| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper algebraic form then partial fractions |
| Difficulty | Standard +0.3 This is a standard partial fractions question with a proper rational function requiring decomposition into A/x + B/x² + C/(3x+2), followed by routine integration. The algebra is straightforward, and the techniques are textbook exercises, making it slightly easier than average for A-level. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\frac{A}{x} + \frac{B}{x^2} + \frac{C}{3x+2}\) | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of the values \(A=3\), \(B=-2\), \(C=-6\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | Mark the form \(\frac{Ax+B}{x^2} + \frac{C}{3x+2}\) using same pattern of marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate and obtain terms \(3\ln x = \frac{2}{x} - 2\ln(3x+2)\) | B3FT | FT is on \(A\), \(B\) and \(C\). Candidates integrating \(\frac{3x-2}{x^2}\) by parts should obtain \(3\ln x + \frac{2}{x} - 3\) or equivalent |
| Use limits correctly, having integrated all partial fractions, in a solution containing terms \(a\ln x + \frac{b}{x} + c\ln(3x+2)\) | M1 | |
| Obtain the given answer following full and exact working | A1 |
## Question 9(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{x} + \frac{B}{x^2} + \frac{C}{3x+2}$ | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of the values $A=3$, $B=-2$, $C=-6$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | Mark the form $\frac{Ax+B}{x^2} + \frac{C}{3x+2}$ using same pattern of marks |
**Total: 5**
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## Question 9(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate and obtain terms $3\ln x = \frac{2}{x} - 2\ln(3x+2)$ | B3FT | FT is on $A$, $B$ and $C$. Candidates integrating $\frac{3x-2}{x^2}$ by parts should obtain $3\ln x + \frac{2}{x} - 3$ or equivalent |
| Use limits correctly, having integrated all partial fractions, in a solution containing terms $a\ln x + \frac{b}{x} + c\ln(3x+2)$ | M1 | |
| Obtain the given answer following full and exact working | A1 | |
**Total: 5**
---9 Let $\mathrm { f } ( x ) = \frac { 3 x ^ { 2 } - 4 } { x ^ { 2 } ( 3 x + 2 ) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence show that $\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 8 } \right) - 1$.\\
\hfill \mbox{\textit{CAIE P3 2017 Q9 [10]}}