| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2013 |
| Session | November |
| Topic | Polynomial Division & Manipulation |
| Type | Integration Using Polynomial Division |
| Difficulty | Standard +0.3 This is a straightforward multi-part question requiring polynomial long division (a standard A-level technique) followed by routine integration. Part (i) is basic coordinate finding, and part (ii) guides students through the algebraic manipulation before asking for a definite integral with a given answer to verify. The question is slightly easier than average because it's highly scaffolded and uses standard methods without requiring problem-solving insight. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals1.08j Integration using partial fractions |
12 The diagram shows the curve $y = \frac { x ^ { 2 } - 3 } { x + 1 }$ for $x > - 1$.\\
\includegraphics[max width=\textwidth, alt={}, center]{806dc286-416e-4785-8d13-0d524f808cb0-3_435_874_897_639}\\
(i) Find the coordinates of the points where the curve crosses the axes.\\
(ii) Express $\frac { x ^ { 2 } - 3 } { x + 1 }$ in the form $A x + B + \frac { C } { x + 1 }$, where $A , B$ and $C$ are constants, and hence show that the exact area enclosed by the $x$-axis, the curve $y = \frac { x ^ { 2 } - 3 } { x + 1 }$ and the lines $x = 2$ and $x = 4$ is $4 + \ln \frac { 9 } { 25 }$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2013 Q12}}