| Exam Board | AQA |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solve f(x) > g(x) using sketch |
| Difficulty | Standard +0.3 This is a standard FP1 rational function question with routine steps: identifying asymptotes from denominator zeros, solving a cubic that factors nicely (x-1 is clearly a common factor), sketching with given information, and reading off an inequality from the sketch. While it requires multiple techniques, each step follows predictable patterns with no novel insight needed, making it slightly easier than average. |
| 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 |
|---|---|---|
| (a) \(x = 2\); \(x = 0.5\); \(y = 0\) | B2,1,0 | OE , B1 for two correct equations and no more than one incorrect equation |
| Answer | Marks | Guidance |
|---|---|---|
| (b) \(\frac{1}{2}(x-1) = \frac{x-1}{(x-2)(2x-1)}\) | M1 | Correct elimination of \(y\) |
| \(\frac{1}{2}(x-1)(x-2)(2x-1) = x-1\) | A1 | Any correct cubic |
| \((x-1)(x-2)(2x-1) = 2(x-1)\) | ||
| \((x-1)[2x^2-5x] = 0\) | A1 | From a relevant factorised form or from \(2x^3 - 7x^2 + 5x = 0\) obtained from correct working |
| \(x = 0\), \(x = 1\), \(x = 2.5\) | A1 | From a relevant factorised form or from \(2x^3 - 7x^2 + 5x = 0\) obtained from correct working |
| Answer | Marks |
|---|---|
| (c) C: 3-branch curve, no parabolas, no branch having positive slopes. Condone slight deviations at the two horizontal extremes. | B1 |
| C: correct curve with correct asymptotic behaviour with correct asymptotes seen/implied | B1 |
| L: correct 'line', intersecting a 3-branch curve at 3 points, two of which are on the coordinate axes | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| (d) \(x \le 0\), \(0.5 < x \le 1\), \(2 < x \le 2.5\) | M1 | Three inequalities consistent with the \(c\)'s 3-branch curve \(C\) and line \(L\) with positive slope drawn in part (c), ft three values of \(x\) obtained in (b) used with correct values for vertical asymptotes, condoning \(<\) for \(\le\) and vice versa |
| A2,1 | A2 all three inequalities correct. A1 if only error is either one or both '<' replaced by '\(\le\)' or one '\(\le\)' replaced by '<' |
**(a)** $x = 2$; $x = 0.5$; $y = 0$ | B2,1,0 | OE , B1 for two correct equations and no more than one incorrect equation
**Total for (a): 2 marks**
**(b)** $\frac{1}{2}(x-1) = \frac{x-1}{(x-2)(2x-1)}$ | M1 | Correct elimination of $y$
$\frac{1}{2}(x-1)(x-2)(2x-1) = x-1$ | A1 | Any correct cubic
$(x-1)(x-2)(2x-1) = 2(x-1)$ | |
$(x-1)[2x^2-5x] = 0$ | A1 | From a relevant factorised form or from $2x^3 - 7x^2 + 5x = 0$ obtained from correct working
$x = 0$, $x = 1$, $x = 2.5$ | A1 | From a relevant factorised form or from $2x^3 - 7x^2 + 5x = 0$ obtained from correct working
**Total for (b): 3 marks**
**(c)** C: 3-branch curve, no parabolas, no branch having positive slopes. Condone slight deviations at the two horizontal extremes. | B1 |
C: correct curve with correct asymptotic behaviour with correct asymptotes seen/implied | B1 |
L: correct 'line', intersecting a 3-branch curve at 3 points, two of which are on the coordinate axes | B1 |
**Total for (c): 3 marks**
**(d)** $x \le 0$, $0.5 < x \le 1$, $2 < x \le 2.5$ | M1 | Three inequalities consistent with the $c$'s 3-branch curve $C$ and line $L$ with positive slope drawn in part (c), ft three values of $x$ obtained in (b) used with correct values for vertical asymptotes, condoning $<$ for $\le$ and vice versa
| A2,1 | A2 all three inequalities correct. A1 if only error is either one or both '<' replaced by '$\le$' or one '$\le$' replaced by '<'
**Total for (d): 3 marks**
**Overall Total: 11 marks**A curve $C$ has equation $y = \frac{x - 1}{(x - 2)(2x - 1)}$.
The line $L$ has equation $y = \frac{1}{2}(x - 1)$.
\begin{enumerate}[label=(\alph*)]
\item Write down the equations of the asymptotes of $C$.
[2 marks]
\item By forming and solving a suitable cubic equation, find the $x$-coordinates of the points of intersection of $L$ and $C$.
[3 marks]
\item Given that $C$ has no stationary points, sketch $C$ and $L$ on the same axes.
[3 marks]
\item Hence solve the inequality $\frac{x - 1}{(x - 2)(2x - 1)} \geqslant \frac{1}{2}(x - 1)$.
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA FP1 2016 Q9 [11]}}