| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Polynomial Division & Manipulation |
| Type | Solving Inequalities with Rational Functions |
| Difficulty | Challenging +1.2 This is a multi-part question on rational functions requiring identification of asymptotes from a graph, sketching behavior, and solving a rational inequality. While it involves several steps (finding constants, analyzing end behavior, solving equations and inequalities), each component uses standard FP1 techniques without requiring novel insight. The inequality solving is slightly elevated above routine A-level work due to the rational function context, but remains a textbook-style question for Further Maths students. |
| 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 |
|---|---|---|
| Vertical asymptotes at \(x = -2\) and \(x = \frac{1}{2}\) occur when \((bx-1)(x+a) = 0\) | M1 | Some evidence of valid reasoning — may be implied |
| \(\Rightarrow a = 2\) and \(b = 2\) | A1 A1 |
| Answer | Marks |
|---|---|
| \(\frac{cx^2}{(2x-1)(x+2)} \to \frac{3}{2} \Rightarrow c = 3\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Valid reasoning seen | M1 | Some evidence of method needed e.g. substitute in 'large' values with result |
| Answer | Marks | Guidance |
|---|---|---|
| Large negative \(x\), \(y \to \frac{3}{2}\) from above | A1 | Both approaches correct (correct \(b,c\)) |
| [Sketch with asymptotes \(x = -2\), \(x = \frac{1}{2}\), \(y = \frac{3}{2}\)] | B1 | LH branch correct |
| B1 | RH branch correct; each one carefully drawn |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{3x^2}{(2x-1)(x+2)} = 1 \Rightarrow 3x^2 = (2x-1)(x+2)\) | M1 | Or other valid method; explicit values of \(x\) |
| Answer | Marks |
|---|---|
| \(\Rightarrow x = 1\) or \(x = 2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| for \(-2 < x < \frac{1}{2}\) or \(1 < x < 2\) | B1 B1 | FT their \(x=1,2\) provided \(>1/2\) |
## Question 7:
### Part (i):
Vertical asymptotes at $x = -2$ and $x = \frac{1}{2}$ occur when $(bx-1)(x+a) = 0$ | M1 | Some evidence of valid reasoning — may be implied
$\Rightarrow a = 2$ and $b = 2$ | A1 A1 |
Horizontal asymptote at $y = \frac{3}{2}$ so when $x$ gets very large,
$\frac{cx^2}{(2x-1)(x+2)} \to \frac{3}{2} \Rightarrow c = 3$ | A1 |
**[4]**
### Part (ii):
Valid reasoning seen | M1 | Some evidence of method needed e.g. substitute in 'large' values with result
Large positive $x$, $y \to \frac{3}{2}$ from below
Large negative $x$, $y \to \frac{3}{2}$ from above | A1 | Both approaches correct (correct $b,c$)
[Sketch with asymptotes $x = -2$, $x = \frac{1}{2}$, $y = \frac{3}{2}$] | B1 | LH branch correct
| B1 | RH branch correct; each one carefully drawn
**[4]**
### Part (iii):
$\frac{3x^2}{(2x-1)(x+2)} = 1 \Rightarrow 3x^2 = (2x-1)(x+2)$ | M1 | Or other valid method; explicit values of $x$
$\Rightarrow 0 = (x-2)(x-1)$
$\Rightarrow x = 1$ or $x = 2$ | A1 |
From the graph $\frac{3x^2}{(2x-1)(x+2)} < 1$
for $-2 < x < \frac{1}{2}$ or $1 < x < 2$ | B1 B1 | FT their $x=1,2$ provided $>1/2$
**[4]**7 Fig. 7 shows an incomplete sketch of $y = \frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) }$ where $a , b$ and $c$ are integers. The asymptotes of the curve are also shown.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{597abea9-6d00-416e-9203-d5bce9bd1af1-3_928_996_493_535}
\captionsetup{labelformat=empty}
\caption{Fig. 7}
\end{center}
\end{figure}
(i) Determine the values of $a , b$ and $c$.
Use these values of $a , b$ and $c$ throughout the rest of the question.\\
(ii) Determine how the curve approaches the horizontal asymptote for large positive values of $x$, and for large negative values of $x$, justifying your answer. On the copy of Fig. 7, sketch the rest of the curve.\\
(iii) Find the $x$ coordinates of the points on the curve where $y = 1$. Write down the solution to the inequality $\frac { c x ^ { 2 } } { ( b x - 1 ) ( x + a ) } < 1$.
\hfill \mbox{\textit{OCR MEI FP1 2013 Q7 [12]}}