| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with repeated linear factor |
| Difficulty | Standard +0.3 This is a standard partial fractions question with a repeated linear factor, followed by routine integration and logarithm simplification. The decomposition form is predictable (A/(3x+1) + B/(x+1) + C/(x+1)²), and the integration is straightforward once decomposed. Slightly above average difficulty due to the repeated factor and algebraic manipulation required, but this is a textbook exercise type that students practice extensively. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(f(x) = \frac{A}{(3x+1)} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)}\) | B1 | |
| State or obtain \(A = -3\) | B1 | |
| State or obtain \(B = 2\) | B1 | |
| Use any relevant method to find \(C\) | M1 | |
| Obtain \(C = 1\) | A1 | |
| Guidance: Special case: allow the form \(\frac{A}{(3x+1)} + \frac{Dx+E}{(x+1)^2}\) and apply the above scheme (\(A = -3, D = 1, E = 3\)). | ||
| SR: if \(f(x)\) is given in incomplete form of partial fractions, give B1 for a form equivalent to the omission of \(C\), or \(E\), or \(B\) in the above, and M1 for finding one coefficient. | Total: 5 marks | |
| (ii) Integrate and obtain terms \(-\ln(3x-1) - \frac{x}{(x+1)} + \ln(x+1)\) | B1 + B1 + B1 | |
| Use limits correctly | M1 | |
| Obtain the given answer correctly | A1 | Total: 5 marks |
**(i)** State or imply $f(x) = \frac{A}{(3x+1)} + \frac{B}{(x+1)^2} + \frac{C}{(x+1)}$ | B1 |
State or obtain $A = -3$ | B1 |
State or obtain $B = 2$ | B1 |
Use any relevant method to find $C$ | M1 |
Obtain $C = 1$ | A1 |
**Guidance:** Special case: allow the form $\frac{A}{(3x+1)} + \frac{Dx+E}{(x+1)^2}$ and apply the above scheme ($A = -3, D = 1, E = 3$). | |
SR: if $f(x)$ is given in incomplete form of partial fractions, give B1 for a form equivalent to the omission of $C$, or $E$, or $B$ in the above, and M1 for finding one coefficient. | | **Total: 5 marks**
**(ii)** Integrate and obtain terms $-\ln(3x-1) - \frac{x}{(x+1)} + \ln(x+1)$ | B1 + B1 + B1 |
Use limits correctly | M1 |
Obtain the given answer correctly | A1 | **Total: 5 marks**
---6 Let $\mathrm { f } ( x ) = \frac { 4 x } { ( 3 x + 1 ) ( x + 1 ) ^ { 2 } }$.\\
(i) Express $f ( x )$ in partial fractions.\\
(ii) Hence show that $\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = 1 - \ln 2$.
\hfill \mbox{\textit{CAIE P3 2002 Q6 [10]}}