| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2012 |
| 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 question requires polynomial long division (since numerator degree equals denominator degree) followed by standard partial fractions decomposition and integration. While it involves multiple steps, each technique is routine for A-level: the algebraic manipulation is straightforward, the partial fractions setup is standard with linear factors, and the integration produces simple logarithms. The 'show that' format removes problem-solving difficulty. Slightly above average due to the improper fraction requiring division first, 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 the form \(A + \frac{B}{x+1} + \frac{C}{2x-3}\) | B1 | |
| State or obtain \(A = 2\) | B1 | |
| Use a correct method for finding a constant | M1 | |
| Obtain \(B = -2\) | A1 | |
| Obtain \(C = -1\) | A1 | [5] |
| (ii) Obtain integral \(2x - 2\ln(x+1) - \frac{1}{2}\ln(2x-3)\) | B3 |
| Answer | Marks | Guidance |
|---|---|---|
| Substitute limits correctly in an expression containing terms \(\ln(x+1)\) and \(\ln(2x-3)\) | M1 | |
| Obtain the given answer following full and exact working | A1 | [5] |
**(i)** State or imply the form $A + \frac{B}{x+1} + \frac{C}{2x-3}$ | B1 |
State or obtain $A = 2$ | B1 |
Use a correct method for finding a constant | M1 |
Obtain $B = -2$ | A1 |
Obtain $C = -1$ | A1 | [5]
**(ii)** Obtain integral $2x - 2\ln(x+1) - \frac{1}{2}\ln(2x-3)$ | B3 |
(Deduct B1 for each error or omission. The f.t. is on $A, B, C$.)
Substitute limits correctly in an expression containing terms $\ln(x+1)$ and $\ln(2x-3)$ | M1 |
Obtain the given answer following full and exact working | A1 | [5]
[SR: If $A$ omitted from the form of fractions, give B0B0M1A0A0 in (i); B1 B1M1A0 in (ii).]
[SR: For a solution starting with $\frac{B}{x+1} + \frac{Dx+E}{2x-3}$, give M1A1 for one of $B = -2, D = 4, E = -7$ and A1 for the other two constants; then give B1B1 for $A = 2, C = -1$.]
[SR: For a solution starting with $\frac{Fx+G}{x+1} + \frac{C}{2x-3}$ or with $\frac{Fx}{x+1} + \frac{C}{2x-3}$, give M1A1 for one of $C = -1, F = 2, G = 0$ and A1 for the other constants or constant; then give B1B1 for $A = 2, B = -2$.]
---8 Let $\mathrm { f } ( x ) = \frac { 4 x ^ { 2 } - 7 x - 1 } { ( x + 1 ) ( 2 x - 3 ) }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Show that $\int _ { 2 } ^ { 6 } \mathrm { f } ( x ) \mathrm { d } x = 8 - \ln \left( \frac { 49 } { 3 } \right)$.\\
\hfill \mbox{\textit{CAIE P3 2012 Q8 [10]}}