| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| 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 routine Further Pure 1 question on rational functions requiring standard techniques: finding intercepts by setting x=0 and y=0, identifying asymptotes from denominator zeros and degree comparison, sketching behavior near asymptotes, and solving an inequality using sign analysis. While it's multi-part with 4 sections, each part follows textbook procedures with no novel insight required. The FP1 context places it slightly above average A-level difficulty, but it remains a standard exercise. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| \((\sqrt{3},\ 0),\ (-\sqrt{3},\ 0),\ \left(0,\ \frac{3}{8}\right)\) | B1, B1 [2] | Intercepts with \(x\) axis (both); intercept with \(y\) axis. SC1 if seen on graph or if \(x=\pm\sqrt{3}\), \(y=3/8\) seen without \(y=0\), \(x=0\) specified |
| Answer | Marks | Guidance |
|---|---|---|
| \(x=4,\ x=-2,\ y=1\) | B3 [3] | Minus 1 for each error. Accept equations written on the graph |
| Answer | Marks | Guidance |
|---|---|---|
| *(Graph described)* | B1 | Correct approaches to vertical asymptotes, LH and RH branches |
| B1B1 | LH and RH branches approaching horizontal asymptote | |
| B1 [4] | On LH branch \(0|
| |
| Answer | Marks | Guidance |
|---|---|---|
| \(-2 | B1 | LH interval and RH interval correct (award even if errors in inclusive/exclusive signs) |
| B2 [3] | All inequality signs correct, minus 1 each error |
# Question 8(i):
$(\sqrt{3},\ 0),\ (-\sqrt{3},\ 0),\ \left(0,\ \frac{3}{8}\right)$ | B1, B1 [2] | Intercepts with $x$ axis (both); intercept with $y$ axis. SC1 if seen on graph or if $x=\pm\sqrt{3}$, $y=3/8$ seen without $y=0$, $x=0$ specified
# Question 8(ii):
$x=4,\ x=-2,\ y=1$ | B3 [3] | Minus 1 for each error. Accept equations written on the graph
# Question 8(iii):
*(Graph described)* | B1 | Correct approaches to vertical asymptotes, LH and RH branches
| B1B1 | LH and RH branches approaching horizontal asymptote
| B1 [4] | On LH branch $0<y<1$ as $x\to-\infty$
# Question 8(iv):
$-2<x\leq-\sqrt{3}$ and $4>x\geq\sqrt{3}$ | B1 | LH interval and RH interval correct (award even if errors in inclusive/exclusive signs)
| B2 [3] | All inequality signs correct, minus 1 each error
---8 Fig. 8 shows part of the graph of $y = \frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) }$. Two sections of the graph have been omitted.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{35094899-149c-438e-b6c8-b333d2fefc0c-3_725_1025_1160_559}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{center}
\end{figure}
(i) Write down the coordinates of the points where the curve crosses the axes.\\
(ii) Write down the equations of the two vertical asymptotes and the one horizontal asymptote.\\
(iii) Copy Fig. 8 and draw in the two missing sections.\\
(iv) Solve the inequality $\frac { x ^ { 2 } - 3 } { ( x - 4 ) ( x + 2 ) } \leqslant 0$.
\hfill \mbox{\textit{OCR MEI FP1 2009 Q8 [12]}}