| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Improper fraction with quadratic factor – division then partial fractions and integrate |
| Difficulty | Moderate -0.3 This is a standard C4 improper fraction question with clear scaffolding through three parts: polynomial division (routine), expressing in partial fraction form (direct application), and integration (straightforward once decomposed). While it requires multiple techniques, each step follows a well-practiced procedure with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Leading term in quotient \(= 2x\) | B1 | |
| Suff evidence of division or identity process | M1 | Stated or in relevant position in division |
| Quotient \(= 2x + 3\) | A1 | Accept \(\frac{x}{x^2+4}\) as remainder |
| Remainder \(= x\) | A1 | |
| (ii) their quotient \(+ \frac{\text{their remainder}}{x^2 + 4}\) | √B1 | \(2x + 3 + \frac{x}{x^2+4}\) |
| (iii) Working with their expression in part (ii) | √B1 | |
| their \(Ax + B\) integrated as \(\frac{1}{2}Ax^2 + Bx\) | √B1 | |
| their \(\frac{Cx}{x^2+4}\) integrated as \(k\ln(x^2+4)\) | M1 | Ignore any integration of \(\frac{D}{x^2+4}\) |
| \(k = \frac{1}{2}C\) | √A1 | |
| Limits used correctly throughout | M1 | |
| \(14 + \frac{1}{2}\ln\frac{13}{5}\) | A1 | logs need not be combined. |
(i) Leading term in quotient $= 2x$ | B1 |
Suff evidence of division or identity process | M1 | Stated or in relevant position in division
Quotient $= 2x + 3$ | A1 | Accept $\frac{x}{x^2+4}$ as remainder | 4
Remainder $= x$ | A1 |
(ii) their quotient $+ \frac{\text{their remainder}}{x^2 + 4}$ | √B1 | $2x + 3 + \frac{x}{x^2+4}$ | 1
(iii) Working with their expression in part (ii) | √B1 |
their $Ax + B$ integrated as $\frac{1}{2}Ax^2 + Bx$ | √B1 |
their $\frac{Cx}{x^2+4}$ integrated as $k\ln(x^2+4)$ | M1 | Ignore any integration of $\frac{D}{x^2+4}$
$k = \frac{1}{2}C$ | √A1 |
Limits used correctly throughout | M1 |
$14 + \frac{1}{2}\ln\frac{13}{5}$ | A1 | logs need not be combined. | 5
---7 (i) Find the quotient and the remainder when $2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12$ is divided by $x ^ { 2 } + 4$.\\
(ii) Hence express $\frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 }$ in the form $A x + B + \frac { C x + D } { x ^ { 2 } + 4 }$, where the values of the constants $A , B , C$ and $D$ are to be stated.\\
(iii) Use the result of part (ii) to find the exact value of $\int _ { 1 } ^ { 3 } \frac { 2 x ^ { 3 } + 3 x ^ { 2 } + 9 x + 12 } { x ^ { 2 } + 4 } \mathrm {~d} x$.
\hfill \mbox{\textit{OCR C4 2007 Q7 [10]}}