| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with repeated linear factor |
| Difficulty | Standard +0.3 This is a standard partial fractions question with a repeated linear factor, requiring routine algebraic manipulation to find constants A, B, C, followed by straightforward integration of logarithmic and rational terms. While it involves multiple steps and careful arithmetic, it follows a well-practiced algorithm with no novel insight required, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08j Integration using partial fractions |
| Answer | Marks |
|---|---|
| 9(i) | A B C |
| Answer | Marks |
|---|---|
| 2−x 3+2x ( 3+2x )2 | B1 |
| Use a correct method to find a constant | M1 |
| Obtain one of A = 1, B = – 1, C = 3 | A1 |
| Obtain a second value | A1 |
| Answer | Marks |
|---|---|
| E = 0, B1M1A1A1A1 as above.] | A1 |
| Answer | Marks |
|---|---|
| 9(ii) | Integrate and obtain terms |
| Answer | Marks | Guidance |
|---|---|---|
| 2 2 ( 3+2x ) | B3ft | The f.t is on A, B, C; or on A, D, E. |
| Answer | Marks |
|---|---|
| b ln (3 + 2x) and c / (3 + 2x) where abc ≠ 0 | M1 |
| Answer | Marks |
|---|---|
| parts is used for the second partial fraction.] | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 9:
--- 9(i) ---
9(i) | A B C
State or imply the form + +
2−x 3+2x ( 3+2x )2 | B1
Use a correct method to find a constant | M1
Obtain one of A = 1, B = – 1, C = 3 | A1
Obtain a second value | A1
Obtain the third value
A Dx+E
[Mark the form + , where A = 1, D = – 2 and
2−x ( 3+2x )2
E = 0, B1M1A1A1A1 as above.] | A1
5
--- 9(ii) ---
9(ii) | Integrate and obtain terms
−ln ( 2−x ) − 1 ln ( 3+2x ) – 3
2 2 ( 3+2x ) | B3ft | The f.t is on A, B, C; or on A, D, E.
Substitute correctly in an integral with terms a ln (2 – x),
b ln (3 + 2x) and c / (3 + 2x) where abc ≠ 0 | M1
Obtain the given answer after full and correct working
[Correct integration of the A, D, E form gives an extra constant term if integration by
parts is used for the second partial fraction.] | A1
5
Question | Answer | Marks | GuidanceLet $f(x) = \frac{6x^2 + 8x + 9}{(2 - x)(3 + 2x)^2}$.
\begin{enumerate}[label=(\roman*)]
\item Express $f(x)$ in partial fractions. [5]
\item Hence, showing all necessary working, show that $\int_{-1}^0 f(x) dx = 1 + \frac{1}{2}\ln\left(\frac{4}{3}\right)$. [5]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2018 Q9 [10]}}