| Exam Board | AQA |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2021 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Sketch with inequalities or regions |
| Difficulty | Moderate -0.3 This is a straightforward curve sketching question requiring students to find intercepts (x=a with multiplicity 2, x=3, y=-3a²) and sketch a cubic with standard shape. Part (b) follows directly from the sketch by reading off where y>0. While it requires multiple steps and understanding of repeated roots, it's a standard AS-level exercise with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.02h Express solutions: using 'and', 'or', set and interval notation1.02n Sketch curves: simple equations including polynomials |
| Answer | Marks | Guidance |
|---|---|---|
| 5(a) | Draws a correctly orientated | |
| cubic graph | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Condone omission of 0 | 1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Condone omission of 0 | 2.2a | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Condone omission of 0 | 1.1b | B1 |
| Subtotal | 4 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 5(b) | Deduces that x < a or a < x < 3 | |
| OE | 2.2a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Or | 2.5 | R1 |
| {𝑥𝑥: 0 < 𝑥𝑥 < 3, 𝑥𝑥 ≠S𝑎𝑎u}btotal | 2 | |
| Question Total | 6 | |
| Q | Marking instructions | AO |
Question 5:
--- 5(a) ---
5(a) | Draws a correctly orientated
cubic graph | 1.1b | B1 | y = 0 at 3 and a (twice)
Draws a graph cutting x axis at
(3, 0) labelled.
Condone omission of 0 | 1.1b | B1
Draws a graph meeting x axis at
(a, 0) labelled.
Condone omission of 0 | 2.2a | B1
Draws a graph cutting y axis at
(0, 3a2) labelled.
Condone omission of 0 | 1.1b | B1
Subtotal | 4
Q | Marking instructions | AO | Marks | Typical solution
--- 5(b) ---
5(b) | Deduces that x < a or a < x < 3
OE | 2.2a | M1 | { x: x < a }
⋃ { 𝑥𝑥: 𝑎𝑎 < 𝑥𝑥 < 3}
Expresses answer in set
notation.
Or | 2.5 | R1
{𝑥𝑥: 0 < 𝑥𝑥 < 3, 𝑥𝑥 ≠S𝑎𝑎u}btotal | 2
Question Total | 6
Q | Marking instructions | AO | Marks | Typical solution\begin{enumerate}[label=(\alph*)]
\item Sketch the curve
$$y = (x - a)^2(3 - x) \quad \text{where } 0 < a < 3$$
indicating the coordinates of the points where the curve and the axes meet.
[4 marks]
\includegraphics{figure_5}
\item Hence, solve
$$(x - a)^2(3 - x) > 0$$
giving your answer in set notation form.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 1 2021 Q5 [6]}}