| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2007 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Rational curve sketching with asymptotes and inequalities |
| Difficulty | Standard +0.3 This is a straightforward Further Pure 1 question on rational functions requiring identification of vertical asymptotes (x = ±1) and horizontal asymptote (y = 0), a standard sketch, and solving an inequality using sign analysis. While FP1 content, it's routine application of asymptote rules and sign diagrams with no novel problem-solving required. |
| 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 |
|---|---|---|
| Asymptotes \(y = 0\), \(x = -1\), \(x = 1\) | B1 × 3 | 3 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Three branches approaching two vertical asymptotes | B1 | |
| Middle branch passing through \(O\) | B1 | |
| Curve approaching \(y = 0\) as \(x \to \pm \infty\) | B1 | |
| All correct | B1 | 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Critical values \(x = -1, 0\) and \(1\) | B1 | |
| Solution set \(-1 < x < 0, x > 1\) | M1A1 | 3 marks |
### Part (a)
Asymptotes $y = 0$, $x = -1$, $x = 1$ | B1 × 3 | 3 marks |
### Part (b)
Three branches approaching two vertical asymptotes | B1 | | Asymptotes not necessarily drawn
Middle branch passing through $O$ | B1 | | with no stationary points
Curve approaching $y = 0$ as $x \to \pm \infty$ | B1 | |
All correct | B1 | 4 marks | with asymptotes shown and curve approaching all asymptotes correctly
### Part (c)
Critical values $x = -1, 0$ and $1$ | B1 | |
Solution set $-1 < x < 0, x > 1$ | M1A1 | 3 marks | M1 if one part correct or consistent with c's graph
### **Total for Question 5: 10 marks**
---5 A curve has equation
$$y = \frac { x } { x ^ { 2 } - 1 }$$
\begin{enumerate}[label=(\alph*)]
\item Write down the equations of the three asymptotes to the curve.
\item Sketch the curve.\\
(You are given that the curve has no stationary points.)
\item Solve the inequality
$$\frac { x } { x ^ { 2 } - 1 } > 0$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2007 Q5 [10]}}