| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Session | Specimen |
| Marks | 8 |
| Topic | Partial Fractions |
| Type | Partial fractions with quadratic in denominator |
| Difficulty | Standard +0.3 This is a standard partial fractions question with a linear and irreducible quadratic factor, followed by routine integration. The decomposition form is given, requiring only algebraic manipulation to find constants, then straightforward application of logarithmic and arctangent integration formulas. Slightly easier than average due to the provided form and standard techniques. |
| Spec | 1.02y Partial fractions: decompose rational functions1.08d Evaluate definite integrals: between limits |
$A = 1$ B1
Method to determine $B$ and $C$ with at least fractions removed M1
$B = -3$ A1
$C = 0$ A1
Attempt to integrate separate terms and obtain at least two log expressions of the form $m\ln(x+2)$ *OR* $n\ln(x^2+1)$ M1
Obtain $\ln(x+2) - \frac{3}{2}\ln(x^2+1)$ A1
Attempt to evaluate $F(1) - F(0)$ M1
Obtain $\ln 3 - \frac{3}{2}\ln 2 - \ln 2$ at least and hence $\ln 3 - \frac{5}{2}\ln 2$ AG A1
**Total: 8**7 Express $\frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) }$ in the form $\frac { A } { x + 2 } + \frac { B x + C } { x ^ { 2 } + 1 }$ where the numerical values of $A , B$ and $C$ are to be found.
Hence show that $\int _ { 0 } ^ { 1 } \frac { 1 - 6 x - 2 x ^ { 2 } } { ( x + 2 ) \left( x ^ { 2 } + 1 \right) } \mathrm { d } x = \ln 3 - \frac { 5 } { 2 } \ln 2$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 Q7 [8]}}