| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2006 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Inequalities |
| Type | Rational inequality algebraically |
| Difficulty | Standard +0.3 This is a straightforward FP1 question involving a rational function. Part (a) requires identifying vertical and horizontal asymptotes by inspection, part (b) is a standard sketch, and part (c) involves solving a rational inequality by rearranging to standard form and considering sign changes—all routine techniques for Further Maths students with no novel problem-solving required. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Asymptotes \(x = 1\), \(y = 6\) | B1B1 | SC Only one branch: |
| Curve (correct general shape) | M1 | B1 for origin |
| Curve passing through origin | A1 | B1 for approaching both asymptotes |
| Both branches approach \(x = 1\) | A1 | ( Max 2/4) |
| Both branches approaching \(y = 6\) | A1 | |
| (c) Correct method | M1 | |
| Critical values \(\pm 1\) | B1B1 | From graph or calculation |
| Solution set \(-1 < x < 1\) | A1√ | ft one error in CVs; NMS 4/4 after a good graph |
**(a)** Asymptotes $x = 1$, $y = 6$ | B1B1 | SC Only one branch:
Curve (correct general shape) | M1 | B1 for origin
Curve passing through origin | A1 | B1 for approaching both asymptotes
Both branches approach $x = 1$ | A1 | ( Max 2/4)
Both branches approaching $y = 6$ | A1 |
**(c)** Correct method | M1
Critical values $\pm 1$ | B1B1 | From graph or calculation
Solution set $-1 < x < 1$ | A1√ | ft one error in CVs; NMS 4/4 after a good graph
**Total for Q4: 10 marks**
---4 A curve has equation
$$y = \frac { 6 x } { x - 1 }$$
\begin{enumerate}[label=(\alph*)]
\item Write down the equations of the two asymptotes to the curve.
\item Sketch the curve and the two asymptotes.
\item Solve the inequality
$$\frac { 6 x } { x - 1 } < 3$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2006 Q4 [10]}}