| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | March |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions with linear factors – decompose and integrate (definite) |
| Difficulty | Standard +0.3 This is a straightforward partial fractions question with standard integration. Part (a) requires routine decomposition into two linear factors, and part (b) involves integrating logarithmic terms and simplifying using log laws. The algebra is clean with the constant 'a', and the final answer simplifies nicely. Slightly easier than average due to the predictable structure and minimal algebraic manipulation required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks |
|---|---|
| 6 | Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear. |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
| Answer | Marks |
|---|---|
| 6(a) | Carry out a relevant method to determine constants A and B such that |
| Answer | Marks |
|---|---|
| ( 2x−a )( 3a−x ) 2x−a 3a−x | M1 |
| Obtain A = 2 | A1 |
| Obtain B = 1 | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| 6(b) | Integrate and obtain terms ln(2x−a)−ln ( 3a−x ) | B1 FT |
| B1 FT | The FT is on the values of A and B. |
| Answer | Marks |
|---|---|
| bln ( 2x−a ) and cln ( 3a−x ) , where bc≠0 | M1 |
| Obtain the given answer showing full and correct working | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 6:
6 | Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear.
PPMMTT
9709/32 Cambridge International A Level – Mark Scheme March 2021
PUBLISHED
Abbreviations
AEF/OE Any Equivalent Form (of answer is equally acceptable) / Or Equivalent
AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid)
CAO Correct Answer Only (emphasising that no ‘follow through’ from a previous error is allowed)
CWO Correct Working Only
ISW Ignore Subsequent Working
SOI Seen Or Implied
SC Special Case (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the
light of a particular circumstance)
WWW Without Wrong Working
AWRT Answer Which Rounds To
© UCLES 2021 Page 5 of 15
Question | Answer | Marks | Guidance
--- 6(a) ---
6(a) | Carry out a relevant method to determine constants A and B such that
5a A B
= +
( 2x−a )( 3a−x ) 2x−a 3a−x | M1
Obtain A = 2 | A1
Obtain B = 1 | A1
3
--- 6(b) ---
6(b) | Integrate and obtain terms ln(2x−a)−ln ( 3a−x ) | B1 FT
B1 FT | The FT is on the values of A and B.
Substitute limits correctly in a solution containing terms of the form
bln ( 2x−a ) and cln ( 3a−x ) , where bc≠0 | M1
Obtain the given answer showing full and correct working | A1
4
Question | Answer | Marks | GuidanceLet $\text{f}(x) = \frac{5a}{(2x - a)(3a - x)}$, where $a$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Express f$(x)$ in partial fractions. [3]
\item Hence show that $\int_a^{2a} \text{f}(x) \, dx = \ln 6$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q6 [7]}}