| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2020 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Solving quadratics and applications |
| Type | Quadratic trajectory/projectile model |
| Difficulty | Moderate -0.8 This is a straightforward application of quadratic functions requiring only routine techniques: solving a quadratic equation (given y=0), completing the square or using calculus for maximum height, and substituting a value. All methods are standard AS-level procedures with no problem-solving insight required, making it easier than average but not trivial due to the multi-part structure and arithmetic involved. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02f Solve quadratic equations: including in a function of unknown1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| 11(a)(i) | Substitutes y = 0 to give a | |
| quadratic equation (PI) | 3.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Solves quadratic equation | 1.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Condone x=6 if clearly chosen. | 3.2a | A1 |
| Subtotal | 3 |
| Answer | Marks |
|---|---|
| 11(a)(ii) | Subtracts their two values for a to |
| Answer | Marks | Guidance |
|---|---|---|
| was chosen for a | 3.2a | B1F |
| Subtotal | 1 |
| Answer | Marks |
|---|---|
| 11(b) | Differentiates, at least one term |
| Answer | Marks | Guidance |
|---|---|---|
| (PI) | 3.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| solutions to (a)(i) (PI) | 1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| PI by correct value of y | 1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| include units | 3.2a | A1 |
| Subtotal | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| 11(c) | Substitutes 11 + a into equation | |
| to find y. | 3.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| No air resistance | 3.5b | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| assumption given | 2.2b | R1 |
| Subtotal | 3 | |
| Question Total | 11 | |
| Q | Marking Instructions | AO |
Question 11:
--- 11(a)(i) ---
11(a)(i) | Substitutes y = 0 to give a
quadratic equation (PI) | 3.1b | M1 | 0 = β0.0125x2 + 0.5x β 2.55
x2 β 40x +204 = 0
(x β 6)(x β 34)
x = 6 or 34
a = 6
Solves quadratic equation | 1.1b | M1
Selects lower root to obtain the
correct value of a.
Condone x=6 if clearly chosen. | 3.2a | A1
Subtotal | 3
--- 11(a)(ii) ---
11(a)(ii) | Subtracts their two values for a to
find the correct distance travelled
ie 28m
FT provided both values for their
a are positive and the smallest
was chosen for a | 3.2a | B1F | 34 β 6 = 2 8 metres
Subtotal | 1
--- 11(b) ---
11(b) | Differentiates, at least one term
correct
Or
Uses the symmetry of the curve
(PI) | 3.1b | M1 | = β0.025x + 0.5
πππ¦π¦
πππ₯π₯
β0.025x + 0.5 = 0
x = 20
Max height = 2.45 m
Sets = 0 to find maximum
πππ¦π¦
Or
πππ₯π₯
Identifies that the maximum value
will be halfway between βtheirβ
solutions to (a)(i) (PI) | 1.1a | M1
Obtains correct value of x = 20
PI by correct value of y | 1.1b | A1
Substitutes back into y to find the
max height = 2.45m CAO must
include units | 3.2a | A1
Subtotal | 4
--- 11(c) ---
11(c) | Substitutes 11 + a into equation
to find y. | 3.1b | M1 | Using x = 17, y = 2.3375
I have assumed the jet of water has no
size
2.3375 > 2.3 so passes over the wall
Explains a limitation of the model
eg
that the model assumes that jet
has no size or a size less than
3.75cm
or
Wall has no width or has some
width
or
No air resistance | 3.5b | E1
Compares correct value with 2.3
to infer that jet passes over wall /
fails to pass over the wall.
Inference must be consistent with
stated assumption.
Condone a valid comparison if no
assumption given | 2.2b | R1
Subtotal | 3
Question Total | 11
Q | Marking Instructions | AO | Marks | Typical SolutionA fire crew is tackling a grass fire on horizontal ground.
The crew directs a single jet of water which flows continuously from point $A$.
\includegraphics{figure_11}
The path of the jet can be modelled by the equation
$$y = -0.0125x^2 + 0.5x - 2.55$$
where $x$ metres is the horizontal distance of the jet from the fire truck at $O$ and $y$ metres is the height of the jet above the ground.
The coordinates of point $A$ are $(a, 0)$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find the value of $a$.
[3 marks]
\item Find the horizontal distance from $A$ to the point where the jet hits the ground.
[1 mark]
\end{enumerate}
\item Calculate the maximum vertical height reached by the jet.
[4 marks]
\item A vertical wall is located 11 metres horizontally from $A$ in the direction of the jet. The height of the wall is 2.3 metres.
Using the model, determine whether the jet passes over the wall, stating any necessary modelling assumption.
[3 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 2020 Q11 [11]}}