| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2010 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions with algebraic division first |
| Difficulty | Standard +0.3 This is a standard C4 partial fractions question requiring algebraic division to extract the polynomial part, then decomposition into partial fractions, followed by routine integration of logarithmic terms. While it requires multiple steps, each technique is straightforward and commonly practiced, making it slightly easier than the average A-level question which typically involves more problem-solving or conceptual challenge. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| \(10x^2+8=2(x+1)(5x-1)+A(5x-1)+B(x+1)\) | M1 | \(A\) and \(B\) terms correct |
| A1 | ||
| \(x=-1\quad x=\frac{1}{5}\) | m1 | Use two values of \(x\) to find \(A\) and \(B\), or set up and solve: \(8+5A+B=0\); \(-2-A+B=8\) |
| \(A=-3\quad B=7\) | A1 (4 marks) | SC1 NWS \(A\) & \(B\) correct \(\frac{4}{4}\); SC2 NWS \(A\) or \(B\) correct \(\frac{1}{4}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int\frac{10x^2+8}{(x+1)(5x-1)}\,dx=\int2-\frac{3}{x+1}+\frac{7}{5x-1}\,dx\) | M1 | Use the partial fractions |
| \(=2x+C\) | B1 | |
| \(a\ln(x+1)+b\ln(5x-1)\), condone missing brackets | M1 | |
| \(-3\ln(x+1)+\frac{7}{5}\ln(5x-1)\) | A1F (4 marks) | F on \(A\) and \(B\) |
## Question 4:
### Part (a)
| $10x^2+8=2(x+1)(5x-1)+A(5x-1)+B(x+1)$ | M1 | $A$ and $B$ terms correct |
| | A1 | |
| $x=-1\quad x=\frac{1}{5}$ | m1 | Use two values of $x$ to find $A$ and $B$, or set up and solve: $8+5A+B=0$; $-2-A+B=8$ |
| $A=-3\quad B=7$ | A1 (4 marks) | SC1 NWS $A$ & $B$ correct $\frac{4}{4}$; SC2 NWS $A$ or $B$ correct $\frac{1}{4}$ |
### Part (b)
| $\int\frac{10x^2+8}{(x+1)(5x-1)}\,dx=\int2-\frac{3}{x+1}+\frac{7}{5x-1}\,dx$ | M1 | Use the partial fractions |
| $=2x+C$ | B1 | |
| $a\ln(x+1)+b\ln(5x-1)$, condone missing brackets | M1 | |
| $-3\ln(x+1)+\frac{7}{5}\ln(5x-1)$ | A1F (4 marks) | F on $A$ and $B$ |
**Total: 8 marks**
---4 The expression $\frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) }$ can be written in the form $2 + \frac { A } { x + 1 } + \frac { B } { 5 x - 1 }$, where $A$ and $B$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $A$ and $B$.
\item Hence find $\int \frac { 10 x ^ { 2 } + 8 } { ( x + 1 ) ( 5 x - 1 ) } \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2010 Q4 [8]}}