| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Rational curve sketching with asymptotes and inequalities |
| Difficulty | Moderate -0.8 This is a straightforward FP1 question testing basic rational function properties. Part (a) requires identifying standard asymptotes (x=3, y=0) and sketching a simple reciprocal function plus a linear function—routine tasks. Part (b) involves solving a quadratic equation from the intersection and reading inequality solutions from the sketch. All techniques are standard with no novel problem-solving required, making this easier than average A-level questions. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02p Interpret algebraic solutions: graphically1.02q Use intersection points: of graphs to solve equations |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 3\) and \(y = 0\) | B1, B1 | One mark each |
| Answer | Marks | Guidance |
|---|---|---|
| Correct shape of rectangular hyperbola with asymptotes shown; passes through \((-\frac{1}{3}, 0)\) — wait, intersection with x-axis: none; y-axis: \(x=0 \Rightarrow y = \frac{1}{0-3} = -\frac{1}{3}\), so \((0, -\frac{1}{3})\) | B1, B1 | B1 correct shape/asymptotes, B1 for \((0, -\frac{1}{3})\) marked |
| Answer | Marks | Guidance |
|---|---|---|
| Straight line \(y = 2x-5\); passes through \((\frac{5}{2}, 0)\) and \((0,-5)\) | B1 | B1 for correct line with intercepts marked |
| Answer | Marks | Guidance |
|---|---|---|
| \(1 = (2x-5)(x-3)\); \(2x^2 - 11x + 15 = 1\); \(2x^2 - 11x + 14 = 0\); \((2x-7)(x-2)=0\); \(x = \frac{7}{2}\) or \(x=2\) | M1, M1, A1 | M1 multiply through, M1 forming quadratic, A1 both solutions |
| Answer | Marks | Guidance |
|---|---|---|
| \(x > \frac{7}{2}\) or \(x < 2\) (intersected with \(x > 3\) and \(x < 3\)); Answer: \(x > \frac{7}{2}\) or \(2 < x < 3\) | M1, A1 | M1 for using solutions from (b)(i), A1 correct inequalities |
# Question 7:
**(a)(i)**
| $x = 3$ and $y = 0$ | B1, B1 | One mark each |
**(a)(ii)**
| Correct shape of rectangular hyperbola with asymptotes shown; passes through $(-\frac{1}{3}, 0)$ — wait, intersection with x-axis: none; y-axis: $x=0 \Rightarrow y = \frac{1}{0-3} = -\frac{1}{3}$, so $(0, -\frac{1}{3})$ | B1, B1 | B1 correct shape/asymptotes, B1 for $(0, -\frac{1}{3})$ marked |
**(a)(iii)**
| Straight line $y = 2x-5$; passes through $(\frac{5}{2}, 0)$ and $(0,-5)$ | B1 | B1 for correct line with intercepts marked |
**(b)(i)**
| $1 = (2x-5)(x-3)$; $2x^2 - 11x + 15 = 1$; $2x^2 - 11x + 14 = 0$; $(2x-7)(x-2)=0$; $x = \frac{7}{2}$ or $x=2$ | M1, M1, A1 | M1 multiply through, M1 forming quadratic, A1 both solutions |
**(b)(ii)**
| $x > \frac{7}{2}$ or $x < 2$ (intersected with $x > 3$ and $x < 3$); Answer: $x > \frac{7}{2}$ or $2 < x < 3$ | M1, A1 | M1 for using solutions from (b)(i), A1 correct inequalities |
---7
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write down the equations of the two asymptotes of the curve $y = \frac { 1 } { x - 3 }$.
\item Sketch the curve $y = \frac { 1 } { x - 3 }$, showing the coordinates of any points of intersection with the coordinate axes.
\item On the same axes, again showing the coordinates of any points of intersection with the coordinate axes, sketch the line $y = 2 x - 5$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Solve the equation
$$\frac { 1 } { x - 3 } = 2 x - 5$$
\item Find the solution of the inequality
$$\frac { 1 } { x - 3 } < 2 x - 5$$
□\\
\includegraphics[max width=\textwidth, alt={}, center]{763d89e4-861a-4754-a93c-d0902987673f-08_367_197_2496_155}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2010 Q7 [10]}}