| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Parameter values from curve properties |
| Difficulty | Challenging +1.8 This is a comprehensive Further Maths curve sketching question requiring differentiation of a rational function (quotient rule), analysis of stationary points via discriminant conditions, asymptote analysis, and multiple sketches for different parameter ranges. While systematic, it demands sustained multi-step reasoning across calculus, algebra, and coordinate geometry—significantly above standard A-level but typical for FP1. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.02z Models in context: use functions in modelling1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
The curve $C$ has equation
$$y = \frac { x ^ { 2 } + 2 \lambda x } { x ^ { 2 } - 2 x + \lambda }$$
where $\lambda$ is a constant and $\lambda \neq - 1$.\\
(i) Show that $C$ has at most two stationary points.\\
(ii) Show that if $C$ has exactly two stationary points then $\lambda > - \frac { 5 } { 4 }$.\\
(iii) Find the set of values of $\lambda$ such that $C$ has two vertical asymptotes.\\
(iv) Find the $x$-coordinates of the points of intersection of $C$ with
\begin{enumerate}[label=(\alph*)]
\item the $x$-axis,
\item the horizontal asymptote.\\
(v) Sketch $C$ in each of the cases\\
(a) $\lambda < - 2$,\\
(b) $\lambda > 2$.
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2010 Q12 EITHER}}