| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Stationary Points of Rational Functions |
| Difficulty | Challenging +1.2 This is a multi-part Further Maths question requiring differentiation of a rational function using the quotient rule, finding asymptotes, and curve sketching. While it involves several steps and techniques, each part follows standard procedures: asymptotes are routine, the intersection is algebraic manipulation, finding stationary points requires quotient rule differentiation and discriminant analysis, and sketching uses standard rational function behavior. The question is methodical rather than requiring novel insight, making it moderately above average difficulty for A-level but standard for FP1. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.07n Stationary points: find maxima, minima using derivatives |
The curve $C$ has equation
$$y = \frac { ( x - 2 ) ( x - a ) } { ( x - 1 ) ( x - 3 ) } ,$$
where $a$ is a constant not equal to 1,2 or 3 .\\
(i) Write down the equations of the asymptotes of $C$.\\
(ii) Show that $C$ meets the asymptote parallel to the $x$-axis at the point where $x = \frac { 2 a - 3 } { a - 2 }$.\\
(iii) Show that the $x$-coordinates of any stationary points on $C$ satisfy
$$( a - 2 ) x ^ { 2 } + ( 6 - 4 a ) x + ( 5 a - 6 ) = 0$$
and hence find the set of values of $a$ for which $C$ has stationary points.\\
(iv) Sketch the graph of $C$ for
\begin{enumerate}[label=(\alph*)]
\item $a > 3$,
\item $2 < a < 3$.
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2008 Q12 EITHER}}