| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic translation and transformation |
| Difficulty | Standard +0.8 This is a multi-step Further Maths question requiring understanding of rational function asymptotes, algebraic manipulation to find constants, and then sketching a related implicit curve involving a square root transformation. Part (i) is moderately routine for FP2, but part (ii) requires careful analysis of how squaring y affects the curve's behavior, asymptotes, and symmetry—this demands solid conceptual understanding beyond standard techniques. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x) |
4\\
\includegraphics[max width=\textwidth, alt={}, center]{074597e7-5bb1-4249-9cfa-784974a6fd2b-2_947_1305_986_420}
The diagram shows the curve with equation
$$y = \frac { a x + b } { x + c }$$
where $a , b$ and $c$ are constants.\\
(i) Given that the asymptotes of the curve are $x = - 1$ and $y = - 2$ and that the curve passes through $( 3,0 )$, find the values of $a , b$ and $c$.\\
(ii) Sketch the curve with equation
$$y ^ { 2 } = \frac { a x + b } { x + c }$$
for the values of $a , b$ and $c$ found in part (i). State the coordinates of any points where the curve crosses the axes, and give the equations of any asymptotes.
\hfill \mbox{\textit{OCR FP2 2010 Q4 [7]}}