| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Improper algebraic form then partial fractions |
| Difficulty | Moderate -0.8 This is a straightforward algebraic division followed by standard integration. Part (a) requires polynomial long division (a routine C1/C2 skill), and part (b) involves integrating polynomials and 1/(x-2), then substituting limits. The 'show that' format provides the answer form, making it easier than an open-ended question. This is below average difficulty for A-level as it's purely procedural with no problem-solving or insight required. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08d Evaluate definite integrals: between limits1.08h Integration by substitution |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sets \(2x^2-5x+8=(Ax+B)(x-2)+C\) with attempt at one constant | M1 | Via identity or division looking for quotient as far as \(2x\). |
| Attempts all 3 constants: sets \(x=2 \Rightarrow C=...\), \(-2B+C=8 \Rightarrow B=...\), compares \(x^2\) terms gives \(A=...\) | dM1 | Via division: look for quotient of \(2x \pm c\) and a constant remainder. Dependent on previous mark. |
| \(2x - 1 + \dfrac{6}{x-2}\) | A1 | Expression must be written; cannot award for just stating correct values of \(A\), \(B\), \(C\) unless later written correctly. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int 2x-1+\dfrac{6}{x-2}\,dx = x^2 - x + 6\ln(x-2)\) | M1 A1ft | M1: attempts to integrate \(\frac{C}{x-2}\) proceeding to \(D\ln |
| \(\left[x^2 - x + 6\ln(x-2)\right]_4^8 = 56 + 6\ln 6 - 12 - 6\ln 2\) | dM1 | Attempts to use limits 8 and 4, subtracts either way round, and combines ln terms correctly: \(\ln 6 - \ln 2 = \ln 3\). |
| \(= 44 + 6\ln 3\) | A1 | Withhold if integral sign and \(dx\) remain, or \(+c\) present. |
## Question 2:
### Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sets $2x^2-5x+8=(Ax+B)(x-2)+C$ with attempt at one constant | M1 | Via identity or division looking for quotient as far as $2x$. |
| Attempts all 3 constants: sets $x=2 \Rightarrow C=...$, $-2B+C=8 \Rightarrow B=...$, compares $x^2$ terms gives $A=...$ | dM1 | Via division: look for quotient of $2x \pm c$ and a constant remainder. Dependent on previous mark. |
| $2x - 1 + \dfrac{6}{x-2}$ | A1 | Expression must be written; cannot award for just stating correct values of $A$, $B$, $C$ unless later written correctly. |
### Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int 2x-1+\dfrac{6}{x-2}\,dx = x^2 - x + 6\ln(x-2)$ | M1 A1ft | M1: attempts to integrate $\frac{C}{x-2}$ proceeding to $D\ln|x-2|$. A1ft: correct integration of $g(x)$ for their $A$, $B$, $C$ from part (a). |
| $\left[x^2 - x + 6\ln(x-2)\right]_4^8 = 56 + 6\ln 6 - 12 - 6\ln 2$ | dM1 | Attempts to use limits 8 and 4, subtracts either way round, and combines ln terms correctly: $\ln 6 - \ln 2 = \ln 3$. |
| $= 44 + 6\ln 3$ | A1 | Withhold if integral sign and $dx$ remain, or $+c$ present. |
---2.
$$g ( x ) = \frac { 2 x ^ { 2 } - 5 x + 8 } { x - 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Write $g ( x )$ in the form
$$A x + B + \frac { C } { x - 2 }$$
where $A , B$ and $C$ are integers to be found.
\item Hence use algebraic integration to show that
$$\int _ { 4 } ^ { 8 } \mathrm {~g} ( x ) \mathrm { d } x = \alpha + \beta \ln 3$$
where $\alpha$ and $\beta$ are integers to be found.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2024 Q2 [7]}}