| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Rational functions with parameters: finding parameter values from conditions |
| Difficulty | Standard +0.3 This is a straightforward rational function question requiring identification of asymptotes (giving a=3, b=2), a simple graphical argument about intersections, and solving a rational inequality. All steps are routine for Further Maths students with no novel insight required, 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 |
|---|---|---|
| \(a = 3\) | B1 | Accept \(y = \frac{3x}{x+b}\) with any non-zero \(b\) |
| \(b = 2\); giving \(y = \frac{3x}{x+2}\) | B1 | Accept \(y = \frac{ax}{x+2}\) with any non-zero \(a\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = \frac{3x}{2}\) is a tangent to the hyperbola | B1 | Condone 'tangent' only |
| Any tangent to a hyperbola only intersects once, so there is exactly one root of the equation | B1 | Accept 'conic' instead of 'hyperbola'; condone no conclusion given |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Forms simplified quadratic from \(ax - (1-x)(x+b) = 0\); e.g. \(x^2 + 4x - 2 = 0\) or \(-x^2 - 4x + 2 = 0\) | M1 (1.1a) | Accept any inequality sign instead of \(=\); PI by \(-2+\sqrt{6}\) or \(-2-\sqrt{6}\); Or forms simplified cubic from \(ax(x+2)-(1-x)(x+b)(x+2)=0\) |
| \(x = \frac{-4 \pm \sqrt{16 + 4 times 2}}{2} = -2 \pm \sqrt{6}\) | A1F (1.1b) | Identifies \(-2+\sqrt{6}\) or \(-2-\sqrt{6}\) as a critical value; follow through on \(a\) and \(b\) |
| Identifies \(-2\) as a critical value | B1 (1.1b) | |
| \(x \leq -2-\sqrt{6}\), \(-2 < x \leq -2+\sqrt{6}\) | A1 (3.2a) | Correct set of values of \(x\) |
## Question 16(a):
$a = 3$ | B1 | Accept $y = \frac{3x}{x+b}$ with any non-zero $b$
$b = 2$; giving $y = \frac{3x}{x+2}$ | B1 | Accept $y = \frac{ax}{x+2}$ with any non-zero $a$
---
## Question 16(b):
$y = \frac{3x}{2}$ is a tangent to the hyperbola | B1 | Condone 'tangent' only
Any tangent to a hyperbola only intersects once, so there is exactly one root of the equation | B1 | Accept 'conic' instead of 'hyperbola'; condone no conclusion given
## Question 16(c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Forms simplified quadratic from $ax - (1-x)(x+b) = 0$; e.g. $x^2 + 4x - 2 = 0$ or $-x^2 - 4x + 2 = 0$ | M1 (1.1a) | Accept any inequality sign instead of $=$; PI by $-2+\sqrt{6}$ or $-2-\sqrt{6}$; Or forms simplified cubic from $ax(x+2)-(1-x)(x+b)(x+2)=0$ |
| $x = \frac{-4 \pm \sqrt{16 + 4 times 2}}{2} = -2 \pm \sqrt{6}$ | A1F (1.1b) | Identifies $-2+\sqrt{6}$ or $-2-\sqrt{6}$ as a critical value; follow through on $a$ and $b$ |
| Identifies $-2$ as a critical value | B1 (1.1b) | |
| $x \leq -2-\sqrt{6}$, $-2 < x \leq -2+\sqrt{6}$ | A1 (3.2a) | Correct set of values of $x$ |
---16 Curve $C$ has equation $y = \frac { a x } { x + b }$ where $a$ and $b$ are constants.\\
The equations of the asymptotes to $C$ are $x = - 2$ and $y = 3$\\
\includegraphics[max width=\textwidth, alt={}, center]{f7e7c21b-6e72-4c20-92fc-ba0336a11136-20_796_819_459_609}
16
\begin{enumerate}[label=(\alph*)]
\item Write down the value of $a$ and the value of $b$
16
\item The gradient of $C$ at the origin is $\frac { 3 } { 2 }$\\
With reference to the graph, explain why there is exactly one root of the equation
$$\frac { a x } { x + b } = \frac { 3 x } { 2 }$$
16
\item Using the values found in part (a), solve the inequality
$$\frac { a x } { x + b } \leq 1 - x$$
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2021 Q16 [8]}}