| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | January |
| Marks | 13 |
| 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 routine techniques. Part (a) involves simple polynomial division (2 marks), and part (b) is a textbook repeated factor decomposition followed by straightforward integration. While part (b) requires careful algebra with the repeated factor, it follows a completely standard method with no novel insight required, making it slightly easier than average overall. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02y Partial fractions: decompose rational functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08j Integration using partial fractions |
| Answer | Marks |
|---|---|
| \(\frac{2x+7}{x+2} = 2 + \frac{3}{x+2}\) | B1, B1 (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int\frac{2x+7}{x+2} = 3\ln(x+2) + 2x + C\) | B1F, B1F (2 marks) | Either term correct; both correct, constant required; condone missing bracket; ft on \(A\), \(B\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(28 + 4x^2 = P(5-x)^2 + Q(1+3x)(5-x) + R(1+3x)\) | M1 | |
| \(x=5 \Rightarrow R=8\); \(x=-\frac{1}{3} \Rightarrow P=1\) | m1, A1 | Two values of \(x\) used to find \(R\) and \(P\); SC \(R=8\), \(P=1\) NMS can score B1,B1 |
| \(x=0 \Rightarrow 28 = 25P + 5Q + R\), \(Q = -1\) | m1, A1 (5 marks) | Third value of \(x\) used to find \(Q\) |
| Answer | Marks |
|---|---|
| Collect terms and form equations; correct equations; solve for \(P\), \(Q\), \(R\): \(P=1\), \(Q=-1\), \(R=8\) | (M1), (m1), (A1), (m1), (A1) (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int\frac{1}{1+3x} - \frac{1}{5-x} + \frac{8}{(5-x)^2}\,dx\) | M1 | Use partial fractions |
| \(= \frac{1}{3}\ln(1+3x) + \ln(5-x) + \frac{8}{5-x} + (C)\) | m1, A1F, A1F (4 marks) | \(a\ln(1+3x) + b\ln(5-x)\); OE, both ln integrals correct, needs \(()\); other term correct; ft on their \(P\), \(Q\), \(R\); SC: if no \(P,Q,R\) found in (b)(i), can gain method marks by inserting other values or retaining letters (max 2/4) |
## Question 3:
### Part (a)(i)
| $\frac{2x+7}{x+2} = 2 + \frac{3}{x+2}$ | B1, B1 (2 marks) | |
### Part (a)(ii)
| $\int\frac{2x+7}{x+2} = 3\ln(x+2) + 2x + C$ | B1F, B1F (2 marks) | Either term correct; both correct, constant required; condone missing bracket; ft on $A$, $B$ |
### Part (b)(i)
| $28 + 4x^2 = P(5-x)^2 + Q(1+3x)(5-x) + R(1+3x)$ | M1 | |
| $x=5 \Rightarrow R=8$; $x=-\frac{1}{3} \Rightarrow P=1$ | m1, A1 | Two values of $x$ used to find $R$ and $P$; SC $R=8$, $P=1$ NMS can score B1,B1 |
| $x=0 \Rightarrow 28 = 25P + 5Q + R$, $Q = -1$ | m1, A1 (5 marks) | Third value of $x$ used to find $Q$ |
**Alternative:**
| Collect terms and form equations; correct equations; solve for $P$, $Q$, $R$: $P=1$, $Q=-1$, $R=8$ | (M1), (m1), (A1), (m1), (A1) (5 marks) | |
### Part (b)(ii)
| $\int\frac{1}{1+3x} - \frac{1}{5-x} + \frac{8}{(5-x)^2}\,dx$ | M1 | Use partial fractions |
| $= \frac{1}{3}\ln(1+3x) + \ln(5-x) + \frac{8}{5-x} + (C)$ | m1, A1F, A1F (4 marks) | $a\ln(1+3x) + b\ln(5-x)$; OE, both ln integrals correct, needs $()$; other term correct; ft on their $P$, $Q$, $R$; SC: if no $P,Q,R$ found in (b)(i), can gain method marks by inserting other values or retaining letters (max 2/4) |
---3
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Express $\frac { 2 x + 7 } { x + 2 }$ in the form $A + \frac { B } { x + 2 }$, where $A$ and $B$ are integers. (2 marks)
\item Hence find $\int \frac { 2 x + 7 } { x + 2 } \mathrm {~d} x$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Express $\frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } }$ in the form $\frac { P } { 1 + 3 x } + \frac { Q } { 5 - x } + \frac { R } { ( 5 - x ) ^ { 2 } }$, where $P , Q$ and $R$ are constants.
\item Hence find $\int \frac { 28 + 4 x ^ { 2 } } { ( 1 + 3 x ) ( 5 - x ) ^ { 2 } } \mathrm {~d} x$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2009 Q3 [13]}}