| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | November |
| Marks | 11 |
| 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. Part (a) requires setting up A/(3x+2) + (Bx+C)/(x²+4) and solving for constants—routine algebraic manipulation. Part (b) involves integrating ln and arctan terms, which are standard results at this level. The question is slightly above average difficulty due to the irreducible quadratic requiring two constants in the numerator, but it follows a well-practiced template with no novel problem-solving required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\dfrac{A}{3x+2} + \dfrac{Bx+C}{x^2+4}\) | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A=3,\; B=-1,\; C=3\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate and obtain \(\ln(3x+2)\ldots\) | B1 FT | The FT is on \(A\) |
| State a term of the form \(k\ln(x^2+4)\) | M1 | From \(\int \dfrac{\lambda x}{x^2+4}\,dx\) |
| \(\ldots - \frac{1}{2}\ln(x^2+4)\ldots\) | A1 FT | The FT is on \(B\) |
| \(\ldots + \frac{3}{2}\tan^{-1}\dfrac{x}{2}\) | B1 FT | The FT is on \(C\) |
| Substitute limits correctly in an integral with at least two terms of the form \(a\ln(3x+2)\), \(b\ln(x^2+4)\) and \(c\tan^{-1}\!\left(\frac{x}{2}\right)\), and subtract in correct order | M1 | Using terms that have been obtained correctly from completed integrals |
| Obtain answer \(\frac{3}{2}\ln 2 + \frac{3}{8}\pi\), or exact 2-term equivalent | A1 | |
| Total: 6 |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\dfrac{A}{3x+2} + \dfrac{Bx+C}{x^2+4}$ | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of $A=3,\; B=-1,\; C=3$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| **Total: 5** | | |
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## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate and obtain $\ln(3x+2)\ldots$ | B1 FT | The FT is on $A$ |
| State a term of the form $k\ln(x^2+4)$ | M1 | From $\int \dfrac{\lambda x}{x^2+4}\,dx$ |
| $\ldots - \frac{1}{2}\ln(x^2+4)\ldots$ | A1 FT | The FT is on $B$ |
| $\ldots + \frac{3}{2}\tan^{-1}\dfrac{x}{2}$ | B1 FT | The FT is on $C$ |
| Substitute limits correctly in an integral with at least two terms of the form $a\ln(3x+2)$, $b\ln(x^2+4)$ and $c\tan^{-1}\!\left(\frac{x}{2}\right)$, and subtract in correct order | M1 | Using terms that have been obtained correctly from completed integrals |
| Obtain answer $\frac{3}{2}\ln 2 + \frac{3}{8}\pi$, or exact 2-term equivalent | A1 | |
| **Total: 6** | | |9 Let $\mathrm { f } ( x ) = \frac { 7 x + 18 } { ( 3 x + 2 ) \left( x ^ { 2 } + 4 \right) }$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Hence find the exact value of $\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q9 [11]}}