| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions with linear factors – decompose and integrate (definite) |
| Difficulty | Moderate -0.3 This is a standard C4 partial fractions question with straightforward integration. Part (a) involves routine algebraic manipulation with distinct linear factors, and part (b) requires integrating the resulting logarithmic terms and simplifying—all textbook techniques with no novel insight required. Slightly easier than average due to the predictable structure and clean numbers. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks |
|---|---|
| \(\frac{x+11}{(x+4)(x-3)} \equiv \frac{A}{x+4} + \frac{B}{x-3}\) | M1 |
| \(x + 11 \equiv A(x-3) + B(x+4)\) | |
| \(x = -4 \Rightarrow 7 = -7A \Rightarrow A = -1\) | A1 |
| \(x = 3 \Rightarrow 14 = 7B \Rightarrow B = 2\) | A1 |
| \(\frac{x+11}{(x+4)(x-3)} = \frac{-1}{x-3} + \frac{2}{x+4}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(= \int_0^2 \left(\frac{2}{x-3} - \frac{1}{x+4}\right) dx\) | M1 | |
| \(= [2\ln | x-3 | - \ln |
| \(= (0 - \ln 6) - (2\ln 3 - \ln 4)\) | M1 | |
| \(= \ln \frac{2}{27}\) | M1 A1 | (8) |
**(a)**
$\frac{x+11}{(x+4)(x-3)} \equiv \frac{A}{x+4} + \frac{B}{x-3}$ | M1 |
$x + 11 \equiv A(x-3) + B(x+4)$ | |
$x = -4 \Rightarrow 7 = -7A \Rightarrow A = -1$ | A1 |
$x = 3 \Rightarrow 14 = 7B \Rightarrow B = 2$ | A1 |
$\frac{x+11}{(x+4)(x-3)} = \frac{-1}{x-3} + \frac{2}{x+4}$ | |
**(b)**
$= \int_0^2 \left(\frac{2}{x-3} - \frac{1}{x+4}\right) dx$ | M1 |
$= [2\ln|x-3| - \ln|x+4|]_0^2$ | M1 A1 |
$= (0 - \ln 6) - (2\ln 3 - \ln 4)$ | M1 |
$= \ln \frac{2}{27}$ | M1 A1 | (8)\begin{enumerate}[label=(\alph*)]
\item Express $\frac{x+11}{(x+4)(x-3)}$ as a sum of partial fractions. [3]
\item Evaluate
$$\int_0^2 \frac{x+11}{(x+4)(x-3)} \, dx,$$
giving your answer in the form $\ln k$, where $k$ is an exact simplified fraction. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q3 [8]}}