| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | March |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with parameter |
| Difficulty | Challenging +1.2 This is a standard partial fractions question with a parameter that adds mild algebraic complexity. Part (a) requires routine partial fraction decomposition with three linear factors, while part (b) involves integrating logarithmic terms and simplifying using symmetry. The parameter 'a' makes algebra slightly more involved than typical examples, but the techniques are entirely standard for Further Maths Pure 3 students. No novel insight required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply the form \(\frac{A}{2a+x} + \frac{B}{2a-x} + \frac{C}{5a-2x}\) | B1 | Allow if seen prior to assigning a value for \(a\). |
| Use a correct method for finding a coefficient | M1 | |
| Obtain one of \(A = 1\), \(B = 9\), \(C = -16\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| Total | 5 | SC \(\frac{Dx+E}{4a^2-x^2} + \frac{C}{5a-2x}\) B0 M1 and \(C = -16\), A1 Max 2/5. SC Allow M1 only for other incorrect partial fraction. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate and obtain one of the terms \(\ln | 2a+x | - 9\ln |
| Obtain a second correct term | B1 FT | |
| Obtain the third correct term | B1 FT | Max 3/5 if value is assigned for \(a\) (award M0 A0). |
| Substitute limits correctly in an integral of the form \(p\ln | 2a+x | + q\ln |
| Obtain \(18\ln 3 - 8\ln 7\) from correct working | A1 | A0 if the solution involves logarithms of negative numbers. |
## Question 10:
### Part 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\frac{A}{2a+x} + \frac{B}{2a-x} + \frac{C}{5a-2x}$ | B1 | Allow if seen prior to assigning a value for $a$. |
| Use a correct method for finding a coefficient | M1 | |
| Obtain one of $A = 1$, $B = 9$, $C = -16$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| **Total** | **5** | SC $\frac{Dx+E}{4a^2-x^2} + \frac{C}{5a-2x}$ B0 M1 and $C = -16$, A1 Max 2/5. SC Allow M1 only for other incorrect partial fraction. |
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate and obtain one of the terms $\ln|2a+x| - 9\ln|2a-x| + 8\ln|5a-2x|$ | **B1 FT** | Condone missing modulus signs. Use *their* $A$, $B$ and $C$. |
| Obtain a second correct term | **B1 FT** | |
| Obtain the third correct term | **B1 FT** | Max 3/5 if value is assigned for $a$ (award M0 A0). |
| Substitute limits correctly in an integral of the form $p\ln|2a+x| + q\ln|2a-x| + r\ln|5a-2x|$ and remove all $a$'s | **M1** | Either (i) collect terms with same coefficient and remove all $a$'s e.g. $p\ln 3a - p\ln a + q\ln a - q\ln 3a + r\ln 3a - r\ln 7a$, hence $p\ln 3 - q\ln 3 + r\ln 3 - r\ln 7$, or (ii) collect same ln terms and remove all $a$'s e.g. $(p-q+r)\ln 3a - (p-q)\ln a - r\ln 7a$ and $-(p-q)\ln a = (-p+q-r)\ln a + r\ln a$, hence $p\ln 3 - q\ln 3 + r\ln 3 - r\ln 7$. |
| Obtain $18\ln 3 - 8\ln 7$ from correct working | **A1** | A0 if the solution involves logarithms of negative numbers. |
---10 Let $f ( x ) = \frac { 36 a ^ { 2 } } { ( 2 a + x ) ( 2 a - x ) ( 5 a - 2 x ) }$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Hence find the exact value of $\int _ { - a } ^ { a } f ( x ) d x$, giving your answer in the form plnq+rlns where $p$ and $r$ are integers and $q$ and $s$ are prime numbers.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q10 [10]}}