| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential growth/decay model setup |
| Difficulty | Moderate -0.3 This is a straightforward exponential model question requiring standard techniques: substituting values to find k, evaluating the function at given points, and differentiating to find rate of change. All steps are routine C3 material with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06i Exponential growth/decay: in modelling context1.07b Gradient as rate of change: dy/dx notation1.07j Differentiate exponentials: e^(kx) and a^(kx) |
| \(t\) | 0 | 21 | 63 |
| \(M\) | 80 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State 40 | B1 | |
| Attempt value of \(k\) using 21 and 80 | M1 | or equiv |
| Obtain \(40e^{21k} = 80\) and hence \(0.033\) | A1 | or equiv such as \(\frac{1}{21}\ln 2\) |
| Attempt value of \(M\) for \(t = 63\) | M1 | using established formula or using exponential property |
| Obtain 320 | A1 | 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Differentiate to obtain \(ce^{0.033t}\) or \(40ke^{kt}\) | M1 | any constant \(c\) different from 40 |
| Obtain \(40 \times 0.033e^{0.033t}\) | A1\(\checkmark\) | following their value of \(k\) |
| Obtain 2.64 | A1 | 3 |
# Question 5:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State 40 | B1 | |
| Attempt value of $k$ using 21 and 80 | M1 | or equiv |
| Obtain $40e^{21k} = 80$ and hence $0.033$ | A1 | or equiv such as $\frac{1}{21}\ln 2$ |
| Attempt value of $M$ for $t = 63$ | M1 | using established formula or using exponential property |
| Obtain 320 | A1 | **5** | or value rounding to this |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Differentiate to obtain $ce^{0.033t}$ or $40ke^{kt}$ | M1 | any constant $c$ different from 40 |
| Obtain $40 \times 0.033e^{0.033t}$ | A1$\checkmark$ | following their value of $k$ |
| Obtain 2.64 | A1 | **3** | allow 2.6 or $2.64 \pm 0.01$ or greater accuracy $(2.64056\ldots)$ |
---5 The mass, $M$ grams, of a certain substance is increasing exponentially so that, at time $t$ hours, the mass is given by
$$M = 40 \mathrm { e } ^ { k t }$$
where $k$ is a constant. The following table shows certain values of $t$ and $M$.
\begin{center}
\begin{tabular}{ | l | l | l | l | }
\hline
$t$ & 0 & 21 & 63 \\
\hline
$M$ & & 80 & \\
\hline
\end{tabular}
\end{center}
(i) In either order,
\begin{enumerate}[label=(\alph*)]
\item find the values missing from the table,
\item determine the value of $k$.\\
(ii) Find the rate at which the mass is increasing when $t = 21$.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2009 Q5 [8]}}