| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2018 |
| 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 modelling question requiring basic exponential function manipulation and interpretation. Part (i) involves simple substitution (A=500, k=1.044), part (ii) is a single calculation to check consistency, part (iii) requires recognizing exponential growth tends to infinity, parts (iv-v) involve reading features from a given model rather than deriving anything. All steps are routine with no problem-solving insight required, making it easier than average A-level questions. |
| 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 |
|---|---|---|
| 12 | (i) | A = 500 |
| k = 1.044 | B1 |
| Answer | Marks |
|---|---|
| [2] | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| 12 | (ii) | 500 × 1.04410 |
| Answer | Marks |
|---|---|
| consistent | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.4 |
| 1.1 | Ft |
| www | May use 650 and show k or A |
| Answer | Marks | Guidance |
|---|---|---|
| 12 | (iii) | exponential growth oe |
| [1] | 3.5a | OR increase for ever, oe |
| 12 | (iv) | t = 0, N = 500 |
| Answer | Marks |
|---|---|
| in the long run | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3.5a | Oe ‘N will not go beyond 750’ | Allow shown using large |
| Answer | Marks | Guidance |
|---|---|---|
| 12 | (v) | (A) |
| [1] | 3.5b | oe |
| (B) | cosine or sine oe | B1 |
| [1] | 3.5c | Eg Introducing a multiplier of (–1)t |
Question 12:
12 | (i) | A = 500
k = 1.044 | B1
B1
[2] | 3.3
1.1
12 | (ii) | 500 × 1.04410
= 769.1 which is not close to 650 so not
consistent | M1
A1
[2] | 3.4
1.1 | Ft
www | May use 650 and show k or A
different
12 | (iii) | exponential growth oe | B1
[1] | 3.5a | OR increase for ever, oe
12 | (iv) | t = 0, N = 500
t = 10, N = 650.37 ≈ 650
model predicts number of fruit flies tends to 750
in the long run | B1
B1
B1
[3] | 1.1
1.1
3.5a | Oe ‘N will not go beyond 750’ | Allow shown using large
value of t
12 | (v) | (A) | Oscillations | B1
[1] | 3.5b | oe
(B) | cosine or sine oe | B1
[1] | 3.5c | Eg Introducing a multiplier of (–1)t
if t is a large integer.
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© OCR 2018In an experiment 500 fruit flies were released into a controlled environment. After 10 days there were 650 fruit flies present.
Munirah believes that $N$, the number of fruit flies present at time $t$ days after the fruit flies are released, will increase at the rate of 4.4% per day. She proposes that the situation is modelled by the formula $N = Ak^t$.
\begin{enumerate}[label=(\roman*)]
\item Write down the values of $A$ and $k$. [2]
\item Determine whether the model is consistent with the value of $N$ at $t = 10$. [2]
\item What does the model suggest about the number of fruit flies in the long run? [1]
\end{enumerate}
Subsequently it is found that for large values of $t$ the number of fruit flies in the controlled environment oscillates about 750. It is also found that as $t$ increases the oscillations decrease in magnitude.
Munirah proposes a second model in the light of this new information.
$$N = 750 - 250 \times e^{-0.092t}$$
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Identify three ways in which this second model is consistent with the known data. [3]
\item \begin{enumerate}[label=(\Alph*)]
\item Identify one feature which is not accounted for by the second model. [1]
\item Give an example of a mathematical function which needs to be incorporated in the model to account for this feature. [1]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 2018 Q12 [10]}}