| Exam Board | OCR MEI |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Conic sections |
| Type | Conic translation and transformation |
| Difficulty | Standard +0.8 This question requires polynomial division to rewrite the equation, identifying when a rational function becomes linear (requiring algebraic manipulation), recognizing a rectangular hyperbola from its equation form, finding asymptotes, and producing an accurate sketch with specific features. While individual steps are accessible, the multi-part nature, need to connect algebraic form to conic type, and requirement for precise sketching with labeled points makes this moderately challenging for Further Pure students. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.02q Use intersection points: of graphs to solve equations1.03a Straight lines: equation forms y=mx+c, ax+by+c=0 |
5 The curve with equation $y = \frac { x ^ { 2 } - k x + 2 k } { x + k }$ is to be investigated for different values of $k$.
\begin{enumerate}[label=(\roman*)]
\item Use your graphical calculator to obtain rough sketches of the curve in the cases $k = - 2$, $k = - 0.5$ and $k = 1$.
\item Show that the equation of the curve may be written as $y = x - 2 k + \frac { 2 k ( k + 1 ) } { x + k }$.
Hence find the two values of $k$ for which the curve is a straight line.
\item When the curve is not a straight line, it is a conic.\\
(A) Name the type of conic.\\
(B) Write down the equations of the asymptotes.
\item Draw a sketch to show the shape of the curve when $1 < k < 8$. This sketch should show where the curve crosses the axes and how it approaches its asymptotes. Indicate the points A and B on the curve where $x = 1$ and $x = k$ respectively.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP2 2007 Q5 [18]}}