| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2009 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Rational curve sketching with asymptotes and inequalities |
| Difficulty | Moderate -0.3 This is a routine Further Maths curve sketching question requiring identification of vertical asymptotes (from denominator zeros), horizontal asymptote (from degree comparison), x-intercepts (from numerator zeros), and sign analysis. While it involves multiple steps, each is standard technique with no novel insight required. The 'no stationary points' hint simplifies the sketch, and part (b) follows directly from the sketch. |
| 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 | Marks | Guidance |
| Asymptotes \(x=0\), \(x=2\), \(y=1\) | \(\text{B1}\times3\) | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Intersections at \((1,0)\) and \((3,0)\) | B1 | |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| At least one branch approaching asymptotes | B1 | |
| Each branch | \(\text{B1}\times3\) | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(0 < x < 1,\ 2 < x < 3\) | B1, B1 | Allow B1 if one repeated error occurs, eg \(\leq\) for \(<\) |
| Alternative: Complete correct algebraic method | M1A1 | (2) |
| Total | 10 |
## Question 6(a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Asymptotes $x=0$, $x=2$, $y=1$ | $\text{B1}\times3$ | |
| **Total** | **3** | |
## Question 6(a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Intersections at $(1,0)$ and $(3,0)$ | B1 | |
| **Total** | **1** | |
## Question 6(a)(iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| At least one branch approaching asymptotes | B1 | |
| Each branch | $\text{B1}\times3$ | |
| **Total** | **4** | |
## Question 6(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $0 < x < 1,\ 2 < x < 3$ | B1, B1 | Allow B1 if one repeated error occurs, eg $\leq$ for $<$ |
| **Alternative:** Complete correct algebraic method | M1A1 | (2) |
| **Total** | **10** | |
---6 A curve has equation
$$y = \frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) }$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the equations of the three asymptotes of this curve.
\item State the coordinates of the points at which the curve intersects the $x$-axis.
\item Sketch the curve.\\
(You are given that the curve has no stationary points.)
\end{enumerate}\item Hence, or otherwise, solve the inequality
$$\frac { ( x - 1 ) ( x - 3 ) } { x ( x - 2 ) } < 0$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2009 Q6 [10]}}