| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2020 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential model with shifted asymptote |
| Difficulty | Moderate -0.3 This is a standard exponential modelling question requiring substitution of given conditions to find constants A and B, followed by basic model evaluation. The algebra is straightforward (substituting t=0 and t=10), and part (b) requires only recognizing that A represents the asymptotic temperature. Slightly easier than average due to the routine nature of the problem-solving steps. |
| Spec | 1.02z Models in context: use functions in modelling1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| \(t=0, \theta=18 \Rightarrow 18=A-B\) | M1 | 3.1b - Makes first key step, substitutes \(t=0, \theta=18\) or \(t=10, \theta=44\) to obtain equation connecting \(A\) and \(B\) |
| \(t=10, \theta=44 \Rightarrow 44=A-Be^{-0.7}\) and \(\Rightarrow A=..., B=...\) | M1 | 3.1a - Substitutes both conditions and solves simultaneously |
| At least one of: \(A=69.6\), \(B=51.6\) (allow awrt 70/awrt 52) | A1, M1 on EPEN | 1.1b |
| \(\theta = 69.6 - 51.6e^{-0.07t}\) | A1 | 3.3 - Must be fully correct equation as shown |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Mark | Guidance |
| Maximum temperature is \(69.6°C\) according to the model; upper limit of \(69.6°C\); boiling point suggested is \(69.6°C\) | B1ft | 3.4 - Identifies \(A\) as boiling point/maximum temperature |
| Model is not appropriate as \(69.6°C\) is much lower than \(78°C\) | B1ft | 3.5a - Valid conclusion referencing significant difference from 78 |
# Question 9:
## Part (a):
| Working | Mark | Guidance |
|---------|------|----------|
| $t=0, \theta=18 \Rightarrow 18=A-B$ | M1 | 3.1b - Makes first key step, substitutes $t=0, \theta=18$ **or** $t=10, \theta=44$ to obtain equation connecting $A$ and $B$ |
| $t=10, \theta=44 \Rightarrow 44=A-Be^{-0.7}$ and $\Rightarrow A=..., B=...$ | M1 | 3.1a - Substitutes both conditions and solves simultaneously |
| At least one of: $A=69.6$, $B=51.6$ (allow awrt 70/awrt 52) | A1, M1 on EPEN | 1.1b |
| $\theta = 69.6 - 51.6e^{-0.07t}$ | A1 | 3.3 - Must be fully correct equation as shown |
## Part (b):
| Working | Mark | Guidance |
|---------|------|----------|
| Maximum temperature is $69.6°C$ according to the model; upper limit of $69.6°C$; boiling point suggested is $69.6°C$ | B1ft | 3.4 - Identifies $A$ as boiling point/maximum temperature |
| Model is not appropriate as $69.6°C$ is much lower than $78°C$ | B1ft | 3.5a - Valid conclusion referencing significant difference from 78 |
---\begin{enumerate}
\item A quantity of ethanol was heated until it reached boiling point.
\end{enumerate}
The temperature of the ethanol, $\theta ^ { \circ } \mathrm { C }$, at time $t$ seconds after heating began, is modelled by the equation
$$\theta = A - B \mathrm { e } ^ { - 0.07 t }$$
where $A$ and $B$ are positive constants.\\
Given that
\begin{itemize}
\item the initial temperature of the ethanol was $18 ^ { \circ } \mathrm { C }$
\item after 10 seconds the temperature of the ethanol was $44 ^ { \circ } \mathrm { C }$\\
(a) find a complete equation for the model, giving the values of $A$ and $B$ to 3 significant figures.
\end{itemize}
Ethanol has a boiling point of approximately $78 ^ { \circ } \mathrm { C }$\\
(b) Use this information to evaluate the model.
\hfill \mbox{\textit{Edexcel Paper 2 2020 Q9 [6]}}