| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch rational with linear numerator |
| Difficulty | Moderate -0.3 This is a standard FP1 rational function question requiring identification of asymptotes (vertical at x=-2, horizontal at y=3), finding intercepts, and sketching. Part (c) uses the sketch to solve an inequality. While it requires multiple techniques, these are routine procedures for Further Maths students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02i Represent inequalities: graphically on coordinate plane1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.02o Sketch reciprocal curves: y=a/x and y=a/x^2 |
| Answer | Marks |
|---|---|
| (a) Asymptotes \(x = -2\), \(y = 3\) | B1,B1 |
| Answer | Marks |
|---|---|
| (b) Curve approaching asymptotes | B1 |
| Passing through \(\left(\frac{1}{3}, 0\right)\) and \(\left(0, -\frac{1}{2}\right)\) | B1,B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (c) Solution set is \(x > \frac{1}{3}\) | B2,1F | B1 for good attempt; ft wrong point of intersection |
**(a)** Asymptotes $x = -2$, $y = 3$ | B1,B1 |
**Total: 2 marks**
**(b)** Curve approaching asymptotes | B1 |
Passing through $\left(\frac{1}{3}, 0\right)$ and $\left(0, -\frac{1}{2}\right)$ | B1,B1 |
**Total: 5 marks**
**(c)** Solution set is $x > \frac{1}{3}$ | B2,1F | B1 for good attempt; ft wrong point of intersection
**Total: 2 marks**
---7 A curve has equation
$$y = \frac { 3 x - 1 } { x + 2 }$$
\begin{enumerate}[label=(\alph*)]
\item Write down the equations of the two asymptotes to the curve.
\item Sketch the curve, indicating the coordinates of the points where the curve intersects the coordinate axes.
\item Hence, or otherwise, solve the inequality
$$0 < \frac { 3 x - 1 } { x + 2 } < 3$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q7 [9]}}