| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2011 |
| Session | June |
| 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 FP1 curve sketching question requiring identification of vertical asymptotes at x=±2 and horizontal asymptote at y=0, plus a basic sketch. Part (b) involves solving a rational inequality using the sketch, which is routine once the curve is drawn. The techniques are standard and require no novel insight. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02o Sketch reciprocal curves: y=a/x and y=a/x^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Asymptotes \(x = -2\), \(x = 2\), \(y = 0\) | \(\text{B1} \times 3\) | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Middle branch generally correct | B1 | Allow if max pt not in right place |
| Other branches generally correct | B1 | |
| All branches approaching asymps | B1 | Asymps must be shown correctly on diagram or elsewhere; B0 if any other intersections are shown |
| Intersection at \(\left(0, -\frac{1}{4}\right)\) indicated | B1 | |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = -2\) when \(x = \pm\sqrt{3.5}\) | B1 | Allow NMS |
| Sol'n \(-2 < x < -\sqrt{3.5}\), \(\sqrt{3.5} < x < 2\) | B2,1 | Condone dec approx'n for \(\sqrt{3.5}\); B1 if \(\leq\) used instead of \(<\) |
| Total | 3 |
## Question 8:
### Part (a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Asymptotes $x = -2$, $x = 2$, $y = 0$ | $\text{B1} \times 3$ | |
| **Total** | **3** | |
### Part (a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Middle branch generally correct | B1 | Allow if max pt not in right place |
| Other branches generally correct | B1 | |
| All branches approaching asymps | B1 | Asymps must be shown correctly on diagram or elsewhere; B0 if any other intersections are shown |
| Intersection at $\left(0, -\frac{1}{4}\right)$ indicated | B1 | |
| **Total** | **4** | |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = -2$ when $x = \pm\sqrt{3.5}$ | B1 | Allow NMS |
| Sol'n $-2 < x < -\sqrt{3.5}$, $\sqrt{3.5} < x < 2$ | B2,1 | Condone dec approx'n for $\sqrt{3.5}$; B1 if $\leq$ used instead of $<$ |
| **Total** | **3** | |
---8 A curve has equation $y = \frac { 1 } { x ^ { 2 } - 4 }$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the equations of the three asymptotes of the curve.
\item Sketch the curve, showing the coordinates of any points of intersection with the coordinate axes.
\end{enumerate}\item Hence, or otherwise, solve the inequality
$$\frac { 1 } { x ^ { 2 } - 4 } < - 2$$
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2011 Q8 [10]}}