| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Partial fractions with algebraic division first |
| Difficulty | Moderate -0.3 This is a standard two-part C4 question combining algebraic division with partial fractions and integration. While it requires multiple techniques, each step follows routine procedures: polynomial long division, factorizing the denominator, splitting into partial fractions, and integrating using standard forms. The question is slightly easier than average because it's methodical with clear signposting and no conceptual surprises. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Clear start to algebraic division | M1 | At least as far as \(x\) term in quot & subseq mult back & attempt at subtraction |
| Quotient \(= x - 1\) | A1 | |
| Remainder \(= x + 7\) | A1 | |
| Final answer: \(x - 1 + \frac{x+7}{x^2 - x - 6}\) | A1 | Final answer in correct form; must be shown in part (i) or implied in part (ii); Accept \(A=1, B=-1, C=1, D=7\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Convert their \(\frac{Cx+D}{x^2-x-6}\) to Partial Fractions | M1 | |
| \(\frac{x+7}{x^2-x-6} = \frac{2}{x-3} - \frac{1}{x+2}\) | A1A1 | Correct fraction converted to correct PFs |
| \(\int Ax + B\,dx = \frac{1}{2}Ax^2 + Bx\) or \(\frac{(Ax+B)^2}{2A}\) | B1 ft | |
| \(\int \frac{E}{x-3} + \frac{F}{x+2}\,dx = E\ln(x-3) + F\ln(x+2)\) | B1 ft | |
| Using limits in a correct manner | M1 | Tolerate some wrong signs provided intention clear |
| \(8 + \ln\frac{27}{4}\ \left(8 + \ln\frac{54}{8}\right)\) isw | A1 | Answer required in the form \(a + \ln b\); giving only a decimalised form is awarded A0 |
## Question 10:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Clear start to algebraic division | M1 | At least as far as $x$ term in quot & subseq mult back & attempt at subtraction |
| Quotient $= x - 1$ | A1 | |
| Remainder $= x + 7$ | A1 | |
| Final answer: $x - 1 + \frac{x+7}{x^2 - x - 6}$ | A1 | Final answer in correct form; must be shown in part (i) or implied in part (ii); Accept $A=1, B=-1, C=1, D=7$ |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Convert their $\frac{Cx+D}{x^2-x-6}$ to Partial Fractions | M1 | |
| $\frac{x+7}{x^2-x-6} = \frac{2}{x-3} - \frac{1}{x+2}$ | A1A1 | Correct fraction converted to correct PFs |
| $\int Ax + B\,dx = \frac{1}{2}Ax^2 + Bx$ or $\frac{(Ax+B)^2}{2A}$ | B1 ft | |
| $\int \frac{E}{x-3} + \frac{F}{x+2}\,dx = E\ln(x-3) + F\ln(x+2)$ | B1 ft | |
| Using limits in a correct manner | M1 | Tolerate some wrong signs provided intention clear |
| $8 + \ln\frac{27}{4}\ \left(8 + \ln\frac{54}{8}\right)$ isw | A1 | Answer required in the form $a + \ln b$; giving only a decimalised form is awarded A0 |10 (i) Use algebraic division to express $\frac { x ^ { 3 } - 2 x ^ { 2 } - 4 x + 13 } { x ^ { 2 } - x - 6 }$ in the form $A x + B + \frac { C x + D } { x ^ { 2 } - x - 6 }$, where $A , B , C$ and $D$ are constants.\\
(ii) Hence find $\int _ { 4 } ^ { 6 } \frac { x ^ { 3 } - 2 x ^ { 2 } - 4 x + 13 } { x ^ { 2 } - x - 6 } \mathrm {~d} x$, giving your answer in the form $a + \ln b$.
\hfill \mbox{\textit{OCR C4 2013 Q10 [11]}}