| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions with linear factors – decompose and integrate (definite) |
| Difficulty | Moderate -0.8 This is a straightforward partial fractions question with simple linear factors and standard integration. Part (a) requires factorising a difference of squares and finding a single constant by comparing coefficients or substitution. Part (b) involves routine integration of logarithmic forms and simplification. Both parts are mechanical applications of standard techniques with no problem-solving insight required, making this easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| \(3 = k(3 + x + 3 - x)\) | M1 | OE \(\frac{A}{3-x} + \frac{B}{3+x} \Rightarrow 6A = 3\), \(6B = 3\) |
| \(k = \frac{1}{2}\) | A1 | or eg put \(x = 0\), \(3 = k\left(\frac{1}{3} + \frac{1}{3}\right) \Rightarrow k = \frac{1}{2}\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \frac{3}{9-x^2} dx = -\frac{1}{2}\ln(3-x) + \frac{1}{2}\ln(3+x)\) | M1 A1F | \(a\ln(3 \pm x)\) ft on \(k\) |
| \(= \frac{1}{2}[(\ln 5 - \ln 1) - (\ln 4 - \ln 2)] = \frac{1}{2}\ln\left(\frac{5}{2}\right)\) | A1F | accept \(\ln\left(\frac{10}{4}\right)\) |
| ft only for sign error in integral: \(\frac{1}{2}\ln\left(\frac{5}{8}\right)\) |
**1(a)**
| $3 = k(3 + x + 3 - x)$ | M1 | OE $\frac{A}{3-x} + \frac{B}{3+x} \Rightarrow 6A = 3$, $6B = 3$ |
| $k = \frac{1}{2}$ | A1 | or eg put $x = 0$, $3 = k\left(\frac{1}{3} + \frac{1}{3}\right) \Rightarrow k = \frac{1}{2}$ |
**1(b)**
| $\int \frac{3}{9-x^2} dx = -\frac{1}{2}\ln(3-x) + \frac{1}{2}\ln(3+x)$ | M1 A1F | $a\ln(3 \pm x)$ ft on $k$ |
| $= \frac{1}{2}[(\ln 5 - \ln 1) - (\ln 4 - \ln 2)] = \frac{1}{2}\ln\left(\frac{5}{2}\right)$ | A1F | accept $\ln\left(\frac{10}{4}\right)$ |
| | | ft only for sign error in integral: $\frac{1}{2}\ln\left(\frac{5}{8}\right)$ |
**Total: 5 marks**
---1
\begin{enumerate}[label=(\alph*)]
\item Given that $\frac { 3 } { 9 - x ^ { 2 } }$ can be expressed in the form $k \left( \frac { 1 } { 3 + x } + \frac { 1 } { 3 - x } \right)$, find the value of the rational number $k$.
\item Show that $\int _ { 1 } ^ { 2 } \frac { 3 } { 9 - x ^ { 2 } } \mathrm {~d} x = \frac { 1 } { 2 } \ln \left( \frac { a } { b } \right)$, where $a$ and $b$ are integers.
\end{enumerate}
\hfill \mbox{\textit{AQA C4 2008 Q1 [5]}}