| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Solving Inequalities with Rational Functions |
| Difficulty | Standard +0.3 This is a structured multi-part question on rational functions requiring identification of intercepts, asymptotes, sketching, and solving an inequality. While it involves Further Maths content (FP1), the tasks are largely routine: finding zeros/asymptotes by inspection, analyzing sign changes, and applying standard inequality techniques. The main challenge is careful bookkeeping of sign intervals around multiple asymptotes, but no novel insight is required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((1,\ 0)\) and \(\left(0,\ \frac{1}{18}\right)\) | B1, B1 [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x=2,\ x=-3,\ x=-\frac{3}{2},\ y=0\) | B4 [4] | Minus 1 for each error |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Sketch of curve | B1 | Correct approaches to vertical asymptotes |
| B1 [2] | Through clearly marked \((1,0)\) and \(\left(0,\frac{1}{18}\right)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x < -3,\ x > 2\) | B1 | |
| \(-\frac{3}{2} < x \leq 1\) | B2 [3] | B1 for \(-\frac{3}{2} < x < 1\), or \(-\frac{3}{2} \leq x \leq 1\) |
# Question 7(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(1,\ 0)$ and $\left(0,\ \frac{1}{18}\right)$ | B1, B1 **[2]** | |
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# Question 7(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x=2,\ x=-3,\ x=-\frac{3}{2},\ y=0$ | B4 **[4]** | Minus 1 for each error |
---
# Question 7(iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Sketch of curve | B1 | Correct approaches to vertical asymptotes |
| | B1 **[2]** | Through clearly marked $(1,0)$ and $\left(0,\frac{1}{18}\right)$ |
---
# Question 7(iv):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x < -3,\ x > 2$ | B1 | |
| $-\frac{3}{2} < x \leq 1$ | B2 **[3]** | B1 for $-\frac{3}{2} < x < 1$, or $-\frac{3}{2} \leq x \leq 1$ |
---7 The sketch below shows part of the graph of $y = \frac { x - 1 } { ( x - 2 ) ( x + 3 ) ( 2 x + 3 ) }$. One section of the graph has been omitted.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{225bff01-f2c4-421f-ac91-c6a0fcb01e6f-3_842_1198_477_552}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}
(i) Find the coordinates of the points where the curve crosses the axes.\\
(ii) Write down the equations of the three vertical asymptotes and the one horizontal asymptote.\\
(iii) Copy the sketch and draw in the missing section.\\
(iv) Solve the inequality $\frac { x - 1 } { ( x - 2 ) ( x + 3 ) ( 2 x + 3 ) } \geqslant 0$.
\hfill \mbox{\textit{OCR MEI FP1 2008 Q7 [11]}}