| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Critique single model appropriateness |
| Difficulty | Moderate -0.8 This is a straightforward guided question testing basic exponential function properties. Part (a) requires simple differentiation recall, (b) uses the derivative to identify decreasing behavior, (c) involves routine substitution, and (d) asks for elementary comparison with data. All steps are standard textbook exercises with no problem-solving or novel insight required, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.07i Differentiate x^n: for rational n and sums1.07j Differentiate exponentials: e^(kx) and a^(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(-0.03e^{-0.03t}\) | B1 | 1.2 |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Decreasing function because \(e^{-0.03t}\) is positive [for all values of \(t\)] so the gradient is negative | E1 | 2.2a — Explanation may include a sketch graph of \(70e^{-0.03t}\) but must be clear that the graph is of the function and the answer must clearly refer to the gradient of the function and not the trend in the data |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| 70 | B1 | 1.1 |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(38.[4168...]\) | B1 | 1.1 |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Data values decreasing so decreasing function is suitable | E1 | 3.5a |
| At \(t = 0\), calculated \(D = 70\) and this matches the data | B1 | 3.5a |
| At \(t = 20\), data value is 40 which is not exact but close | B1 | 3.5b |
| [3] |
# Question 8:
## Part (a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $-0.03e^{-0.03t}$ | B1 | 1.2 |
| **[1]** | | |
## Part (b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Decreasing function because $e^{-0.03t}$ is positive [for all values of $t$] so the gradient is negative | E1 | 2.2a — Explanation may include a sketch graph of $70e^{-0.03t}$ but must be clear that the graph is of the function and the answer must clearly refer to the gradient of the function and not the trend in the data |
| **[1]** | | |
## Part (c)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| 70 | B1 | 1.1 |
| **[1]** | | |
## Part (c)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $38.[4168...]$ | B1 | 1.1 |
| **[1]** | | |
## Part (d):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Data values decreasing so decreasing function is suitable | E1 | 3.5a |
| At $t = 0$, calculated $D = 70$ and this matches the data | B1 | 3.5a |
| At $t = 20$, data value is 40 which is not exact but close | B1 | 3.5b |
| **[3]** | | |
---8 In an experiment, the temperature of a hot liquid is measured every minute.\\
The difference between the temperature of the hot liquid and room temperature is $D ^ { \circ } \mathrm { C }$ at time $t$ minutes.
Fig. 8 shows the experimental data.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{05376a51-e768-4b45-9c18-c98255a4bd70-07_1144_1541_497_276}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{center}
\end{figure}
It is thought that the model $D = 70 \mathrm { e } ^ { - 0.03 t }$ might fit the data.
\begin{enumerate}[label=(\alph*)]
\item Write down the derivative of $\mathrm { e } ^ { - 0.03 t }$.
\item Explain how you know that $70 \mathrm { e } ^ { - 0.03 t }$ is a decreasing function of $t$.
\item Calculate the value of $70 \mathrm { e } ^ { - 0.03 t }$ when
\begin{enumerate}[label=(\roman*)]
\item $\quad t = 0$,
\item $t = 20$.
\end{enumerate}\item Using your answers to parts (b) and (c), discuss how well the model $D = 70 \mathrm { e } ^ { - 0.03 t }$ fits the data.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 Q8 [7]}}