| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Partial fractions with irreducible quadratic |
| Difficulty | Standard +0.8 This is a standard partial fractions question with an irreducible quadratic factor, requiring decomposition, integration of both logarithmic and arctangent forms, and evaluation of a definite integral. While methodical, it involves multiple techniques (partial fractions setup, solving for constants, integrating ln and arctan terms) and careful algebraic manipulation, placing it moderately above average difficulty but still within standard A-level Further Maths scope. |
| Spec | 1.02y Partial fractions: decompose rational functions1.06d Natural logarithm: ln(x) function and properties1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Obtain one of the values \(A = 3, B = 4, C = 0\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Integrate and obtain term \(k\ln(4 + x^2)\) | B1√ M1 | |
| Obtain term \(2\ln(4 + x^2)\) | A1√ | |
| Substitute correct limits correctly in a complete integral of the form \(a\ln(2 - x) + b\ln(4 + x^2), ab \neq 0\) | M1 | |
| Obtain given answer following full and correct working | A1 | [5] |
**(i) Use any relevant method to determine a constant**
Obtain one of the values $A = 3, B = 4, C = 0$ | A1 |
Obtain a second value | A1 |
Obtain the third value | A1 | [4]
**(ii) Integrate and obtain term $-3\ln(2 - x)$**
Integrate and obtain term $k\ln(4 + x^2)$ | B1√ M1 |
Obtain term $2\ln(4 + x^2)$ | A1√ |
Substitute correct limits correctly in a complete integral of the form $a\ln(2 - x) + b\ln(4 + x^2), ab \neq 0$ | M1 |
Obtain given answer following full and correct working | A1 | [5]8 Let $f ( x ) = \frac { 12 + 8 x - x ^ { 2 } } { ( 2 - x ) \left( 4 + x ^ { 2 } \right) }$.\\
(i) Express $\mathrm { f } ( x )$ in the form $\frac { A } { 2 - x } + \frac { B x + C } { 4 + x ^ { 2 } }$.\\
(ii) Show that $\int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x = \ln \left( \frac { 25 } { 2 } \right)$.
\hfill \mbox{\textit{CAIE P3 2011 Q8 [9]}}