| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2004 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Find asymptotes and sketch rational curve |
| Difficulty | Standard +0.8 This Further Pure 1 question requires finding vertical and oblique/horizontal asymptotes for different parameter values, then sketching a curve with a removable discontinuity (when λ=-1, a factor cancels). It combines algebraic manipulation, asymptote analysis, and curve sketching with multiple cases—more demanding than standard C3 rational functions but routine for FP1 students familiar with parametric variations. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02y Partial fractions: decompose rational functions |
10 The curve $C$ has equation
$$y = \frac { x ^ { 2 } + 2 x - 3 } { ( \lambda x + 1 ) ( x + 4 ) }$$
where $\lambda$ is a constant.\\
(i) Find the equations of the asymptotes of $C$ for the case where $\lambda = 0$.\\
(ii) Find the equations of the asymptotes of $C$ for the case where $\lambda$ is not equal to any of $- 1,0 , \frac { 1 } { 4 } , \frac { 1 } { 3 }$.\\
(iii) Sketch $C$ for the case where $\lambda = - 1$. Show, on your diagram, the equations of the asymptotes and the coordinates of the points of intersection of $C$ with the coordinate axes.
\hfill \mbox{\textit{CAIE FP1 2004 Q10 [11]}}