| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Compare or choose between models |
| Difficulty | Moderate -0.3 This is a straightforward applied exponential modeling question requiring reading values from a graph, forming an exponential equation from two given points using standard substitution methods, and evaluating the model. While multi-part, each component uses routine A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(10750\) barrels | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Valid limitation, e.g. model gives negative \(V\) as \(t\) increases; \(t=10\) gives \(V=-1500\); model only works for \(0 \leqslant t \leqslant \frac{64}{7}\) | B1 | See scheme for acceptable examples |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Suggests exponential model e.g. \(V = Ae^{kt}\) | M1 | Also accept \(V=Ar^t\) or \(V=Ae^{kt}+b\) with candidate-chosen \(b\) |
| Uses \((0,16000)\) and \((4,9000)\): \(9000 = 16000e^{4k}\) | dM1 | Uses both points in their model |
| \(\Rightarrow k = \frac{1}{4}\ln\!\left(\frac{9}{16}\right) \approx -0.144\) | M1 | Uses correct method to find all constants |
| \(V = 16000e^{\frac{1}{4}\ln\!\left(\frac{9}{16}\right)t}\) or \(V = 16000e^{-0.144t}\) | A1 | Equation passing through (or approximately through) both given points |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Uses exponential model with \(t=3 \Rightarrow V \approx 10400\) barrels | B1ft | Follow through on their exponential model |
## Question 6:
**Part (a)(i):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $10750$ barrels | B1 | |
**Part (a)(ii):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Valid limitation, e.g. model gives negative $V$ as $t$ increases; $t=10$ gives $V=-1500$; model only works for $0 \leqslant t \leqslant \frac{64}{7}$ | B1 | See scheme for acceptable examples |
**Part (b)(i):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Suggests exponential model e.g. $V = Ae^{kt}$ | M1 | Also accept $V=Ar^t$ or $V=Ae^{kt}+b$ with candidate-chosen $b$ |
| Uses $(0,16000)$ and $(4,9000)$: $9000 = 16000e^{4k}$ | dM1 | Uses both points in their model |
| $\Rightarrow k = \frac{1}{4}\ln\!\left(\frac{9}{16}\right) \approx -0.144$ | M1 | Uses correct method to find all constants |
| $V = 16000e^{\frac{1}{4}\ln\!\left(\frac{9}{16}\right)t}$ or $V = 16000e^{-0.144t}$ | A1 | Equation passing through (or approximately through) both given points |
**Part (b)(ii):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses exponential model with $t=3 \Rightarrow V \approx 10400$ barrels | B1ft | Follow through on their exponential model |
---6. A company plans to extract oil from an oil field.
The daily volume of oil $V$, measured in barrels that the company will extract from this oil field depends upon the time, $t$ years, after the start of drilling.
The company decides to use a model to estimate the daily volume of oil that will be extracted. The model includes the following assumptions:
\begin{itemize}
\item The initial daily volume of oil extracted from the oil field will be 16000 barrels.
\item The daily volume of oil that will be extracted exactly 4 years after the start of drilling will be 9000 barrels.
\item The daily volume of oil extracted will decrease over time.
\end{itemize}
The diagram below shows the graphs of two possible models.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-08_629_716_918_292}
\captionsetup{labelformat=empty}
\caption{Model $A$}
\end{center}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f7994129-07ee-4f6d-9531-08a15a38b794-08_574_711_918_1064}
\captionsetup{labelformat=empty}
\caption{Model $B$}
\end{center}
\end{figure}
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Use model $A$ to estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.
\item Write down a limitation of using model $A$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Using an exponential model and the information given in the question, find a possible equation for model $B$.
\item Using your answer to (b)(i) estimate the daily volume of oil that will be extracted exactly 3 years after the start of drilling.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 1 Q6 [7]}}