| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Repeated linear factor with distinct linear factor – decompose and integrate |
| Difficulty | Standard +0.3 This is a standard C4 partial fractions question with a repeated linear factor, followed by routine integration. Part (i) requires the standard technique for decomposing with repeated factors, and part (ii) involves straightforward integration of logarithmic and algebraic terms. While it requires careful algebraic manipulation and multiple steps, it follows a well-practiced procedure with no novel insight needed, making it slightly easier than average. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{(x-2)^2}\) | B1 | Correct partial fractions format |
| \(A(x-2)^2 + B(x+1)(x-2) + C(x+1) = x^2 - x - 11\) | M1 | Equivalent identity or method |
| \(A = -1\) | A1 | B1 if cover-up method used |
| \(B = 2\) | A1 | |
| \(C = -3\) | A1 | B1 if cover-up method used |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int \frac{\lambda}{x+1}\, dx = \left(\lambda \text{ or } \frac{1}{\lambda}\right)\ln(x+1)\) | B1 | \(\int \frac{\lambda}{x-2}\, dx = \left(\lambda \text{ or } \frac{1}{\lambda}\right)\ln(x-2)\) |
| \(\int \frac{\mu}{(x-2)^2}\, dx = -\left(\mu \text{ or } \frac{1}{\mu}\right)\frac{1}{x-2}\) | B1 | |
| \(-\frac{3}{2}\) | B1 ft | ft \(\frac{C}{2}\) |
| \(\cdots + \ln\frac{16}{5}\) (ISW for either term) | B1 ft | ft \(\cdots + \ln\left\{\left(\frac{5}{4}\right)^A \cdot 2^B\right\}\) |
## Question 9:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{A}{x+1} + \frac{B}{x-2} + \frac{C}{(x-2)^2}$ | B1 | Correct partial fractions format |
| $A(x-2)^2 + B(x+1)(x-2) + C(x+1) = x^2 - x - 11$ | M1 | Equivalent identity or method |
| $A = -1$ | A1 | B1 if cover-up method used |
| $B = 2$ | A1 | |
| $C = -3$ | A1 | B1 if cover-up method used |
**Special Cases:**
- $\frac{A}{x+1} + \frac{Bx+C}{(x-2)^2}$: allow B1 for PF format, M1 for identity, B1 for $A=-1$ (max 3)
- $\frac{A}{x+1} + \frac{B}{x-2} + \frac{Cx+D}{(x-2)^2}$: allow B1 for PF format, M1 for identity, B1 for $A=-1$ (max 3)
- $\frac{A}{x+1} + \frac{Bx}{(x-2)^2}$: allow B0 for PF format, M1 for identity (max 1, even if $A=-1$)
- $\frac{A}{x+1} + \frac{B}{(x-2)^2}$: allow B0 for PF format, M1 for identity (max 1, even if $A=-1$)
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int \frac{\lambda}{x+1}\, dx = \left(\lambda \text{ or } \frac{1}{\lambda}\right)\ln(x+1)$ | B1 | $\int \frac{\lambda}{x-2}\, dx = \left(\lambda \text{ or } \frac{1}{\lambda}\right)\ln(x-2)$ |
| $\int \frac{\mu}{(x-2)^2}\, dx = -\left(\mu \text{ or } \frac{1}{\mu}\right)\frac{1}{x-2}$ | B1 | |
| $-\frac{3}{2}$ | B1 ft | ft $\frac{C}{2}$ |
| $\cdots + \ln\frac{16}{5}$ (ISW for either term) | B1 ft | ft $\cdots + \ln\left\{\left(\frac{5}{4}\right)^A \cdot 2^B\right\}$ |
---9 (i) Express $\frac { x ^ { 2 } - x - 11 } { ( x + 1 ) ( x - 2 ) ^ { 2 } }$ in partial fractions.\\
(ii) Find the exact value of $\int _ { 3 } ^ { 4 } \frac { x ^ { 2 } - x - 11 } { ( x + 1 ) ( x - 2 ) ^ { 2 } } \mathrm {~d} x$, giving your answer in the form $a + \ln b$, where $a$ and $b$ are rational numbers.
\hfill \mbox{\textit{OCR C4 2012 Q9 [9]}}