| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with irreducible quadratic |
| Difficulty | Standard +0.8 This question requires decomposing a rational function with an irreducible quadratic factor into partial fractions, then integrating the result involving both logarithmic and arctangent terms. While the technique is standard for Further Maths Pure 3, it demands careful algebraic manipulation, knowledge of the arctan integral, and exact value simplification including logarithms and inverse trig functions—making it moderately challenging but within expected FM curriculum scope. |
| Spec | 1.02y Partial fractions: decompose rational functions1.06f Laws of logarithms: addition, subtraction, power rules1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply the form \(\dfrac{A}{3-x} + \dfrac{Bx+C}{1+3x^2}\) | B1 | |
| Use a correct method to find a constant | M1 | |
| Obtain one of \(A = 2\), \(B = 0\) and \(C = 1\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate and obtain term \(-2\ln(3-x)\) | B1 FT | |
| Obtain term of the form \(b\tan^{-1}(\sqrt{3}x)\) | M1 | |
| Obtain term \(\dfrac{1}{\sqrt{3}}\tan^{-1}(\sqrt{3}x)\) | A1 FT | |
| Substitute limits correctly in an integral with terms \(a\ln(3-x)\) and \(b\tan^{-1}(\sqrt{3}x)\), where \(ab \neq 0\) | M1 | |
| Obtain answer \(2\ln\dfrac{3}{2} + \dfrac{1}{3\sqrt{3}}\pi\), or equivalent | A1 | |
| Total | 5 |
## Question 11(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\dfrac{A}{3-x} + \dfrac{Bx+C}{1+3x^2}$ | B1 | |
| Use a correct method to find a constant | M1 | |
| Obtain one of $A = 2$, $B = 0$ and $C = 1$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| **Total** | **5** | |
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## Question 11(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate and obtain term $-2\ln(3-x)$ | B1 FT | |
| Obtain term of the form $b\tan^{-1}(\sqrt{3}x)$ | M1 | |
| Obtain term $\dfrac{1}{\sqrt{3}}\tan^{-1}(\sqrt{3}x)$ | A1 FT | |
| Substitute limits correctly in an integral with terms $a\ln(3-x)$ and $b\tan^{-1}(\sqrt{3}x)$, where $ab \neq 0$ | M1 | |
| Obtain answer $2\ln\dfrac{3}{2} + \dfrac{1}{3\sqrt{3}}\pi$, or equivalent | A1 | |
| **Total** | **5** | |11 Let $\mathrm { f } ( x ) = \frac { 5 - x + 6 x ^ { 2 } } { ( 3 - x ) \left( 1 + 3 x ^ { 2 } \right) }$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Find the exact value of $\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$, simplifying your answer.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q11 [10]}}