| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2012 |
| Session | Specimen |
| Marks | 9 |
| 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 standard integration application. Part (i) requires routine algebraic manipulation (cover-up method or equating coefficients), and part (ii) involves integrating ln and a power function with substitution of limits—all standard A-level techniques with no novel problem-solving required. Slightly easier than average due to the simple two-term denominator and clear structure. |
| Spec | 1.08j Integration using partial fractions4.05c Partial fractions: extended to quadratic denominators |
| Answer | Marks | Guidance |
|---|---|---|
| - Obtain \(3\ln | x-2 | - \frac{8}{x-2}\) [B1] |
**(i)**
- $3x + 2 = A(x-2) + B$ [M1]
- $A = 3$ [B1]
- $B = 8$ [B1]
**(ii)**
- Obtain $k\ln(x-2)$ [B1]
- Obtain $-\frac{P}{x-2}$ [B1]
- Obtain $3\ln|x-2| - \frac{8}{x-2}$ [B1]
- Use limits in correct order [M1]
- Attempt use of log law [M1]
- Obtain $3\ln 2 + 1$ [A1]
**Total: 9 marks**8 (i) Express $\frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } }$ 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 $\int _ { 6 } ^ { 10 } \frac { 3 x + 2 } { ( x - 2 ) ^ { 2 } } \mathrm {~d} x$, giving your answer in the form $a + b \ln c$, where $a , b$ and $c$ are integers.
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2012 Q8 [9]}}