| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Repeated linear factor with distinct linear factor – decompose and integrate |
| Difficulty | Standard +0.3 This is a standard C4 partial fractions question with a repeated linear factor. Part (a) requires routine algebraic manipulation to find constants A, B, C using the cover-up method or equating coefficients. Part (b) involves integrating the partial fractions form, which is straightforward once decomposed. The techniques are well-practiced at this level, though the repeated factor and definite integral evaluation add minor complexity beyond the most basic examples. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = -2 \Rightarrow 49 = 49A \Rightarrow A = 1\) | B1 | |
| \(x = \frac{1}{3} \Rightarrow \frac{28}{3} = \frac{7}{3}C \Rightarrow C = 4\) | B1 | |
| coeffs \(x^2 \Rightarrow 0 = 9A - 3B \Rightarrow B = 3\) | M1 A1 | |
| (b) \(= \int_{-1}^0 \left(\frac{1}{2+x} + \frac{3}{1-3x} + \frac{4}{(1-3x)^2}\right) dx\) | M1 A3 | |
| \(= [\ln | 2+x | - \ln |
| \(= (\ln 2 - 0 + \frac{4}{3}) - (0 - \ln 4 + \frac{1}{3})\) | M1 | |
| \(= 1 + \ln 8\) | M1 A1 | (11) |
**(a)** $15 - 17x = A(1-3x)^2 + B(2+x)(1-3x) + C(2+x)$
$x = -2 \Rightarrow 49 = 49A \Rightarrow A = 1$ | B1 |
$x = \frac{1}{3} \Rightarrow \frac{28}{3} = \frac{7}{3}C \Rightarrow C = 4$ | B1 |
coeffs $x^2 \Rightarrow 0 = 9A - 3B \Rightarrow B = 3$ | M1 A1 |
**(b)** $= \int_{-1}^0 \left(\frac{1}{2+x} + \frac{3}{1-3x} + \frac{4}{(1-3x)^2}\right) dx$ | M1 A3 |
$= [\ln|2+x| - \ln|1-3x| + \frac{4}{3}(1-3x)^{-1}]_{-1}^0$ | M1 |
$= (\ln 2 - 0 + \frac{4}{3}) - (0 - \ln 4 + \frac{1}{3})$ | M1 |
$= 1 + \ln 8$ | M1 A1 | **(11)**6.
$$f ( x ) = \frac { 15 - 17 x } { ( 2 + x ) ( 1 - 3 x ) ^ { 2 } } , \quad x \neq - 2 , \quad x \neq \frac { 1 } { 3 }$$
\begin{enumerate}[label=(\alph*)]
\item Find the values of the constants $A , B$ and $C$ such that
$$\mathrm { f } ( x ) = \frac { A } { 2 + x } + \frac { B } { 1 - 3 x } + \frac { C } { ( 1 - 3 x ) ^ { 2 } }$$
\item Find the value of
$$\int _ { - 1 } ^ { 0 } f ( x ) d x$$
giving your answer in the form $p + \ln q$, where $p$ and $q$ are integers.\\
6. continued
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q6 [11]}}