| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | November |
| Paper | 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 polynomial division, asymptote identification, differentiation of rational functions, and curve sketching. While it involves several techniques and is from FP1 (inherently harder), each part follows standard procedures: (i) comparing coefficients for oblique asymptote, (ii) using dy/dx=0 for turning points, (iii) analyzing range restrictions, (iv) applying transformations. The question is methodical rather than requiring novel insight, making it moderately above average difficulty but accessible to well-prepared Further Maths students. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations1.02s Modulus graphs: sketch graph of |ax+b| |
The curve $C$ has equation
$$y = \frac { a x ^ { 2 } + b x + c } { x + 4 }$$
where $a$, $b$ and $c$ are constants. It is given that $y = 2 x - 5$ is an asymptote of $C$.\\
(i) Find the values of $a$ and $b$.\\
(ii) Given also that $C$ has a turning point at $x = - 1$, find the value of $c$.\\
(iii) Find the set of values of $y$ for which there are no points on $C$.\\
(iv) Draw a sketch of the curve with equation
$$y = \frac { 2 ( x - 7 ) ^ { 2 } + 3 ( x - 7 ) - 2 } { x - 3 }$$
[You should state the equations of the asymptotes and the coordinates of the turning points.]
\hfill \mbox{\textit{CAIE FP1 2007 Q12 EITHER}}