| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Integration with Partial Fractions |
| Type | Rational function curve sketching |
| Difficulty | Standard +0.8 This question requires polynomial long division to find the oblique asymptote, identification of the vertical asymptote, analysis of behavior near asymptotes and intercepts, and accurate sketching for two cases with different qualitative behaviors. While the techniques are standard for Further Maths, the multi-step analysis and need to distinguish between two topologically different cases elevates this above routine exercises. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials |
The curve $C$ has equation
$$y = \frac{x^2}{x + \lambda},$$
where $\lambda$ is a non-zero constant. Obtain the equations of the asymptotes of $C$. [3]
In separate diagrams, sketch $C$ for the cases where
\begin{enumerate}[label=(\roman*)]
\item $\lambda > 0$,
\item $\lambda < 0$.
\end{enumerate}
[4]
\hfill \mbox{\textit{CAIE FP1 2005 Q3 [7]}}