| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Calculus with exponential models |
| Difficulty | Moderate -0.3 This is a straightforward exponential growth question requiring standard techniques: (i) solving e^(kt)=2 using logarithms and recognizing 8=2³, (ii) differentiating and substituting m=400. Both parts are routine applications of C3 content with no problem-solving insight needed, making it slightly easier than average but not trivial due to the multi-step nature. |
| Spec | 1.06i Exponential growth/decay: in modelling context1.07j Differentiate exponentials: e^(kx) and a^(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt process involving logarithm to solve \(e^{0.021t} = 2\) | M1 | with \(t\) the only variable; at least as far as \(0.021t = \ln 2\); must be \(…= 2\) |
| Obtain 33 | A1 | or greater accuracy; ignore absence of, or wrong, units; final answer \(\frac{\ln 2}{0.021}\) is A0 |
| State (or calculate separately to obtain) 99 | B1√ | following previous answer; no need to include units |
| Answer | Marks | Guidance |
|---|---|---|
| Differentiate to obtain \(ke^{0.021t}\) | M1 | where \(k\) is any constant not equal to 250 |
| Obtain \(250 \times 0.021 e^{0.021t}\) | A1 | or simplified equiv 5.25\(e^{0.021t}\) |
| Substitute to obtain 8.4 or \(\frac{42}{5}\) | A1 | or value rounding to 8.4 with no obvious error |
## (i)
Attempt process involving logarithm to solve $e^{0.021t} = 2$ | M1 | with $t$ the only variable; at least as far as $0.021t = \ln 2$; must be $…= 2$
Obtain 33 | A1 | or greater accuracy; ignore absence of, or wrong, units; final answer $\frac{\ln 2}{0.021}$ is A0
State (or calculate separately to obtain) 99 | B1√ | following previous answer; no need to include units
**Total: [3]**
## (ii)
Differentiate to obtain $ke^{0.021t}$ | M1 | where $k$ is any constant not equal to 250
Obtain $250 \times 0.021 e^{0.021t}$ | A1 | or simplified equiv 5.25$e^{0.021t}$
Substitute to obtain 8.4 or $\frac{42}{5}$ | A1 | or value rounding to 8.4 with no obvious error
**Total: [3]**
---The mass, $m$ grams, of a substance is increasing exponentially so that the mass at time $t$ hours is given by
$$m = 250e^{0.02t}.$$
\begin{enumerate}[label=(\roman*)]
\item Find the time taken for the mass to increase to twice its initial value, and deduce the time taken for the mass to increase to 8 times its initial value. [3]
\item Find the rate at which the mass is increasing at the instant when the mass is 400 grams. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2013 Q4 [6]}}