| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | March |
| Marks | 10 |
| 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. Part (i) requires routine algebraic manipulation to find constants A, B, C, D. Part (ii) involves straightforward integration of each term (ln for 1/x, ln for Cx/(x²+2), arctan for constant/(x²+2)) followed by substitution of limits. While it requires multiple techniques, all are standard A-level methods with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or obtain \(A = 3\) | B1 | |
| Use a relevant method to find a constant | M1 | |
| Obtain one of \(B = -4, C = 4\) and \(D = 0\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value | A1 | [5] |
| (ii) Integrate and obtain \(3x - 4\ln x\) | B1* | |
| Integrate and obtain term of the form \(k\ln(x^2 + 2)\) | M1 | |
| Obtain term \(2\ln(x^2 + 2)\) | A1* | |
| Substitute limits in an integral of the form \(ax + b\ln x + c\ln(x^2 + 2)\), where \(abc \neq 0\) | M1 | |
| Obtain given answer \(3 - \ln 4\) after full and correct working | A1 | [5] |
(i) State or obtain $A = 3$ | B1 |
Use a relevant method to find a constant | M1 |
Obtain one of $B = -4, C = 4$ and $D = 0$ | A1 |
Obtain a second value | A1 |
Obtain the third value | A1 | [5]
(ii) Integrate and obtain $3x - 4\ln x$ | B1* |
Integrate and obtain term of the form $k\ln(x^2 + 2)$ | M1 |
Obtain term $2\ln(x^2 + 2)$ | A1* |
Substitute limits in an integral of the form $ax + b\ln x + c\ln(x^2 + 2)$, where $abc \neq 0$ | M1 |
Obtain given answer $3 - \ln 4$ after full and correct working | A1 | [5]9 Let $\mathrm { f } ( x ) = \frac { 3 x ^ { 3 } + 6 x - 8 } { x \left( x ^ { 2 } + 2 \right) }$.\\
(i) Express $\mathrm { f } ( x )$ in the form $A + \frac { B } { x } + \frac { C x + D } { x ^ { 2 } + 2 }$.\\
(ii) Show that $\int _ { 1 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 3 - \ln 4$.
\hfill \mbox{\textit{CAIE P3 2016 Q9 [10]}}