| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Square of partial fractions expression |
| Difficulty | Challenging +1.2 This is a structured multi-part question requiring partial fractions decomposition, algebraic manipulation to square the result, and integration of standard forms. While it involves several steps and careful algebra, each component uses routine A-level techniques with clear scaffolding. The squaring in part (b) requires attention to detail but no novel insight, and part (c) integrates standard rational functions. Slightly above average due to length and algebraic complexity, but well within typical Further Maths Pure content. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08d Evaluate definite integrals: between limits |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\frac{A}{2x-1}+\frac{B}{2x+1}\) and use a relevant method to find \(A\) or \(B\) | M1 | |
| Obtain \(A=1\), \(B=-1\) | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Square the result of part (a) and substitute the fractions of part (a) | M1 | |
| Obtain the given answer correctly | A1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate and obtain \(-\frac{1}{2(2x-1)}-\frac{1}{2}\ln(2x-1)+\frac{1}{2}\ln(2x+1)-\frac{1}{2(2x+1)}\), or equivalent | B3, 2, 1, 0 | |
| Substitute limits correctly | M1 | |
| Obtain the given answer correctly | A1 | |
| 5 |
## Question 7:
### Part 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{2x-1}+\frac{B}{2x+1}$ and use a relevant method to find $A$ or $B$ | M1 | |
| Obtain $A=1$, $B=-1$ | A1 | |
| | **2** | |
### Part 7(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Square the result of part (a) and substitute the fractions of part (a) | M1 | |
| Obtain the given answer correctly | A1 | |
| | **2** | |
### Part 7(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate and obtain $-\frac{1}{2(2x-1)}-\frac{1}{2}\ln(2x-1)+\frac{1}{2}\ln(2x+1)-\frac{1}{2(2x+1)}$, or equivalent | B3, 2, 1, 0 | |
| Substitute limits correctly | M1 | |
| Obtain the given answer correctly | A1 | |
| | **5** | |
---\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Using your answer to part (a), show that
$$( f ( x ) ) ^ { 2 } = \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } - \frac { 1 } { 2 x - 1 } + \frac { 1 } { 2 x + 1 } + \frac { 1 } { ( 2 x + 1 ) ^ { 2 } }$$
\item Hence show that $\int _ { 1 } ^ { 2 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x = \frac { 2 } { 5 } + \frac { 1 } { 2 } \ln \left( \frac { 5 } { 9 } \right)$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q7 [9]}}