| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with irreducible quadratic |
| Difficulty | Standard +0.3 This is a standard partial fractions question with an irreducible quadratic factor, followed by routine integration. The decomposition is straightforward (degree of numerator equals degree of denominator, so constant term A appears), and the integration involves only logarithms and arctangent—all textbook techniques. Slightly above average difficulty due to the irreducible quadratic and the need to verify a specific numerical result, but no novel insight required. |
| Spec | 1.08j Integration using partial fractions4.05c Partial fractions: extended to quadratic denominators |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EITHER: Divide by denominator and obtain a quadratic remainder | M1 | |
| Obtain \(A = 1\) | A1 | |
| Use any relevant method to obtain \(B\), \(C\) or \(D\) | M1 | |
| Obtain one correct answer | A1 | |
| Obtain \(B = -1,\ C = 2,\ D = 0\) | A1 | |
| OR: Reduce RHS to a single fraction and identify numerator with that of \(f(x)\) | M1 | |
| Obtain \(A = 1\) | A1 | |
| Use any relevant method to obtain \(B\), \(C\) or \(D\) | M1 | |
| Obtain one correct answer | A1 | |
| Obtain \(B = -1,\ C = 2,\ D = 0\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Integrate and obtain terms \(x - \ln(x-1)\), or equivalent | B1\(\sqrt{}\) | If \(B = 0\) the first B1\(\sqrt{}\) is not available. SR: If \(A\) is omitted in part (i), treat as if \(A = 0\). Thus only M1M1 and B1\(\sqrt{}\)B1\(\sqrt{}\)M1 are available. |
| Obtain third term \(\ln(x^2+1)\), or equivalent | B1\(\sqrt{}\) | |
| Substitute correct limits correctly in the complete integral | M1 | |
| Obtain given answer following full and exact working | A1 |
## Question 8(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **EITHER:** Divide by denominator and obtain a quadratic remainder | M1 | |
| Obtain $A = 1$ | A1 | |
| Use any relevant method to obtain $B$, $C$ or $D$ | M1 | |
| Obtain one correct answer | A1 | |
| Obtain $B = -1,\ C = 2,\ D = 0$ | A1 | |
| **OR:** Reduce RHS to a single fraction and identify numerator with that of $f(x)$ | M1 | |
| Obtain $A = 1$ | A1 | |
| Use any relevant method to obtain $B$, $C$ or $D$ | M1 | |
| Obtain one correct answer | A1 | |
| Obtain $B = -1,\ C = 2,\ D = 0$ | A1 | |
**Total: [5]**
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## Question 8(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Integrate and obtain terms $x - \ln(x-1)$, or equivalent | B1$\sqrt{}$ | If $B = 0$ the first B1$\sqrt{}$ is not available. SR: If $A$ is omitted in part (i), treat as if $A = 0$. Thus only M1M1 and B1$\sqrt{}$B1$\sqrt{}$M1 are available. |
| Obtain third term $\ln(x^2+1)$, or equivalent | B1$\sqrt{}$ | |
| Substitute correct limits correctly in the complete integral | M1 | |
| Obtain given answer following full and exact working | A1 | |
**Total: [4]**
---8 Let $\mathrm { f } ( x ) = \frac { x ^ { 3 } - x - 2 } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) }$.\\
(i) Express $\mathrm { f } ( x )$ in the form
$$A + \frac { B } { x - 1 } + \frac { C x + D } { x ^ { 2 } + 1 }$$
where $A , B , C$ and $D$ are constants.\\
(ii) Hence show that $\int _ { 2 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = 1$.
\hfill \mbox{\textit{CAIE P3 2003 Q8 [9]}}