| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch rational with linear numerator |
| Difficulty | Standard +0.3 This is a straightforward Further Maths question on rational function sketching with standard techniques: finding asymptotes (vertical at x=-5/3, horizontal at y=2/3), intercepts, solving an inequality from the sketch, and finding the reflection in y=-x (which is the inverse function). All parts are routine applications of well-practiced methods with no novel problem-solving required, making it slightly easier than average even for Further Maths. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02n Sketch curves: simple equations including polynomials1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| At least one correct asymptote | M1 | AO 1.1a |
| \(y = \frac{2}{3}\) and \(x = -\frac{5}{3}\), with no incorrect asymptotes | A1 | AO 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sketches a curve asymptotic to either a horizontal or vertical asymptote; only one branch required | B1 | AO 1.1b |
| Sketches curve with two branches asymptotic to both horizontal and vertical asymptote | M1 | AO 1.1a |
| Correct shape with correct axis intercepts \(\left(-\frac{7}{2}, 0\right)\) and \(\left(0, \frac{7}{5}\right)\); asymptotes must be drawn (condone solid lines) | A1 | AO 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Identifies both critical values and no others; or identifies one correct set of \(x\)-values, FT their asymptote or \(x\)-intercept | M1 | AO 3.1a |
| \(x \leq -\frac{7}{2}\), \(x > -\frac{5}{3}\) | A1 | AO 3.2a |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Selects a method to find the equation of the reflected curve: swaps \(-x\) for \(y\) and \(-y\) for \(x\); or any two of: \(x=-\frac{2}{3}\), \(y=\frac{5}{3}\), \(x\)-intercept \(=-1.4\), \(y\)-intercept \(=3.5\); follow through asymptotes/intercepts from (a) and (b) | M1 | AO 3.1a |
| Multiplies an equation of the form \(\pm x = \frac{\pm 2y+7}{\pm 3y+5}\) throughout by denominator and attempts to isolate \(y\); or any three of the features listed above | M1 | AO 1.1a |
| \(y = \frac{5x+7}{3x+2}\) (correct equation in correct format) | A1 | AO 3.2a |
## Question 13(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| At least one correct asymptote | M1 | AO 1.1a |
| $y = \frac{2}{3}$ and $x = -\frac{5}{3}$, with no incorrect asymptotes | A1 | AO 1.1b |
## Question 13(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketches a curve asymptotic to either a horizontal or vertical asymptote; only one branch required | B1 | AO 1.1b |
| Sketches curve with two branches asymptotic to both horizontal and vertical asymptote | M1 | AO 1.1a |
| Correct shape with correct axis intercepts $\left(-\frac{7}{2}, 0\right)$ and $\left(0, \frac{7}{5}\right)$; asymptotes must be drawn (condone solid lines) | A1 | AO 2.2a |
## Question 13(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Identifies both critical values and no others; or identifies one correct set of $x$-values, FT their asymptote or $x$-intercept | M1 | AO 3.1a |
| $x \leq -\frac{7}{2}$, $x > -\frac{5}{3}$ | A1 | AO 3.2a |
## Question 13(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| Selects a method to find the equation of the reflected curve: swaps $-x$ for $y$ and $-y$ for $x$; **or** any two of: $x=-\frac{2}{3}$, $y=\frac{5}{3}$, $x$-intercept $=-1.4$, $y$-intercept $=3.5$; follow through asymptotes/intercepts from (a) and (b) | M1 | AO 3.1a |
| Multiplies an equation of the form $\pm x = \frac{\pm 2y+7}{\pm 3y+5}$ throughout by denominator and attempts to isolate $y$; **or** any three of the features listed above | M1 | AO 1.1a |
| $y = \frac{5x+7}{3x+2}$ (correct equation in correct format) | A1 | AO 3.2a |
---13
\begin{enumerate}[label=(\alph*)]
\item Write down the equations of the asymptotes of curve $C _ { 1 }$
13 A curve $C _ { 1 }$ has equation
13
\item On the axes below, sketch the graph of curve $C _ { 1 }$\\
Indicate the values of the intercepts of the curve with the axes.\\
\includegraphics[max width=\textwidth, alt={}, center]{fd9715c4-9ce1-4608-aed6-f3d4f71208b5-20_885_898_1192_571}
13
\item Hence, or otherwise, solve the inequality
$$\frac { 2 x + 7 } { 3 x + 5 } \geq 0$$
13
\item Curve $C _ { 2 }$ is a reflection of curve $C _ { 1 }$ in the line $y = - x$\\
Find an equation for curve $C _ { 2 }$ in the form $y = \mathrm { f } ( x )$
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2022 Q13 [10]}}