| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Repeated linear factor only (two-term denominator) |
| Difficulty | Moderate -0.3 This is a straightforward partial fractions question with a repeated linear factor, followed by a routine integration. Part (i) requires standard algebraic manipulation to find constants A and B, and part (ii) applies direct integration formulas (ln and power rule). The question is slightly easier than average because it involves only one repeated factor and the integration is mechanical once the decomposition is complete. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| \(A(x-2) + B = 7 - 2x\) | M1 | or \(A(x-2)^2 + B(x-2) = (7-2x)(x-2)\) |
| \(A = -2\) | A1 | |
| \(B = 3\) | A1 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int \frac{A}{x-2}dx = (A \text{ or } \frac{1}{A})\ln(x-2)\) | B1 | Accept \(\ln |
| \(\int \frac{B}{(x-2)^2}dx = -(B \text{ or } \frac{1}{B})\frac{1}{x-2}\) | B1 | Negative sign is required |
| Correct f.t. of A & B; \(A\ln(x-2) - \frac{B}{x-2}\) | B1∇ | Still accept lns as before |
| Using limits = \(-2\ln 3 + 2\ln 2 + \frac{1}{2}\) | ISW | B1 4 |
**(i)**
$A(x-2) + B = 7 - 2x$ | M1 | or $A(x-2)^2 + B(x-2) = (7-2x)(x-2)$
$A = -2$ | A1 |
$B = 3$ | A1 3 |
**(ii)**
$\int \frac{A}{x-2}dx = (A \text{ or } \frac{1}{A})\ln(x-2)$ | B1 | Accept $\ln|x-2|$, $\ln|2-x|$, $\ln(2-x)$
$\int \frac{B}{(x-2)^2}dx = -(B \text{ or } \frac{1}{B})\frac{1}{x-2}$ | B1 | Negative sign is required
Correct f.t. of A & B; $A\ln(x-2) - \frac{B}{x-2}$ | B1∇ | Still accept lns as before
Using limits = $-2\ln 3 + 2\ln 2 + \frac{1}{2}$ | ISW | B1 4 | No indication of ln(negative)
---2 (i) Express $\frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } }$ in the form $\frac { A } { x - 2 } + \frac { B } { ( x - 2 ) ^ { 2 } }$, where $A$ and $B$ are constants.\\
(ii) Hence find the exact value of $\int _ { 4 } ^ { 5 } \frac { 7 - 2 x } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C4 2011 Q2 [7]}}