| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2022 |
| Session | January |
| Marks | 9 |
| 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, requiring routine decomposition (A/x + B/(x-2) + C/(x-2)²), straightforward integration including ln|x| and ln|x-2| terms plus -1/(x-2), and substitution of limits. While it requires multiple steps and careful algebra, it follows a well-practiced template with no novel insights needed, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.06f Laws of logarithms: addition, subtraction, power rules1.08d Evaluate definite integrals: between limits1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\dfrac{4-4x}{x(x-2)^2} \equiv \dfrac{A}{x}+\dfrac{B}{x-2}+\dfrac{C}{(x-2)^2}\) | B1 | Correct form for the partial fractions |
| \(4-4x = A(x-2)^2+B(x-2)+C \Rightarrow A=...\) or \(B=...\) or \(C=...\) | M1 | Uses a correct strategy to find at least one of their constants |
| \(\dfrac{4-4x}{x(x-2)^2} \equiv \dfrac{1}{x}-\dfrac{1}{x-2}-\dfrac{2}{(x-2)^2}\) | A1 | 2 correct constants |
| All correct | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int\left(\dfrac{1}{x}-\dfrac{1}{x-2}-\dfrac{2}{(x-2)^2}\right)dx = \ln x - \ln(x-2)+\dfrac{2}{x-2}(+c)\) | M1 | M1 for \(\int\dfrac{\alpha}{x}\,dx = \beta\ln x\) or \(\int\dfrac{\alpha}{x-2}\,dx = \beta\ln(x-2)\) |
| M1 | M1 for \(\int\dfrac{\alpha}{(x-2)^2}\,dx = \dfrac{\beta}{x-2}\) | |
| All correct | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\left[\ln x - \ln(x-2)+\dfrac{2}{x-2}\right]_3^5 = \left(\ln 5-\ln 3+\dfrac{2}{3}\right)-(\ln 3-\ln 1+2)\) | M1 | Correct use of limits and reaches required form using log rules |
| \(= \ln\dfrac{5}{9}-\dfrac{4}{3}\) | A1 | Correct answer |
# Question 4:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\dfrac{4-4x}{x(x-2)^2} \equiv \dfrac{A}{x}+\dfrac{B}{x-2}+\dfrac{C}{(x-2)^2}$ | B1 | Correct form for the partial fractions |
| $4-4x = A(x-2)^2+B(x-2)+C \Rightarrow A=...$ or $B=...$ or $C=...$ | M1 | Uses a correct strategy to find at least one of their constants |
| $\dfrac{4-4x}{x(x-2)^2} \equiv \dfrac{1}{x}-\dfrac{1}{x-2}-\dfrac{2}{(x-2)^2}$ | A1 | 2 correct constants |
| All correct | A1 | |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int\left(\dfrac{1}{x}-\dfrac{1}{x-2}-\dfrac{2}{(x-2)^2}\right)dx = \ln x - \ln(x-2)+\dfrac{2}{x-2}(+c)$ | M1 | M1 for $\int\dfrac{\alpha}{x}\,dx = \beta\ln x$ or $\int\dfrac{\alpha}{x-2}\,dx = \beta\ln(x-2)$ |
| | M1 | M1 for $\int\dfrac{\alpha}{(x-2)^2}\,dx = \dfrac{\beta}{x-2}$ |
| All correct | A1 | |
## Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\left[\ln x - \ln(x-2)+\dfrac{2}{x-2}\right]_3^5 = \left(\ln 5-\ln 3+\dfrac{2}{3}\right)-(\ln 3-\ln 1+2)$ | M1 | Correct use of limits and reaches required form using log rules |
| $= \ln\dfrac{5}{9}-\dfrac{4}{3}$ | A1 | Correct answer |4.
$$\mathrm { f } ( x ) = \frac { 4 - 4 x } { x ( x - 2 ) ^ { 2 } } \quad x > 2$$
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in partial fractions.
\item Hence find $\int \mathrm { f } ( x ) \mathrm { d } x$
\item Find
$$\int _ { 3 } ^ { 5 } f ( x ) d x$$
giving your answer in the form $a + \ln b$, where $a$ and $b$ are rational numbers to be found.
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2022 Q4 [9]}}