| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Basic partial fractions then integrate |
| Difficulty | Moderate -0.3 This is a straightforward partial fractions question with linear factors and standard integration. The decomposition is routine (cover-up method works directly), and integrating ln terms is a standard technique. Slightly easier than average due to being a textbook-style exercise with no complications, though the definite integral evaluation requires careful arithmetic. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply the form \(\frac{A}{1+2x} + \frac{B}{4-x}\) and use a correct method to find a constant | M1 | |
| Obtain one of \(A = 4\) and \(B = -1\) | A1 | |
| Obtain the second value | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate and obtain terms \(2\ln(1+2x) + \ln(4-x)\) | B1FT+B1FT | FT is on \(A\) and \(B\) |
| Substitute limits correctly in an integral of the form \(a\ln(1+2x) + b\ln(4-x)\), where \(ab \neq 0\) | M1 | |
| Obtain final answer \(\ln\left(\frac{50}{27}\right)\) | A1 |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\frac{A}{1+2x} + \frac{B}{4-x}$ and use a correct method to find a constant | M1 | |
| Obtain one of $A = 4$ and $B = -1$ | A1 | |
| Obtain the second value | A1 | |
## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate and obtain terms $2\ln(1+2x) + \ln(4-x)$ | B1FT+B1FT | FT is on $A$ and $B$ |
| Substitute limits correctly in an integral of the form $a\ln(1+2x) + b\ln(4-x)$, where $ab \neq 0$ | M1 | |
| Obtain final answer $\ln\left(\frac{50}{27}\right)$ | A1 | |4 Let $f ( x ) = \frac { 15 - 6 x } { ( 1 + 2 x ) ( 4 - x ) }$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Hence find $\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x$, giving your answer in the form $\ln \left( \frac { a } { b } \right)$, where $a$ and $b$ are integers.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q4 [7]}}