| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with irreducible quadratic |
| Difficulty | Standard +0.3 This is a standard partial fractions question with an irreducible quadratic factor, requiring decomposition into A/(3x-1) + (Bx+C)/(x²+3), followed by routine integration using ln and arctan. While it involves multiple steps, the techniques are well-practiced and follow a predictable pattern, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\dfrac{A}{3x-1}+\dfrac{Bx+C}{x^2+3}\) | B1 | |
| Use a correct method for finding a constant | M1 | A maximum of M1 A1 available after B0 |
| Obtain one of \(A=1\), \(B=0\) and \(C=3\) from correct working | A1 | |
| Obtain a second value from correct working | A1 | |
| Obtain the third value from correct working | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate and obtain term \(\dfrac{1}{3}\ln(3x-1)\) | B1 FT | OE e.g. \(\dfrac{1}{3}\ln(x-\frac{1}{3})\). FT on value of \(A\) |
| Obtain term of the form \(k\tan^{-1}\!\left(\dfrac{x}{\sqrt{3}}\right)\) | M1 | |
| Obtain term \(\sqrt{3}\tan^{-1}\!\left(\dfrac{x}{\sqrt{3}}\right)\) | A1 FT | OE. FT on value of \(C\) |
| Substitute correct limits in integral of form \(a\ln(3x-1)+k\tan^{-1}\!\left(\dfrac{x}{\sqrt{3}}\right)\), where \(ak\neq 0\), and evaluate trigonometry | M1 | Must be subtracted right way round; angles in radians; condone decimals |
| Obtain answer \(\dfrac{2}{3}\ln 2+\dfrac{\sqrt{3}\pi}{6}\) from correct working in 8(b) | A1 | Or exact 2-term equivalent e.g. \(\dfrac{1}{3}\ln 4+\dfrac{\pi}{2\sqrt{3}}\). ISW |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\dfrac{A}{3x-1}+\dfrac{Bx+C}{x^2+3}$ | B1 | |
| Use a correct method for finding a constant | M1 | A maximum of M1 A1 available after B0 |
| Obtain one of $A=1$, $B=0$ and $C=3$ from correct working | A1 | |
| Obtain a second value from correct working | A1 | |
| Obtain the third value from correct working | A1 | |
**Total: 5 marks**
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## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate and obtain term $\dfrac{1}{3}\ln(3x-1)$ | B1 FT | OE e.g. $\dfrac{1}{3}\ln(x-\frac{1}{3})$. FT on value of $A$ |
| Obtain term of the form $k\tan^{-1}\!\left(\dfrac{x}{\sqrt{3}}\right)$ | M1 | |
| Obtain term $\sqrt{3}\tan^{-1}\!\left(\dfrac{x}{\sqrt{3}}\right)$ | A1 FT | OE. FT on value of $C$ |
| Substitute correct limits in integral of form $a\ln(3x-1)+k\tan^{-1}\!\left(\dfrac{x}{\sqrt{3}}\right)$, where $ak\neq 0$, and evaluate trigonometry | M1 | Must be subtracted right way round; angles in radians; condone decimals |
| Obtain answer $\dfrac{2}{3}\ln 2+\dfrac{\sqrt{3}\pi}{6}$ from correct working in 8(b) | A1 | Or exact 2-term equivalent e.g. $\dfrac{1}{3}\ln 4+\dfrac{\pi}{2\sqrt{3}}$. ISW |
**Total: 5 marks**
---8 Let $\mathrm { f } ( x ) = \frac { x ^ { 2 } + 9 x } { ( 3 x - 1 ) \left( x ^ { 2 } + 3 \right) }$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Hence find $\int _ { 1 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x$, giving your answer in a simplified exact form.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q8 [10]}}