| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Compare or choose between models |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question on linear vs exponential models requiring only routine substitution and basic logarithms. Parts (a)-(b) involve simple linear equations, (d)-(e) use standard exponential model techniques with ln, and (c)+(f) ask for conceptual explanations that follow directly from the context. No novel problem-solving or insight required—purely procedural application of standard A-level techniques. |
| Spec | 1.02z Models in context: use functions in modelling1.06a Exponential function: a^x and e^x graphs and properties1.06i Exponential growth/decay: in modelling context |
| Answer | Marks |
|---|---|
| Response | Mark |
| Larger values of \(t\) give inaccurate results | B0 |
| Exponential doesn't give the same increase each day | B0 |
| 10% of the new area is not 10% of the original area | B0 |
| Original model predicts an increase of 0.8 each day but the increase 10% each day so the model is an underestimate | B1 |
| This model doesn't grow exponentially but increasing by 10% each day would be modelled by that | B1 |
| The model is linear whereas 10% each day is exponential | B1 |
| Answer | Marks |
|---|---|
| Response | Mark |
| It would get too big | B0 |
| It would get impossibly big | B1 |
| There is no limit to the size of the area, but growth will eventually reduce or stop | B0 |
| The area *would be* too large to be plausible and have to outgrow the lab (would be implies the model) | B1 BOD |
| For large values of \(t\), the area would be too large (large values of \(t\) implies the model being used) | B0 |
| For large values of \(t\), the area would be impossibly large | B1 |
| It suggests that the culture grows to a size that is unrealistic ("it" refers to the model) | B1 |
| The model predicts unlimited growth but this is not possible | B1 |
## Question 10(c) — Exemplar responses:
| Response | Mark |
|---|---|
| Larger values of $t$ give inaccurate results | **B0** |
| Exponential doesn't give the same increase each day | **B0** |
| 10% of the new area is not 10% of the original area | **B0** |
| Original model predicts an increase of 0.8 each day but the increase 10% each day so the model is an underestimate | **B1** |
| This model doesn't grow exponentially but increasing by 10% each day would be modelled by that | **B1** |
| The model is linear whereas 10% each day is exponential | **B1** |
---
## Question 10(f) — Exemplar responses:
| Response | Mark |
|---|---|
| It would get too big | **B0** |
| It would get impossibly big | **B1** |
| There is no limit to the size of the area, but growth will eventually reduce or stop | **B0** |
| The area *would be* too large to be plausible and have to outgrow the lab (would be implies the model) | **B1 BOD** |
| For large values of $t$, the area would be too large (large values of $t$ implies the model being used) | **B0** |
| For large values of $t$, the area would be impossibly large | **B1** |
| It suggests that the culture grows to a size that is unrealistic ("it" refers to the model) | **B1** |
| The model predicts unlimited growth but this is not possible | **B1** |
---10 Zac is measuring the growth of a culture of bacteria in a laboratory. The initial area of the culture is $8 \mathrm {~cm} ^ { 2 }$. The area one day later is $8.8 \mathrm {~cm} ^ { 2 }$.
At first, Zac uses a model of the form $\mathrm { A } = \mathrm { a } + \mathrm { bt }$, where $A \mathrm {~cm} ^ { 2 }$ is the area $t$ days after he begins measuring and $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $a$ and $b$ that best model the initial area and the area one day later.
\item Calculate the value of $t$ for which the model predicts an area of $15 \mathrm {~cm} ^ { 2 }$.
\item Zac notices the area covered by the culture increases by $10 \%$ each day.
Explain why this model may not be suitable after the first day.
Zac decides to use a different model for $A$. His new model is $\mathrm { A } = \mathrm { Pe } ^ { \mathrm { kt } }$, where $P$ and $k$ are constants.
\item Find the values of $P$ and $k$ that best model the initial area and the area one day later.
\item Calculate the value of $t$ for which the area reaches $15 \mathrm {~cm} ^ { 2 }$ according to this model.
\item Explain why this model may not be suitable for large values of $t$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 2024 Q10 [10]}}