| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| 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. While it requires careful algebraic manipulation and knowledge of the ln|ax+b| integration formula, it follows a well-practiced procedure with no novel insights needed. The verification of a specific numerical answer adds minor complexity but remains straightforward. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply the form \(\frac{A}{2-x} + \frac{B}{3+2x} + \frac{C}{(3+2x)^2}\) | B1 | |
| Use a correct method to find a constant | M1 | |
| Obtain one of \(A = 1\), \(B = -1\), \(C = 3\) | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value [Mark the form \(\frac{A}{2-x} + \frac{Dx+E}{(3+2x)^2}\), where \(A=1\), \(D=-2\) and \(E=0\), B1M1A1A1A1 as above.] | A1 | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Integrate and obtain terms \(-\ln(2-x) - \frac{1}{2}\ln(3+2x) - \frac{3}{2(3+2x)}\) | B3ft | The f.t is on \(A\), \(B\), \(C\); or on \(A\), \(D\), \(E\) |
| Substitute correctly in an integral with terms \(a\ln(2-x)\), \(b\ln(3+2x)\) and \(c/(3+2x)\) where \(abc \neq 0\) | M1 | |
| Obtain the given answer after full and correct working [Correct integration of the \(A\), \(D\), \(E\) form gives an extra constant term if integration by parts is used for the second partial fraction.] | A1 | |
| Total: 5 |
## Question 9(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply the form $\frac{A}{2-x} + \frac{B}{3+2x} + \frac{C}{(3+2x)^2}$ | B1 | |
| Use a correct method to find a constant | M1 | |
| Obtain one of $A = 1$, $B = -1$, $C = 3$ | A1 | |
| Obtain a second value | A1 | |
| Obtain the third value [Mark the form $\frac{A}{2-x} + \frac{Dx+E}{(3+2x)^2}$, where $A=1$, $D=-2$ and $E=0$, B1M1A1A1A1 as above.] | A1 | |
| **Total: 5** | | |
## Question 9(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Integrate and obtain terms $-\ln(2-x) - \frac{1}{2}\ln(3+2x) - \frac{3}{2(3+2x)}$ | B3ft | The f.t is on $A$, $B$, $C$; or on $A$, $D$, $E$ |
| Substitute correctly in an integral with terms $a\ln(2-x)$, $b\ln(3+2x)$ and $c/(3+2x)$ where $abc \neq 0$ | M1 | |
| Obtain the given answer after full and correct working [Correct integration of the $A$, $D$, $E$ form gives an extra constant term if integration by parts is used for the second partial fraction.] | A1 | |
| **Total: 5** | | |9 Let $\mathrm { f } ( x ) = \frac { 6 x ^ { 2 } + 8 x + 9 } { ( 2 - x ) ( 3 + 2 x ) ^ { 2 } }$.\\
(i) Express $\mathrm { f } ( x )$ in partial fractions.\\
(ii) Hence, showing all necessary working, show that $\int _ { - 1 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = 1 + \frac { 1 } { 2 } \ln \left( \frac { 3 } { 4 } \right)$.\\
\hfill \mbox{\textit{CAIE P3 2018 Q9 [10]}}