| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions with linear factors – decompose and integrate (indefinite) |
| Difficulty | Moderate -0.3 This is a standard C4 partial fractions question with routine techniques. Part (a) involves straightforward partial fraction decomposition and integration of logarithms, while part (b) requires polynomial long division—both are textbook exercises with no novel problem-solving required. Slightly easier than average due to the mechanical nature of the procedures. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| \(7x - 3 = A(3x - 2) + B(x + 1)\) | M1 | |
| Substitute two values of \(x\) and solve for \(A\) and \(B\): \(x = -1\), \(x = \frac{2}{3}\) | m1 | |
| \(A = 2\), \(B = 1\) | A1 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \frac{7x-3}{(x+1)(3x-2)} dx = p\ln(x+1) + q\ln(3x-2)\) | M1 | Condone missing brackets |
| \(= 2\ln(x+1) + \frac{1}{3}\ln(3x-2) (+c)\) | A1F | 2 marks |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{6x^2 + x + 2}{2x^2 - x + 1} = \frac{6x^2 - 3x + 3 + 4x - 1}{2x^2 - x + 1}\) | M1 | |
| \(= 3 + \frac{4x - 1}{2x^2 - x + 1}\) | B1 A1 | 3 marks |
**3(a)(i)**
$7x - 3 = A(3x - 2) + B(x + 1)$ | M1
Substitute two values of $x$ and solve for $A$ and $B$: $x = -1$, $x = \frac{2}{3}$ | m1
$A = 2$, $B = 1$ | A1 | 3 marks | Or solve $\begin{cases} 7 = 3A + B \\ -3 = -2A + B \end{cases}$ condone one error
**3(a)(ii)**
$\int \frac{7x-3}{(x+1)(3x-2)} dx = p\ln(x+1) + q\ln(3x-2)$ | M1 | Condone missing brackets
$= 2\ln(x+1) + \frac{1}{3}\ln(3x-2) (+c)$ | A1F | 2 marks | F on A and B; constant not required
**3(b)**
$\frac{6x^2 + x + 2}{2x^2 - x + 1} = \frac{6x^2 - 3x + 3 + 4x - 1}{2x^2 - x + 1}$ | M1
$= 3 + \frac{4x - 1}{2x^2 - x + 1}$ | B1 A1 | 3 marks | $P = 3$; $Q = 4$ and $R = -1$
---\begin{enumerate}[label=(\alph*)]
\item
\begin{enumerate}[label=(\roman*)]
\item Express $\frac{7x - 3}{(x + 1)(3x - 2)}$ in the form $\frac{A}{x + 1} + \frac{B}{3x - 2}$. [3 marks]
\item Hence find $\int \frac{7x - 3}{(x + 1)(3x - 2)} dx$. [2 marks]
\end{enumerate}
\item Express $\frac{6x^2 + x + 2}{2x^2 - x + 1}$ in the form $P + \frac{Qx + R}{2x^2 - x + 1}$. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2010 Q3 [8]}}