| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 9 |
| 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 slightly above-average A-level question requiring recognition that the numerator degree equals denominator degree (requiring polynomial division first), followed by standard partial fractions decomposition and logarithmic integration. The 'show that' format and the specific numerical answer add minor complexity, but the techniques are all routine for P3/C4 level. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply the form \(A + \frac{B}{x+1} + \frac{C}{x+3}\) | B1 | |
| State or obtain \(A = 1\) | B1 | |
| Use correct method for finding \(B\) or \(C\) | M1 | |
| Obtain \(B = \frac{1}{2}\) | A1 | |
| Obtain \(C = -\frac{3}{2}\) | A1 | [5 marks] |
| (ii) Obtain integral \(x + \frac{1}{2}\ln(x+1) - \frac{3}{2}\ln(x+3)\) | B2\(\sqrt{\ }\) | [Award B1\(\sqrt{\ }\) if only one error. The f.t. is on A, B, C.] |
| Substitute limits correctly | M1 | |
| Obtain given answer following full and exact working | A1 | [4 marks] |
**(i)** State or imply the form $A + \frac{B}{x+1} + \frac{C}{x+3}$ | B1 |
State or obtain $A = 1$ | B1 |
Use correct method for finding $B$ or $C$ | M1 |
Obtain $B = \frac{1}{2}$ | A1 |
Obtain $C = -\frac{3}{2}$ | A1 | [5 marks]
**(ii)** Obtain integral $x + \frac{1}{2}\ln(x+1) - \frac{3}{2}\ln(x+3)$ | B2$\sqrt{\ }$ | [Award B1$\sqrt{\ }$ if only one error. The f.t. is on A, B, C.]
Substitute limits correctly | M1 |
Obtain given answer following full and exact working | A1 | [4 marks] | [SR: if $A$ omitted, only M1 in part (i) is available; then in part (ii) B1$\sqrt{\ }$ for each correct integral and M1.]
---7 Let $\mathrm { f } ( x ) \equiv \frac { x ^ { 2 } + 3 x + 3 } { ( x + 1 ) ( x + 3 ) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence show that $\int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \frac { 1 } { 2 } \ln 2$.
\hfill \mbox{\textit{CAIE P3 2008 Q7 [9]}}