| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Solving Inequalities with Rational Functions |
| Difficulty | Standard +0.3 This is a standard FP1 rational function question requiring identification of asymptotes, behavior analysis, sketching, and solving an inequality. While it involves multiple parts, each step follows routine procedures: vertical asymptotes from denominators, horizontal from degree comparison, sign analysis for the inequality. The techniques are well-practiced in FP1 with no novel insight required, making it slightly easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02n Sketch curves: simple equations including polynomials1.02y Partial fractions: decompose rational functions |
| Answer | Marks |
|---|---|
| \(x = 3\), \(x = -2\), \(y = 2\) | B1, B1, B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Large positive \(x\), \(y \rightarrow 2^+\) (e.g. consider \(x = 100\)); Large negative \(x\), \(y \rightarrow 2^-\) (e.g. consider \(x = -100\)) | M1, B1, B1 [3] | Evidence of method required |
| Curve: Central and RH branches correct; Asymptotes correct and labelled; LH branch correct, with clear minimum | B1, B1, B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| \(-2 < x < 3\), \(x \neq 0\) | B2, B1 [3] | B2 max if any inclusive inequalities appear; B3 for \(-2 < x < 0\) and \(0 < x < 3\) |
# Question 8:
## Question 8(i):
| $x = 3$, $x = -2$, $y = 2$ | B1, B1, B1 [3] | |
## Question 8(ii) & 8(iii):
| Large positive $x$, $y \rightarrow 2^+$ (e.g. consider $x = 100$); Large negative $x$, $y \rightarrow 2^-$ (e.g. consider $x = -100$) | M1, B1, B1 [3] | Evidence of method required |
| Curve: Central and RH branches correct; Asymptotes correct and labelled; LH branch correct, with clear minimum | B1, B1, B1 [3] | |
## Question 8(iv):
| $-2 < x < 3$, $x \neq 0$ | B2, B1 [3] | B2 max if any inclusive inequalities appear; B3 for $-2 < x < 0$ and $0 < x < 3$ |
---8 A curve has equation $y = \frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) }$.
\begin{enumerate}[label=(\roman*)]
\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.
\item Solve the inequality $\frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) } < 0$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2008 Q8 [12]}}