| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| 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 by setting x=0 and y=0, identifying vertical asymptotes from denominator zeros, finding horizontal asymptote by comparing degrees, and determining approach direction by testing large x values. All steps are 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 | Mark | Guidance |
| \((0, 18)\) | B1 | |
| \((-9, 0)\), \(\left(\frac{8}{3}, 0\right)\) | B1, B1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = 2\), \(x = -2\) and \(y = 3\) | B1, B1, B1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Large positive \(x\), \(y \to 3^+\) from above | B1 | |
| Large negative \(x\), \(y \to 3^-\) from below | B1 | |
| (e.g. consider \(x = 100\), or convincing algebraic argument) | M1 | Must show evidence of working |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Graph with 3 branches correct | B1 | 3 branches correct |
| Asymptotes correct and labelled | B1 | Asymptotes correct and labelled |
| Intercepts correct and labelled | B1 | Intercepts correct and labelled |
| [3] |
# Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $(0, 18)$ | B1 | |
| $(-9, 0)$, $\left(\frac{8}{3}, 0\right)$ | B1, B1 | |
| | **[3]** | |
---
# Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = 2$, $x = -2$ and $y = 3$ | B1, B1, B1 | |
| | **[3]** | |
---
# Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Large positive $x$, $y \to 3^+$ from above | B1 | |
| Large negative $x$, $y \to 3^-$ from below | B1 | |
| (e.g. consider $x = 100$, or convincing algebraic argument) | M1 | Must show evidence of working |
| | **[3]** | |
---
# Question 7(iv):
| Answer | Mark | Guidance |
|--------|------|----------|
| Graph with 3 branches correct | B1 | 3 branches correct |
| Asymptotes correct and labelled | B1 | Asymptotes correct and labelled |
| Intercepts correct and labelled | B1 | Intercepts correct and labelled |
| | **[3]** | |
---7 A curve has equation $y = \frac { ( x + 9 ) ( 3 x - 8 ) } { x ^ { 2 } - 4 }$.
\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 2011 Q7 [12]}}