| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Rational curve sketching with asymptotes and inequalities |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring synthesis of rational function properties (asymptotes, roots, behavior) to construct f(x), then solve an inequality. Part (a) requires systematic reasoning about the form (x+3)/(x+1) and determining the numerator coefficient from the horizontal asymptote. Parts (b) and (c) are standard techniques once the function is found. The multi-step nature and need to connect multiple constraints elevates this above average, but it follows a recognizable pattern for Further Maths rational function questions without requiring particularly deep insight. |
| 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 |
|---|---|
| 13(a) | Writes any rational function with a horizontal asymptote of or one vertical asymptote of |
| Answer | Marks | Guidance |
|---|---|---|
| Accπ¦π¦ep=t ππaπ₯π₯ny+ cπππ₯π₯orre+cππt π₯π₯rear+r aβ―nβ―gement oππf = 2 , where is a function as described above. | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtains a fully correct answer. | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 13(b) | Sketches any rectangular hyperbola, or rational function, tending to the correct vertical and | |
| horizontal asymptotes included or implied. | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Accept un-ruled asymptotes β mark intention. | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| π¦π¦ | 1.1b | A1F |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 13(c) | Forms an equation or inequality with and their rational function. | |
| π¦π¦ = 5 | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| π₯π₯ π¦π¦ = 5 | 1.1b | A1F |
| Answer | Marks | Guidance |
|---|---|---|
| 1 | 2.2a | A1F |
| Answer | Marks | Guidance |
|---|---|---|
| Accept correct regions for their function if M1A1 scored in part (a). | 2.2a | A1 |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| Total | 9 | π₯π₯ < β1 , π₯π₯ β₯ |
| Answer | Marks | Guidance |
|---|---|---|
| Q | Marking instructions | AO |
Question 13:
--- 13(a) ---
13(a) | Writes any rational function with a horizontal asymptote of or one vertical asymptote of
,
π¦π¦ =2
π₯π₯e.=g. β 1 or
πππ₯π₯+ππ 2π₯π₯+ππ
or π¦π¦ = π₯π₯+1 π¦π¦ = π₯π₯+ππ where
ππ ππβ1 ππβ2
πππ₯π₯ +πππ₯π₯ +πππ₯π₯ + β―β― ππ
ππ ππβ1 ππβ2
Accπ¦π¦ep=t ππaπ₯π₯ny+ cπππ₯π₯orre+cππt π₯π₯rear+r aβ―nβ―gement oππf = 2 , where is a function as described above. | 3.1a | M1 | 2π₯π₯+ππ
π¦π¦ =
π₯π₯+1
but when
π₯π₯ = β3 π¦π¦ = 0
2Γβ3+ππ
β΄ 0 =
β3+1
β6+ππ = 0
ππ = 6
2π₯π₯+6
π¦π¦ = f(π₯π₯) f(π₯π₯)
Obtains a fully correct answer. | 1.1b | A1
β΄ π¦π¦ =
π₯π₯+1
but when
(π₯π₯+1)(π¦π¦β2)= ππ
π₯π₯ = β3 π¦π¦ = 0
β΄ (β3+1)(0β2)= ππ
4 = ππ
β΄ (π₯π₯+1)(π¦π¦β2) = 4
4
π¦π¦β2 =
π₯π₯+1
4
Q | Marking instructions | AO | Mark | Typical solution
--- 13(b) ---
13(b) | Sketches any rectangular hyperbola, or rational function, tending to the correct vertical and
horizontal asymptotes included or implied. | 1.1a | M1
Sketches a correct graph, including the asymptotes.
Accept the graph of their function if M1A1 scored in part (a).
Accept un-ruled asymptotes β mark intention. | 1.1b | A1
Indicates correct axis-intercepts.
Follow through their equation if their -intercept matches their graph.
π¦π¦ | 1.1b | A1F
Q | Marking instructions | AO | Mark | Typical solution
--- 13(c) ---
13(c) | Forms an equation or inequality with and their rational function.
π¦π¦ = 5 | 1.1a | M1 | 2π₯π₯+6
5 =
π₯π₯+1
5(π₯π₯+1) = 2π₯π₯+6
5π₯π₯+5 = 2π₯π₯+6
3π₯π₯ = 1
1
π₯π₯ =
3
1
π₯π₯ < β1 , π₯π₯ β₯
3
Obtains correct -intercept with
Follow through their rational function from part (a)
π₯π₯ π¦π¦ = 5 | 1.1b | A1F
Deduces one correct region
or
1
π₯π₯Coβ₯nd3one π₯π₯ < β1 for this mark.
Follow throπ₯π₯ugβ€hβ th1eir if greater than β1
1 | 2.2a | A1F
3
Deduces correct regions.
Accept correct regions for their function if M1A1 scored in part (a). | 2.2a | A1
Q | Marking instructions | AO | Mark | Typical solution
2π₯π₯+6
β€ 5
π₯π₯+1
2
(2π₯π₯+6)(π₯π₯+1)β€ 5(π₯π₯+1)
0 β€ (π₯π₯+1)οΏ½5(π₯π₯+1)β(2π₯π₯ +6)οΏ½
0 β€ (π₯π₯+1)(3π₯π₯β1)
+ β +
1
β1 π₯π₯
3
1
π₯π₯ < β1 , π₯π₯ β₯
3
1
Q | Marking instructions | AO | Mark | Typical solution
2π₯π₯+6
β€ 5
π₯π₯+1
2π₯π₯+6 5(π₯π₯+1)
β β€ 0
π₯π₯+1 π₯π₯+1
β3π₯π₯+1
β€ 0
π₯π₯+1
β + β
1
β1 π₯π₯
3
1
π₯π₯ < β1 , π₯π₯ β₯
3
1
Q | Marking instructions | AO | Mark | Typical solution
For :
π₯π₯ > β1
2π₯π₯+6 β€ 5(π₯π₯ +1)
1 β€ 3π₯π₯
1
π₯π₯ β₯ 3
For :
π₯π₯ < β1
2π₯π₯+6 β₯ 5(π₯π₯ +1)
1 β₯ 3π₯π₯
1
Total | 9 | π₯π₯ < β1 , π₯π₯ β₯
3
Q | Marking instructions | AO | Mark | Typical solutionThe graph of the rational function $y = f(x)$ intersects the $x$-axis exactly once at $(-3, 0)$
The graph has exactly two asymptotes, $y = 2$ and $x = -1$
\begin{enumerate}[label=(\alph*)]
\item Find $f(x)$
[2 marks]
\item Sketch the graph of the function.
[3 marks]
\includegraphics{figure_13b}
\item Find the range of values of $x$ for which $f(x) \leq 5$
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2018 Q13 [9]}}