| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| 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 straightforward partial fractions question with an improper fraction requiring polynomial division first, followed by standard decomposition and integration. While it involves multiple steps (division, partial fractions setup, solving for constants, integration, and logarithm simplification), each step follows routine procedures taught in A-level. The final answer verification adds minimal difficulty. Slightly above average due to the improper fraction complication and multi-step nature, but no novel insight required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply the form \(A + \frac{B}{2x+1} + \frac{C}{x+2}\) | B1 | |
| State or obtain \(A = 2\) | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain one of \(B = 1, C = -2\) | A1 | |
| Obtain the other value | A1 | [5] |
| (ii) Integrate and obtain terms \(2x + \frac{1}{2}\ln(2x+1) - 2\ln(x+2)\) | B3♦ | |
| Substitute correct limits correctly in an integral with terms \(a\ln(2x+1)\) and \(b\ln(x+2)\), where \(ab \neq 0\) | M1 | |
| Obtain the given answer after full and correct working | A1 | [5] |
**(i)** State or imply the form $A + \frac{B}{2x+1} + \frac{C}{x+2}$ | B1 |
State or obtain $A = 2$ | B1 |
Use a correct method for finding a constant | M1 |
Obtain one of $B = 1, C = -2$ | A1 |
Obtain the other value | A1 | [5]
**(ii)** Integrate and obtain terms $2x + \frac{1}{2}\ln(2x+1) - 2\ln(x+2)$ | B3♦ |
Substitute correct limits correctly in an integral with terms $a\ln(2x+1)$ and $b\ln(x+2)$, where $ab \neq 0$ | M1 |
Obtain the given answer after full and correct working | A1 | [5]7 Let $\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } + 7 x + 4 } { ( 2 x + 1 ) ( x + 2 ) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Show that $\int _ { 0 } ^ { 4 } \mathrm { f } ( x ) \mathrm { d } x = 8 - \ln 3$.
\hfill \mbox{\textit{CAIE P3 2016 Q7 [10]}}