| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Stationary Points of Rational Functions |
| Difficulty | Challenging +1.2 This is a comprehensive rational function question requiring multiple techniques (asymptotes, differentiation using quotient rule, stationary points, sketching), but each component is relatively standard for Further Pure 1. The algebraic manipulation and quotient rule differentiation are moderately demanding but follow established procedures without requiring novel insight. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives |
The curve $C$ has equation
$$y = \frac { x ( x + 1 ) } { ( x - 1 ) ^ { 2 } }$$
(i) Obtain the equations of the asymptotes of $C$.\\
(ii) Show that there is exactly one point of intersection of $C$ with the asymptotes and find its coordinates.\\
(iii) Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$ and hence
\begin{enumerate}[label=(\alph*)]
\item find the coordinates of any stationary points of $C$,
\item state the set of values of $x$ for which the gradient of $C$ is negative.\\
(iv) Draw a sketch of $C$.
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2010 Q11 OR}}