| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions with linear factors – decompose and integrate (definite) |
| Difficulty | Moderate -0.3 This is a straightforward partial fractions question with simple linear factors and standard integration. Part (i) requires routine algebraic manipulation to find constants, and part (ii) applies standard logarithm integration. The question is slightly easier than average because it involves only two linear factors with simple coefficients and no complications in the integration step. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{-2}{1-x} + \frac{3}{1-2x}\) (by cover-up rule or any valid method) | B1 | For correct canonical form |
| M1 | ||
| A1 | A1 for each of \(-2\) and \(3\) | |
| A1 | ||
| Total: 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int_2^3 \frac{1+x}{(1-x)(1-2x)}\,dx\) | ||
| \(= \int_2^3 \left(\frac{-2}{1-x} + \frac{3}{1-2x}\right)dx\) | M1 | |
| \(= \left[2\ln | 1-x | - \frac{3}{2}\ln |
| \(= 2\ln\frac{2}{1} - \frac{3}{2}\ln\frac{5}{3}\) | A1 | or equivalent answer |
| \((= 0.620\ldots)\) | ||
| Total: 4 marks |
## Question 5(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{-2}{1-x} + \frac{3}{1-2x}$ (by cover-up rule or any valid method) | B1 | For correct canonical form |
| | M1 | |
| | A1 | A1 for each of $-2$ and $3$ |
| | A1 | |
| **Total: 4 marks** | | |
---
## Question 5(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int_2^3 \frac{1+x}{(1-x)(1-2x)}\,dx$ | | |
| $= \int_2^3 \left(\frac{-2}{1-x} + \frac{3}{1-2x}\right)dx$ | M1 | |
| $= \left[2\ln|1-x| - \frac{3}{2}\ln|1-2x|\right]_2^3$ | A1, B1 | B1 for essential modulus signs |
| $= 2\ln\frac{2}{1} - \frac{3}{2}\ln\frac{5}{3}$ | A1 | or equivalent answer |
| $(= 0.620\ldots)$ | | |
| **Total: 4 marks** | | |
---5 (i) Express $\frac { 1 + x } { ( 1 - x ) ( 1 - 2 x ) }$ in partial fractions.\\
(ii) Hence find $\int _ { 2 } ^ { 3 } \frac { 1 + x } { ( 1 - x ) ( 1 - 2 x ) } \mathrm { d } x$.
\hfill \mbox{\textit{OCR MEI C4 Q5 [8]}}