| Exam Board | AQA |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Classify nature of stationary points |
| Difficulty | Standard +0.3 This question requires integration of g''(x) to find g'(x), then using given conditions to find the constant. Students must evaluate g''(1) and g''(4) to classify turning points (straightforward second derivative test), then solve g'(x) > 0 for part (b). While it involves multiple steps and integration, these are standard AS-level techniques with no novel insight required—slightly easier than average due to the structured guidance. |
| Spec | 1.07e Second derivative: as rate of change of gradient1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| 11(a) | Evaluates g′′(1) or g′′(4) | 3.1a |
| Answer | Marks | Guidance |
|---|---|---|
| at x = 4 | 2.2a | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| any coordinates given | 2.2a | R1 |
| Subtotal | 3 | |
| Q | Marking instructions | AO |
| Answer | Marks | Guidance |
|---|---|---|
| 11(b) | Identifies one correct increasing | |
| region | 2.4 | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Accept x≤1, x≥4 | 2.1 | R1 |
| Subtotal | 2 | |
| Question 11 Total | 5 | |
| Q | Marking instructions | AO |
Question 11:
--- 11(a) ---
11(a) | Evaluates g′′(1) or g′′(4) | 3.1a | M1 | g′′(1) = –3
–3 < 0 therefore maximum at x = 1
g′′(4) = 3
3 > 0 therefore minimum at x = 4
Deduces nature of the turning
point at x = 1 or the turning point
at x = 4 | 2.2a | A1
Obtains g′′(1) = –3 and
g′′(4) = 3 and compares each
value with 0 to correctly deduce
the nature of both turning points
Must link to explicitly stated x
values or coordinates
Condone incorrect y values for
any coordinates given | 2.2a | R1
Subtotal | 3
Q | Marking instructions | AO | Marks | Typical solution
--- 11(b) ---
11(b) | Identifies one correct increasing
region | 2.4 | M1 | x < 1, x > 4
Obtains x < 1, x > 4
Accept x≤1, x≥4 | 2.1 | R1
Subtotal | 2
Question 11 Total | 5
Q | Marking instructions | AO | Marks | Typical solutionIt is given that for the continuous function $g$
• $g'(1) = 0$
• $g'(4) = 0$
• $g''(x) = 2x - 5$
\begin{enumerate}[label=(\alph*)]
\item Determine the nature of each of the turning points of $g$
Fully justify your answer.
[3 marks]
\item Find the set of values of $x$ for which $g$ is an increasing function.
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 1 2024 Q11 [5]}}