| Exam Board | OCR MEI |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2005 |
| Session | January |
| Marks | 14 |
| 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 zeros/asymptotes, analyzing end behavior, curve sketching, and solving a rational inequality. Part (v) requires algebraic manipulation to get a common denominator, factoring a quadratic, and careful sign analysis around discontinuities—more sophisticated than typical A-level pure maths but standard for Further Maths. |
| 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 | Mark | Guidance |
| \(x = \frac{3}{2}\) and \(-1\) | B1, [1] | Both |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x = 2\), \(x = -4\) and \(y = 2\) | B1, B1, B1, [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Large positive \(x\), \(y \to 2^-\) (e.g. consider \(x = 100\)) | B1 | Evidence of method needed for this mark |
| Large negative \(x\), \(y \to 2^+\) (e.g. consider \(x = -100\)) | B1, B1, [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Curve: 3 branches | B1 | Consistent with their (iii) |
| Asymptotes marked | B1 | |
| Correctly located and no extra intercepts | B1, [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(y = 2 \Rightarrow 2 = \frac{(2x-3)(x+1)}{(x+4)(x-2)}\) | M1 | Some attempt at rearrangement |
| \(\Rightarrow 2x^2 + 4x - 16 = 2x^2 - x - 3\) | ||
| \(\Rightarrow x = \frac{13}{5}\) | A1 | May be given retrospectively |
| From sketch, \(y \leq 2\) for \(x \geq \frac{13}{5}\) or \(2 > x > -4\) | A1, B1, [4] | B1 for \(2 > x > -4\) |
# Question 7:
## Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = \frac{3}{2}$ and $-1$ | B1, **[1]** | Both |
## Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x = 2$, $x = -4$ and $y = 2$ | B1, B1, B1, **[3]** | |
## Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Large positive $x$, $y \to 2^-$ (e.g. consider $x = 100$) | B1 | Evidence of method needed for this mark |
| Large negative $x$, $y \to 2^+$ (e.g. consider $x = -100$) | B1, B1, **[3]** | |
## Part (iv):
| Answer | Mark | Guidance |
|--------|------|----------|
| Curve: 3 branches | B1 | Consistent with their (iii) |
| Asymptotes marked | B1 | |
| Correctly located and no extra intercepts | B1, **[3]** | |
## Part (v):
| Answer | Mark | Guidance |
|--------|------|----------|
| $y = 2 \Rightarrow 2 = \frac{(2x-3)(x+1)}{(x+4)(x-2)}$ | M1 | Some attempt at rearrangement |
| $\Rightarrow 2x^2 + 4x - 16 = 2x^2 - x - 3$ | | |
| $\Rightarrow x = \frac{13}{5}$ | A1 | May be given retrospectively |
| From sketch, $y \leq 2$ for $x \geq \frac{13}{5}$ or $2 > x > -4$ | A1, B1, **[4]** | B1 for $2 > x > -4$ |
---7 A curve has equation $y = \frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) }$.
\begin{enumerate}[label=(\roman*)]
\item Write down the values of $x$ for which $y = 0$.
\item Write down the equations of the three asymptotes.
\item Determine whether the curve approaches the horizontal asymptote from above or from below for\\
(A) large positive values of $x$,\\
(B) large negative values of $x$.
\item Sketch the curve.
\item Solve the inequality $\frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) } \leqslant 2$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI FP1 2005 Q7 [14]}}