| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2019 |
| 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, then integrating to produce logarithmic forms. While the partial fractions setup is standard A-level, the irreducible quadratic adds complexity, and the definite integral requires careful algebraic manipulation to reach the exact form ln(c). This is moderately challenging but within typical Further Maths scope. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\frac{A}{2x-1}+\frac{Bx+C}{x^2+2}\) | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(A=4\), \(B=-1\), \(C=0\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| Total | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate and obtain term \(2\ln(2x-1)\) | B1FT | The FT is on \(A\). \(\frac{1}{2}A\ln(2x-1)\) |
| Integrate and obtain term of the form \(k\ln(x^2+2)\) | *M1 | From \(\frac{nx}{x^2+2}\) |
| Obtain term \(-\frac{1}{2}\ln(x^2+2)\) | A1FT | The FT is on \(B\) |
| Substitute limits correctly in an integral of the form \(a\ln(2x-1)+b\ln(x^2+2)\), where \(ab\neq 0\) | DM1 | \(2\ln9(-2\ln1)-\frac{1}{2}\ln27+\frac{1}{2}\ln3\) |
| Obtain answer \(\ln 27\) after full and correct exact working | A1 | ISW |
| Total | 5 |
## Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{2x-1}+\frac{Bx+C}{x^2+2}$ | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of $A=4$, $B=-1$, $C=0$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | |
| **Total** | **5** | |
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## Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate and obtain term $2\ln(2x-1)$ | B1FT | The FT is on $A$. $\frac{1}{2}A\ln(2x-1)$ |
| Integrate and obtain term of the form $k\ln(x^2+2)$ | *M1 | From $\frac{nx}{x^2+2}$ |
| Obtain term $-\frac{1}{2}\ln(x^2+2)$ | A1FT | The FT is on $B$ |
| Substitute limits correctly in an integral of the form $a\ln(2x-1)+b\ln(x^2+2)$, where $ab\neq 0$ | DM1 | $2\ln9(-2\ln1)-\frac{1}{2}\ln27+\frac{1}{2}\ln3$ |
| Obtain answer $\ln 27$ after full and correct exact working | A1 | ISW |
| **Total** | **5** | |
---8 Let $\mathrm { f } ( x ) = \frac { 2 x ^ { 2 } + x + 8 } { ( 2 x - 1 ) \left( x ^ { 2 } + 2 \right) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence, showing full working, find $\int _ { 1 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$, giving the answer in the form $\ln c$, where $c$ is an integer.\\
\hfill \mbox{\textit{CAIE P3 2019 Q8 [10]}}