| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2002 |
| Session | November |
| Paper | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Difficulty | Standard +0.8 This Further Pure 1 question requires polynomial long division, asymptote identification, differentiation using quotient rule, and curve sketching with multiple constraints. While each technique is standard, the multi-step nature, the need to analyze stationary points algebraically, and the constraint a+b>c requiring careful consideration of curve behavior make this moderately challenging for FP1 level. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives |
The curve $C$ has equation
$$y = \frac { ( x - a ) ( x - b ) } { x - c }$$
where $a , b , c$ are constants, and it is given that $0 < a < b < c$.\\
(i) Express $y$ in the form
$$x + P + \frac { Q } { x - c }$$
giving the constants $P$ and $Q$ in terms of $a , b$ and $c$.\\
(ii) Find the equations of the asymptotes of $C$.\\
(iii) Show that $C$ has two stationary points.\\
(iv) Given also that $a + b > c$, sketch $C$, showing the asymptotes and the coordinates of the points of intersection of $C$ with the axes.
\hfill \mbox{\textit{CAIE FP1 2002 Q11 OR}}