| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Quadratic modelling problems |
| Difficulty | Moderate -0.3 This is a straightforward quadratic modelling question requiring students to form a quadratic from given vertex and root (standard technique), then solve a quadratic inequality. The problem-solving is minimal as the approach is clearly signposted, and all calculations are routine A-level methods. Slightly easier than average due to the scaffolded structure and standard techniques involved. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions1.02e Complete the square: quadratic polynomials and turning points |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Translates situation into suitable equation, e.g. \(y = a(x-40)^2 + 12\) | M1 | Must use max at \((40, 12)\) |
| Applies complete strategy with appropriate constraints to find all constants | M1 | Substitutes \((0,0)\) to find \(a = \frac{-3}{400}\) |
| \(Y = -\frac{3}{400}(x-40)^2 + 12\) or \(Y = -\frac{3}{400}x(x-80)\) | A1 | Correct final equation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitutes \(Y = 9\) into quadratic and proceeds to \((x \pm c)^2 = d\) form | M1 | Obtains \((x-40)^2 = 400\) |
| Correct method of solving to give at least one solution; \(x = 20,\ x = 60\) | M1 | Correct solution method |
| Chooses \(x = 20\) | A1 | Correct value selected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Any one valid limitation, e.g. ground is horizontal / no air resistance / ball modelled as particle / no spin / trajectory is a perfect parabola | B1 | One valid limitation required |
## Question 8:
### Part (a): Find a quadratic equation linking $Y$ with $x$
| Answer/Working | Mark | Guidance |
|---|---|---|
| Translates situation into suitable equation, e.g. $y = a(x-40)^2 + 12$ | M1 | Must use max at $(40, 12)$ |
| Applies complete strategy with appropriate constraints to find all constants | M1 | Substitutes $(0,0)$ to find $a = \frac{-3}{400}$ |
| $Y = -\frac{3}{400}(x-40)^2 + 12$ or $Y = -\frac{3}{400}x(x-80)$ | A1 | Correct final equation |
**(3 marks)**
---
### Part (b): Deduce the ball should be placed about 20 m from the first barrier wall
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $Y = 9$ into quadratic and proceeds to $(x \pm c)^2 = d$ form | M1 | Obtains $(x-40)^2 = 400$ |
| Correct method of solving to give at least one solution; $x = 20,\ x = 60$ | M1 | Correct solution method |
| Chooses $x = 20$ | A1 | Correct value selected |
**(3 marks)**
---
### Part (c): Give one limitation of the model
| Answer/Working | Mark | Guidance |
|---|---|---|
| Any one valid limitation, e.g. ground is horizontal / no air resistance / ball modelled as particle / no spin / trajectory is a perfect parabola | B1 | One valid limitation required |
**(1 mark)**
---8.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f9dcb521-6aaa-4496-86e8-2dcd07838e10-14_551_1479_388_365}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
In a competition, competitors are going to kick a ball over the barrier walls. The height of the barrier walls are each 9 metres high and 50 cm wide and stand on horizontal ground. The figure 2 is a graph showing the motion of a ball.
The ball reaches a maximum height of 12 metres and hits the ground at a point 80 metres from where its kicked.\\
a. Find a quadratic equation linking $Y$ with $x$ that models this situation.
The ball pass over the barrier walls.\\
b. Use your equation to deduce that the ball should be placed about 20 m from the first barrier wall.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q8 [7]}}