| 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 | Challenging +1.2 This is a standard partial fractions question with an irreducible quadratic factor, requiring decomposition into A/(1+x) + (Bx+C)/(2+x²), followed by integration using ln and arctan. While it involves multiple steps and careful algebraic manipulation, it follows a well-established procedure taught explicitly in P3/C4 courses with no novel problem-solving required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.06f Laws of logarithms: addition, subtraction, power rules1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply the form \(\frac{A}{1+x} + \frac{Bx+C}{2+x^2}\) | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A = 2\), \(B = -1\) and \(C = 0\) | A1 | SC: A maximum of M1A1 is available for obtaining \(A = 2\) after scoring B0 |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate and obtain term \(2\ln(1+x)\) | B1 FT | \(A\ln(1+x)\) |
| Integrate and obtain term of the form \(k\ln(2+x^2)\) from an integral of the correct form | \*M1 | Ignore any separate working relating to \(C \neq 0\) |
| Obtain term \(-\frac{1}{2}\ln(2+x^2)\) | A1 FT | \(\frac{B}{2}\ln(2+x^2)\) |
| Substitute limits in an integral containing terms of the form \(a\ln(1+x) + b\ln(2+x^2)\), where \(ab \neq 0\) | DM1 | Ignore working relating to \(C \neq 0\). \((2\ln 5 - 2\ln 1 - \frac{1}{2}\ln 18 + \frac{1}{2}\ln 2)\). Dependent on first M1. Must be subtracting the correct way round. Must have an exact substitution |
| Obtain answer \(\ln\!\left(\dfrac{25}{3}\right)\) | A1 | ISW. Any exact equivalent e.g. \(\ln\frac{25\sqrt{2}}{\sqrt{18}}\) with no number in front of the logarithm |
| 5 |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply the form $\frac{A}{1+x} + \frac{Bx+C}{2+x^2}$ | **B1** | |
| Use a correct method for finding a constant | **M1** | |
| Obtain one of $A = 2$, $B = -1$ and $C = 0$ | **A1** | SC: A maximum of M1A1 is available for obtaining $A = 2$ after scoring B0 |
| Obtain a second value | **A1** | |
| Obtain the third value | **A1** | |
| | **5** | |
## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate and obtain term $2\ln(1+x)$ | **B1 FT** | $A\ln(1+x)$ |
| Integrate and obtain term of the form $k\ln(2+x^2)$ from an integral of the correct form | **\*M1** | Ignore any separate working relating to $C \neq 0$ |
| Obtain term $-\frac{1}{2}\ln(2+x^2)$ | **A1 FT** | $\frac{B}{2}\ln(2+x^2)$ |
| Substitute limits in an integral containing terms of the form $a\ln(1+x) + b\ln(2+x^2)$, where $ab \neq 0$ | **DM1** | Ignore working relating to $C \neq 0$. $(2\ln 5 - 2\ln 1 - \frac{1}{2}\ln 18 + \frac{1}{2}\ln 2)$. Dependent on first M1. Must be subtracting the correct way round. Must have an exact substitution |
| Obtain answer $\ln\!\left(\dfrac{25}{3}\right)$ | **A1** | ISW. Any exact equivalent e.g. $\ln\frac{25\sqrt{2}}{\sqrt{18}}$ with no number in front of the logarithm |
| | **5** | |10 Let $\mathrm { f } ( x ) = \frac { 4 - x + x ^ { 2 } } { ( 1 + x ) \left( 2 + x ^ { 2 } \right) }$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Find the exact value of $\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer as a single logarithm.\\
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 Q10 [10]}}