| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Sketching Rational Functions with Horizontal Asymptote Only |
| Difficulty | Standard +0.3 This is a standard FP1 rational function sketching question requiring routine techniques: finding intercepts (factored form given), identifying asymptotes (vertical from denominators, horizontal from degree comparison), and determining approach behavior using limit analysis. While it involves multiple steps, each is algorithmic with no novel insight required, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.02y Partial fractions: decompose rational functions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((0,10),(-2,0),\left(\frac{5}{3},0\right)\) | B1, B1, B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = \frac{-1}{2}\), \(x=1\), \(y=\frac{3}{2}\) | B1, B1, B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Large positive \(x\), \(y \to \frac{3}{2}^+\) (e.g. consider \(x=100\)) | M1, B1 | Clear evidence of method required for full marks |
| Large negative \(x\), \(y \to \frac{3}{2}^-\) (e.g. consider \(x=-100\)) | B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Curve: 3 branches of correct shape | B1 | |
| Asymptotes correct and labelled | B1 | |
| Intercepts correct and labelled | B1 [3] |
# Question 7(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(0,10),(-2,0),\left(\frac{5}{3},0\right)$ | B1, B1, B1 **[3]** | |
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# Question 7(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = \frac{-1}{2}$, $x=1$, $y=\frac{3}{2}$ | B1, B1, B1 **[3]** | |
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# Question 7(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Large positive $x$, $y \to \frac{3}{2}^+$ (e.g. consider $x=100$) | M1, B1 | Clear evidence of method required for full marks |
| Large negative $x$, $y \to \frac{3}{2}^-$ (e.g. consider $x=-100$) | B1 **[3]** | |
---
# Question 7(iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Curve: 3 branches of correct shape | B1 | |
| Asymptotes correct and labelled | B1 | |
| Intercepts correct and labelled | B1 **[3]** | |
---7 A curve has equation $y = \frac { ( x + 2 ) ( 3 x - 5 ) } { ( 2 x + 1 ) ( x - 1 ) }$.
\begin{enumerate}[label=(\roman*)]
\item Write down the coordinates of the points where the curve crosses the axes.
\item Write down the equations of the three asymptotes.
\item Determine whether the curve approaches the horizontal asymptote from above or below for\\
(A) large positive values of $x$,\\
(B) large negative values of $x$.
\item Sketch the curve.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2009 Q7 [12]}}