| Exam Board | AQA |
|---|---|
| Module | Further AS Paper 1 (Further AS Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Find asymptotes and sketch rational curve |
| Difficulty | Standard +0.3 This is a standard Further Maths question on rational functions requiring algebraic manipulation to identify asymptotes and sketch. Part (a) involves routine algebraic rearrangement, part (b) is direct reading from the form, and part (c) is a standard sketch with asymptotes and intercepts. While it's Further Maths content, the techniques are mechanical and well-practiced, making it slightly easier than average overall. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02m Graphs of functions: difference between plotting and sketching1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks |
|---|---|
| 10(a) | Selects a method to find the values of . |
| Answer | Marks | Guidance |
|---|---|---|
| or by dividing the numerator of the(2 eπ₯π₯q+ua4t)ion by its denominator. | 3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| . | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3 11 | 2.1 | R1 |
| Answer | Marks | Guidance |
|---|---|---|
| 10(b) | (π₯π₯+2)οΏ½π¦π¦β2οΏ½= β 2 | |
| Obtains a correct asymptote. | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Obtains the other correct asymptote and no incorrect asymptotes. | 1.1b | B1 |
| Answer | Marks |
|---|---|
| 10(c) | Sketches a curve asymptotic to or . |
| Answer | Marks | Guidance |
|---|---|---|
| FT their asymptotes. | 1.1b | B1F |
| Sketches a curve with two branches, asymptotic to their asymptotes. | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| FT their asymptotes. | 2.2a | A1F |
| Total | 8 | |
| Q | Marking instructions | AO |
Question 10:
--- 10(a) ---
10(a) | Selects a method to find the values of .
e.g. ππ,ππ,ππ
by expanding
or by multiplyin(π₯π₯g +theππ )e(qπ¦π¦u+atππio)n= byππ and expanding
or by dividing the numerator of the(2 eπ₯π₯q+ua4t)ion by its denominator. | 3.1a | M1 | π¦π¦(2π₯π₯+4)= 3π₯π₯β5
2π₯π₯π¦π¦+4π¦π¦ = 3π₯π₯β5
3 5
π₯π₯π¦π¦+2π¦π¦β2π₯π₯+2 = 0
3 11
π₯π₯π¦π¦+2π¦π¦β2π₯π₯β3 = β2
3 11
(π₯π₯+2)οΏ½π¦π¦β2οΏ½= β2
Expresses the original equation in a form that allows comparison with
. | 1.1a | M1
(π₯π₯+ππ)(π¦π¦+ππ)= ππ
Completes a rigorous argument to show that can be written as
3π₯π₯β5
.
π¦π¦ = 2π₯π₯+4
3 11 | 2.1 | R1
--- 10(b) ---
10(b) | (π₯π₯+2)οΏ½π¦π¦β2οΏ½= β 2
Obtains a correct asymptote. | 1.1b | B1 | π₯π₯ = β2
3
Obtains the other correct asymptote and no incorrect asymptotes. | 1.1b | B1
--- 10(c) ---
10(c) | Sketches a curve asymptotic to or .
3
π₯π₯ = β2 π¦π¦ = 2
FT their asymptotes. | 1.1b | B1F
Sketches a curve with two branches, asymptotic to their asymptotes. | 1.1a | M1
Deduces the shape of the curve and sketches it correctly with a root at
5
and -intercept at .
3
5
π¦π¦ β4
FT their asymptotes. | 2.2a | A1F
Total | 8
Q | Marking instructions | AO | Marks | Typical solution\begin{enumerate}[label=(\alph*)]
\item Show that the equation
$$y = \frac{3x - 5}{2x + 4}$$
can be written in the form
$$(x + a)(y + b) = c$$
where $a$, $b$ and $c$ are integers to be found.
[3 marks]
\item Write down the equations of the asymptotes of the graph of
$$y = \frac{3x - 5}{2x + 4}$$
[2 marks]
\item Sketch, on the axes provided, the graph of
$$y = \frac{3x - 5}{2x + 4}$$
\includegraphics{figure_10}
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further AS Paper 1 2020 Q10 [8]}}