| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| 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 straightforward exponential modeling question requiring substitution of given conditions to form two equations, then solving for constants. Part (i) involves basic algebra with exponentials and logarithms (standard C3 content), while part (ii) simply requires recognizing the horizontal asymptote as tââ. Slightly easier than average due to the direct application of standard techniques without requiring problem-solving insight. |
| Spec | 1.06i Exponential growth/decay: in modelling context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| (i) When \(t=0\), \(P = 5+a = 8 \Rightarrow a = 3\) | M1, A1 | Substituting \(t=0\) into equation |
| When \(t=1\), \(5+3e^{-b} = 6 \Rightarrow e^{-b} = \frac{1}{3}\) | M1 | Forming equation using their \(a\) |
| \(\Rightarrow -b = \ln\frac{1}{3} \Rightarrow b = \ln 3 = 1.10\) (3 s.f.) | M1, A1ft | Taking ln on correct re-arrangement (ft their \(a\)) |
| (ii) 5 million | B1 | or \(P=5\) |
## Question 2:
| Answer/Working | Marks | Guidance |
|---|---|---|
| **(i)** When $t=0$, $P = 5+a = 8 \Rightarrow a = 3$ | M1, A1 | Substituting $t=0$ into equation |
| When $t=1$, $5+3e^{-b} = 6 \Rightarrow e^{-b} = \frac{1}{3}$ | M1 | Forming equation using their $a$ |
| $\Rightarrow -b = \ln\frac{1}{3} \Rightarrow b = \ln 3 = 1.10$ (3 s.f.) | M1, A1ft | Taking ln on correct re-arrangement (ft their $a$) |
| **(ii)** 5 million | B1 | or $P=5$ |
---2 A population is $P$ million at time $t$ years. $P$ is modelled by the equation
$$P = 5 + a \mathrm { e } ^ { - b t }$$
where $a$ and $b$ are constants.\\
The population is initially 8 million, and declines to 6 million after 1 year.\\
(i) Use this information to calculate the values of $a$ and $b$, giving $b$ correct to 3 significant figures.\\
(ii) What is the long-term population predicted by the model?
\hfill \mbox{\textit{OCR MEI C3 2006 Q2 [6]}}