)} - \dfrac{1}{(x+1)} + \dfrac{2}{(x+1)^2}$
**Total: [4]**
**Notes:**
- Candidates assign their own $A$, $B$, $C$
- M1 can be implied by correct working
- If no partial fraction decomposition shown: 2nd B1 can follow from correct identity; A1 can be awarded for correct $B$ if partial fractions written at end
- Correct partial fraction from no working scores B1B1M1A1
- The correct partial fraction from no working scores **B1B1M1A1**\begin{enumerate}
\item Express in partial fractions
\end{enumerate}
$$\frac { 5 x + 3 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }$$
\hfill \mbox{\textit{Edexcel C4 2013 Q1 [4]}}