| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Solving Inequalities with Rational Functions |
| Difficulty | Standard +0.8 This FP1 question requires multiple techniques: finding intercepts and asymptotes (routine), analyzing end behavior with justification (moderate), and solving a rational inequality requiring algebraic manipulation and sign analysis across multiple intervals defined by discontinuities. The inequality in part (iv) is non-trivial as students must rearrange, find a common denominator, and carefully consider signs around vertical asymptotes—more demanding than standard A-level pure questions. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02k 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 |
| \(\left(0, -\frac{1}{2}\right)\) | B1 | |
| \((-3, 0),\ \left(\frac{1}{2}, 0\right)\) | B1 [2] | For both |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = 3,\ x = 2\) and \(y = 2\) | B1, B1, B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Large positive \(x\), \(y \to 2^+\) (e.g. substitute \(x=100\) to give \(2.15\ldots\), or convincing algebraic argument) | M1, A1 | Must show evidence of method; A0 if no valid method |
| Correct RH branch (sketch) | B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\frac{(2x-1)(x+3)}{(x-3)(x-2)} = 2\) | M1 | Or other valid method to find intersection with horizontal asymptote |
| \(\Rightarrow (2x-1)(x+3) = 2(x-3)(x-2)\) | ||
| \(\Rightarrow x = 1\) | A1 | |
| From graph \(x < 1\) or \(2 < x < 3\) | B1, B1 [4] | For \(x < 1\); for \(2 < x < 3\) |
# Question 7(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\left(0, -\frac{1}{2}\right)$ | B1 | |
| $(-3, 0),\ \left(\frac{1}{2}, 0\right)$ | B1 **[2]** | For both |
---
# Question 7(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = 3,\ x = 2$ and $y = 2$ | B1, B1, B1 **[3]** | |
---
# Question 7(iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Large positive $x$, $y \to 2^+$ (e.g. substitute $x=100$ to give $2.15\ldots$, or convincing algebraic argument) | M1, A1 | Must show evidence of method; A0 if no valid method |
| Correct RH branch (sketch) | B1 **[3]** | |
---
# Question 7(iv):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\frac{(2x-1)(x+3)}{(x-3)(x-2)} = 2$ | M1 | Or other valid method to find intersection with horizontal asymptote |
| $\Rightarrow (2x-1)(x+3) = 2(x-3)(x-2)$ | | |
| $\Rightarrow x = 1$ | A1 | |
| From graph $x < 1$ or $2 < x < 3$ | B1, B1 **[4]** | For $x < 1$; for $2 < x < 3$ |
---7 Fig. 7 shows an incomplete sketch of $y = \frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-3_786_1376_450_386}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}
(i) Find the coordinates of the points where the curve cuts the axes.\\
(ii) Write down the equations of the three asymptotes.\\
(iii) Determine whether the curve approaches the horizontal asymptote from above or below for large positive values of $x$, justifying your answer. Copy and complete the sketch.\\
(iv) Solve the inequality $\frac { ( 2 x - 1 ) ( x + 3 ) } { ( x - 3 ) ( x - 2 ) } < 2$.
\hfill \mbox{\textit{OCR MEI FP1 2010 Q7 [12]}}