| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2006 |
| Session | November |
| 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 using logarithms. The decomposition form is predictable (A/(2x+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 definite integral evaluation, but still a textbook exercise requiring no novel insight. |
| Spec | 4.02b Express complex numbers: cartesian and modulus-argument forms4.02e Arithmetic of complex numbers: add, subtract, multiply, divide4.02k Argand diagrams: geometric interpretation4.02o Loci in Argand diagram: circles, half-lines |
| Answer | Marks |
|---|---|
| (i) EITHER: State or imply \(f(x) = \frac{A}{2x+1} + \frac{B}{x+1} + \frac{C}{(x+1)^2}\) | B1 |
| Use any relevant method to obtain a constant | M1 |
| Obtain one of the values \(A = 2, B = -1, C = 3\) | A1 |
| Obtain the remaining two values | A1 + A1 |
| [A correct solution starting with third term \(\frac{Cx}{(x+1)^2}\) or \(\frac{Cx+D}{(x+1)^2}\) is also possible.] | |
| OR: State or imply \(f(x) = \frac{A}{2x+1} + \frac{Dx+E}{(x+1)^2}\) | B1 |
| Use any relevant method to obtain a constant | M1 |
| Obtain one of the values \(A = 2, D = -1, E = 2\) | A1 |
| Obtain the remaining two values | A1 + A1 |
| 5 | |
| (ii) Integrate and obtain terms \(\frac{1}{2}\ln(2x+1) - \ln(x+1) - \frac{3}{x+1}\), or equivalent | B1/' + B1/' - B1/' |
| Use limits correctly, having integrated all the partial fractions | M1 |
| Obtain given answer following full and exact working | A1 |
| [The f.t. is on the value of u.] | |
| [SR: If \(b, C,\) or \(E\) are omitted, give B1M1 in part (i) and B1'/B1'/M1 in part (ii): max 5/10.] | |
| 5 |
(i) EITHER: State or imply $f(x) = \frac{A}{2x+1} + \frac{B}{x+1} + \frac{C}{(x+1)^2}$ | B1 |
Use any relevant method to obtain a constant | M1 |
Obtain one of the values $A = 2, B = -1, C = 3$ | A1 |
Obtain the remaining two values | A1 + A1 |
[A correct solution starting with third term $\frac{Cx}{(x+1)^2}$ or $\frac{Cx+D}{(x+1)^2}$ is also possible.] | | |
OR: State or imply $f(x) = \frac{A}{2x+1} + \frac{Dx+E}{(x+1)^2}$ | B1 |
Use any relevant method to obtain a constant | M1 |
Obtain one of the values $A = 2, D = -1, E = 2$ | A1 |
Obtain the remaining two values | A1 + A1 |
| | 5 |
(ii) Integrate and obtain terms $\frac{1}{2}\ln(2x+1) - \ln(x+1) - \frac{3}{x+1}$, or equivalent | B1/' + B1/' - B1/' |
Use limits correctly, having integrated all the partial fractions | M1 |
Obtain given answer following full and exact working | A1 |
[The f.t. is on the value of u.] | | |
[SR: If $b, C,$ or $E$ are omitted, give B1M1 in part (i) and B1'/B1'/M1 in part (ii): max 5/10.] | | |
| | 5 |8 Let $\mathrm { f } ( x ) = \frac { 7 x + 4 } { ( 2 x + 1 ) ( x + 1 ) ^ { 2 } }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence show that $\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 2 + \ln \frac { 5 } { 3 }$.
\hfill \mbox{\textit{CAIE P3 2006 Q8 [10]}}