| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2010 |
| 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 standard partial fractions question with an improper fraction requiring polynomial division first, followed by routine integration. While it has multiple steps (finding A,B,C,D then integrating), each step follows well-practiced techniques with no novel insight required. The improper form adds slight complexity above a basic partial fractions question, but this remains a textbook exercise slightly easier than average A-level difficulty. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: Divide by denominator and obtain quadratic remainder | M1 | |
| Obtain \(A = 1\) | A1 | |
| Use any relevant method to obtain \(B\), \(C\), \(D\) | M1 | |
| Obtain one correct answer | A1 | |
| Obtain \(B = 2\), \(C = 1\) and \(D = -3\) | A1 | |
| OR: Reduce RHS to a single fraction and equate numerators, or equivalent | M1 | |
| Obtain \(A = 1\) | A1 | |
| Use any relevant method to obtain \(B\), \(C\), \(D\) | M1 | |
| Obtain one correct answer | A1 | |
| Obtain \(B = 2\), \(C = 1\) and \(D = -3\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| (ii) Integrate and obtain \(x + 2 \ln x - \frac{1}{x} - \frac{3}{2}\ln(2x - 1)\), or equivalent | B3√ | |
| (The f.t. is on \(A, B, C, D\). Give B2√ if only one error in integration; B1√ if two.) | M1 | |
| Substitute limits correctly in the complete integral | M1 | |
| Obtain given answer correctly following full and exact working | A1 | [5] |
**(i)** **EITHER:** Divide by denominator and obtain quadratic remainder | M1 |
Obtain $A = 1$ | A1 |
Use any relevant method to obtain $B$, $C$, $D$ | M1 |
Obtain one correct answer | A1 |
Obtain $B = 2$, $C = 1$ and $D = -3$ | A1 |
**OR:** Reduce RHS to a single fraction and equate numerators, or equivalent | M1 |
Obtain $A = 1$ | A1 |
Use any relevant method to obtain $B$, $C$, $D$ | M1 |
Obtain one correct answer | A1 |
Obtain $B = 2$, $C = 1$ and $D = -3$ | A1 | [5]
[SR: If $A = 1$ stated without working give B1.]
**(ii)** Integrate and obtain $x + 2 \ln x - \frac{1}{x} - \frac{3}{2}\ln(2x - 1)$, or equivalent | B3√ |
(The f.t. is on $A, B, C, D$. Give B2√ if only one error in integration; B1√ if two.) | M1 |
Substitute limits correctly in the complete integral | M1 |
Obtain given answer correctly following full and exact working | A1 | [5]10 (i) Find the values of the constants $A , B , C$ and $D$ such that
$$\frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \equiv A + \frac { B } { x } + \frac { C } { x ^ { 2 } } + \frac { D } { 2 x - 1 }$$
(ii) Hence show that
$$\int _ { 1 } ^ { 2 } \frac { 2 x ^ { 3 } - 1 } { x ^ { 2 } ( 2 x - 1 ) } \mathrm { d } x = \frac { 3 } { 2 } + \frac { 1 } { 2 } \ln \left( \frac { 16 } { 27 } \right)$$
\hfill \mbox{\textit{CAIE P3 2010 Q10 [10]}}