| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Topic | Partial Fractions |
| Type | Factor polynomial then partial fractions |
| Difficulty | Standard +0.8 This question requires factorising a cubic (likely by trial of factors), decomposing into partial fractions with three linear factors, then integrating. While each step is standard A-level technique, the combination of cubic factorisation plus partial fractions with three terms makes this moderately harder than typical textbook exercises, though still within standard A-level scope. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.02y Partial fractions: decompose rational functions1.08j Integration using partial fractions |
Identify a correct factor — B1
Attempt division or coefficient matching for their factor — M1
Obtain a quadratic quotient — M1
Obtain $(x+3)(x-1)^2$ — A1
State partial fractions of form $\dfrac{A}{x+3} + \dfrac{B}{x-1} + \dfrac{C}{(x-1)^2}$ — B1
Attempt to remove fractions from partial fractions in the form above or as in the SR — M1*
Attempt to find $A$, $B$ and $C$ — M1*
Obtain any two of $A = 1$, $B = 1$ and $C = 1$ — A1*
Obtain all three values — A1*
Obtain $A\ln(x+3)$ — B1
$B\ln(x-1)$ — B1
$-\dfrac{c}{x-1}$ — B1
SR partial fractions may also be of the form $\dfrac{A}{x+3} + \dfrac{Bx+c}{(x-1)^2}$ **[12]**13 By first factorising completely $x ^ { 3 } + x ^ { 2 } - 5 x + 3$, find $\int \frac { 2 x ^ { 2 } + x + 1 } { x ^ { 3 } + x ^ { 2 } - 5 x + 3 } \mathrm {~d} x$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2013 Q13 [12]}}