| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Session | Specimen |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Quadratic modelling problems |
| Difficulty | Moderate -0.8 This is a straightforward quadratic modelling question requiring students to form a quadratic from three given points (vertex and two x-intercepts), then substitute a value to check clearance. The setup is clear, the algebra is routine (completing the square or using vertex form), and part (b) is a simple substitution. Part (c) tests basic modelling awareness but requires minimal mathematical work. Easier than average A-level content. |
| Spec | 1.02d Quadratic functions: graphs and discriminant conditions |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts appropriate model e.g. \(y = A(3-x)(3+x)\) or \(y = A(9-x^2)\) | M1 | AO 3.3 |
| Substitutes \(x=0, y=5 \Rightarrow 5 = A(9-0) \Rightarrow A = \frac{5}{9}\) | M1 | AO 3.1b |
| \(y = \frac{5}{9}(9-x^2)\) or \(y = \frac{5}{9}(3-x)(3+x),\ \{-3 \leqslant x \leqslant 3\}\) | A1 | AO 1.1b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitutes \(x = \frac{2.4}{2}\) into \(y = \frac{5}{9}(9-x^2)\) | M1 | AO 3.4 |
| \(y = \frac{5}{9}(9-x^2) = 4.2 > 4.1 \Rightarrow\) Coach can enter the tunnel | A1 | AO 2.2b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(4.1 = \frac{5}{9}(9-x^2) \Rightarrow x = \frac{9\sqrt{2}}{10}\), maximum width \(= 2\left(\frac{9\sqrt{2}}{10}\right)\) | M1 | AO 3.4 |
| \(= 2.545... > 2.4 \Rightarrow\) Coach can enter the tunnel | A1 | AO 2.2b |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Valid limitation e.g. road may be cambered; curve only models entrance not full length; overhead lights may block path; must enter through centre if one-way | B1 | AO 3.5b |
## Question 6:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts appropriate model e.g. $y = A(3-x)(3+x)$ or $y = A(9-x^2)$ | M1 | AO 3.3 |
| Substitutes $x=0, y=5 \Rightarrow 5 = A(9-0) \Rightarrow A = \frac{5}{9}$ | M1 | AO 3.1b |
| $y = \frac{5}{9}(9-x^2)$ or $y = \frac{5}{9}(3-x)(3+x),\ \{-3 \leqslant x \leqslant 3\}$ | A1 | AO 1.1b |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes $x = \frac{2.4}{2}$ into $y = \frac{5}{9}(9-x^2)$ | M1 | AO 3.4 |
| $y = \frac{5}{9}(9-x^2) = 4.2 > 4.1 \Rightarrow$ Coach can enter the tunnel | A1 | AO 2.2b |
### Part (b) Alt 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $4.1 = \frac{5}{9}(9-x^2) \Rightarrow x = \frac{9\sqrt{2}}{10}$, maximum width $= 2\left(\frac{9\sqrt{2}}{10}\right)$ | M1 | AO 3.4 |
| $= 2.545... > 2.4 \Rightarrow$ Coach can enter the tunnel | A1 | AO 2.2b |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Valid limitation e.g. road may be cambered; curve only models entrance not full length; overhead lights may block path; must enter through centre if one-way | B1 | AO 3.5b |
---6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-12_554_780_246_223}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{96e004d9-c6b6-474b-9b67-06e1771c609e-12_554_706_246_1133}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 2 shows the entrance to a road tunnel. The maximum height of the tunnel is measured as 5 metres and the width of the base of the tunnel is measured as 6 metres.
Figure 3 shows a quadratic curve $B C A$ used to model this entrance.\\
The points $A , O , B$ and $C$ are assumed to lie in the same vertical plane and the ground $A O B$ is assumed to be horizontal.
\begin{enumerate}[label=(\alph*)]
\item Find an equation for curve $B C A$.
A coach has height 4.1 m and width 2.4 m .
\item Determine whether or not it is possible for the coach to enter the tunnel.
\item Suggest a reason why this model may not be suitable to determine whether or not the coach can pass through the tunnel.
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 Q6 [6]}}