| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2015 |
| 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. While it requires careful algebraic manipulation to find the constants and proper handling of the repeated factor, it follows a well-established procedure taught in all A-level Further Maths courses. The integration step is straightforward once partial fractions are obtained, and verifying a given answer is easier than finding it independently. Slightly above average difficulty due to the repeated factor, but still a textbook exercise. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply \(f(x) = \frac{A}{2x-1} + \frac{B}{x+2} + \frac{C}{(x+2)^2}\) | B1 | |
| Use a relevant method to determine a constant | M1 | |
| Obtain one of the values \(A = 2, B = -1, C = 3\) | A1 | |
| Obtain the remaining values A1 + | A1 | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Integrate and obtain terms \(\frac{1}{2} \cdot 2\ln(2x-1) - \ln(x+2) - \frac{3}{x+2}\) | B1* + B1* + B1* | |
| Use limits correctly, namely substitution must be seen in at least two of the partial fractions to obtain M1 | M1 | |
| Integrate all 3 partial fractions and substitute in all three partial fractions for A1 since AG | M1 | |
| Obtain the given answer following full and exact working | A1 | 5 |
| Answer | Marks |
|---|---|
| [NB: Candidates who follow the \(A, D, E\) scheme in part (i) and then integrate \(\frac{-x+1}{(x+2)^2}\) by parts should obtain \(\frac{1}{2} \cdot 2\ln(2x-1) - \ln(x+2) + \frac{x-1}{x+2}\) (the third term is equivalent to \(-\frac{3}{x+2} + 1\)).] |
**(i)** State or imply $f(x) = \frac{A}{2x-1} + \frac{B}{x+2} + \frac{C}{(x+2)^2}$ | B1 |
Use a relevant method to determine a constant | M1 |
Obtain one of the values $A = 2, B = -1, C = 3$ | A1 |
Obtain the remaining values A1 + | A1 | 5 |
[Apply an analogous scheme to the form $\frac{A}{2x-1} + \frac{Dx+E}{(x+2)^2}$; the values being $A = 2, D = -1, E = 1.$]
**(ii)** Integrate and obtain terms $\frac{1}{2} \cdot 2\ln(2x-1) - \ln(x+2) - \frac{3}{x+2}$ | B1* + B1* + B1* |
Use limits correctly, namely substitution must be seen in at least two of the partial fractions to obtain M1 | M1 |
Integrate all 3 partial fractions and substitute in all three partial fractions for A1 since AG | M1 |
Obtain the given answer following full and exact working | A1 | 5 |
[The t marks are dependent on $A, B, C$ etc.]
[SR: If $B, C$ or $E$ omitted, give B1M1 in part (i) and B1✓B1✓M1 in part (ii).]
[NB: Candidates who follow the $A, D, E$ scheme in part (i) and then integrate $\frac{-x+1}{(x+2)^2}$ by parts should obtain $\frac{1}{2} \cdot 2\ln(2x-1) - \ln(x+2) + \frac{x-1}{x+2}$ (the third term is equivalent to $-\frac{3}{x+2} + 1$).] |10 Let $\mathrm { f } ( x ) = \frac { 11 x + 7 } { ( 2 x - 1 ) ( x + 2 ) ^ { 2 } }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Show that $\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = \frac { 1 } { 4 } + \ln \left( \frac { 9 } { 4 } \right)$.
\hfill \mbox{\textit{CAIE P3 2015 Q10 [10]}}