| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | State appropriate partial fraction form |
| Difficulty | Moderate -0.8 Part (a) requires only recall of standard partial fraction forms for linear factors, irreducible quadratics, and repeated factors—no calculation or problem-solving. Part (b) is a routine integration using partial fractions with straightforward logarithm simplification. This is easier than average A-level content, being primarily pattern recognition and standard technique application. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| (a)(i) State answer \(\frac{A}{x+4} + \frac{Bx+C}{x^2+3}\) | B1 | Total: 1 mark |
| (ii) State answer \(\frac{A}{x-2} + \frac{Bx+C}{(x+2)^2}\) or \(\frac{A}{x-2} + \frac{B}{x+2} + \frac{C}{(x+2)^2}\) | B2 | Total: 2 marks |
| [Award B1 if the B term is omitted or for the form \(\frac{A}{x-2} + \frac{B}{x+2} + \frac{Cx+D}{(x+2)^2}\).] | ||
| (b) Stating or implying \(f(x) = \frac{A}{x+1} + \frac{B}{x-2}\), use a relevant method to determine A or B | M1 | |
| Obtain \(A = 1\) and \(B = 2\) | A1 | [SR: If A = 1 and B = 2 stated without working, award B1 + B1.] |
| Integrate and obtain terms \(\ln(x+1) + 2\ln(x-2)\) | A1√ + A1√ | |
| Use correct limits correctly in the complete integral | M1 | |
| Obtain given answer in 5 following full and exact working | A1 | Total: 6 marks |
**(a)(i)** State answer $\frac{A}{x+4} + \frac{Bx+C}{x^2+3}$ | B1 | **Total: 1 mark** |
**(ii)** State answer $\frac{A}{x-2} + \frac{Bx+C}{(x+2)^2}$ or $\frac{A}{x-2} + \frac{B}{x+2} + \frac{C}{(x+2)^2}$ | B2 | **Total: 2 marks** |
| | | [Award B1 if the B term is omitted or for the form $\frac{A}{x-2} + \frac{B}{x+2} + \frac{Cx+D}{(x+2)^2}$.] |
**(b)** Stating or implying $f(x) = \frac{A}{x+1} + \frac{B}{x-2}$, use a relevant method to determine A or B | M1 | |
Obtain $A = 1$ and $B = 2$ | A1 | [SR: If A = 1 and B = 2 stated without working, award B1 + B1.] |
Integrate and obtain terms $\ln(x+1) + 2\ln(x-2)$ | A1√ + A1√ | |
Use correct limits correctly in the complete integral | M1 | |
Obtain given answer in 5 following full and exact working | A1 | **Total: 6 marks** |8 An appropriate form for expressing $\frac { 3 x } { ( x + 1 ) ( x - 2 ) }$ in partial fractions is
$$\frac { A } { x + 1 } + \frac { B } { x - 2 }$$
where $A$ and $B$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Without evaluating any constants, state appropriate forms for expressing the following in partial fractions:
\begin{enumerate}[label=(\roman*)]
\item $\frac { 4 x } { ( x + 4 ) \left( x ^ { 2 } + 3 \right) }$,
\item $\frac { 2 x + 1 } { ( x - 2 ) ( x + 2 ) ^ { 2 } }$.
\end{enumerate}\item Show that $\int _ { 3 } ^ { 4 } \frac { 3 x } { ( x + 1 ) ( x - 2 ) } \mathrm { d } x = \ln 5$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2004 Q8 [9]}}