| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential growth/decay model setup |
| Difficulty | Standard +0.3 This is a straightforward exponential decay question requiring standard techniques: solving M = 0.25M₀ using logarithms in (i)(a), differentiation and substitution in (i)(b), and finding the decay constant from two data points in (ii). All steps are routine C3 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06i Exponential growth/decay: in modelling context1.07j Differentiate exponentials: e^(kx) and a^(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \(e^{-0.132t} = 0.25\) | B1 | or equiv such as \(40e^{-0.132t} = 10\) |
| Attempt solution of eqn of form \(e^{-0.132t} = k\) | M1 | using sound process; implied by correct ans; allow trial and improvement attempt |
| Obtain \(10.5\) | A1 | or greater accuracy |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Differentiate to obtain \(ke^{-0.132t}\) | M1 | where \(k\) is a constant not equal to 40 (allow even if process looks like integration) |
| Obtain \(5.28e^{-0.132t}\) or \(-5.28e^{-0.132t}\) | A1 | or (unsimplified) equiv |
| Substitute 5 to obtain \(2.73\) or \(-2.73\) | A1 | accept \(2.7\) or \(-2.7\) or greater accuracy; allow \(2.73\) or \(-2.73\) whatever it is claimed to be |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| EITHER: Attempt to solve \(40e^{2\lambda} = 31.4\) or \(40e^{-2\lambda} = 31.4\) | M1 | using sound process; method implied by correct formula for mass of \(B\) obtained |
| Obtain or imply \(40e^{-0.121t}\) | A1 | or greater accuracy \((-0.12103...)\) or \(0.5\ln 0.785\) |
| Substitute 3 to obtain \(27.8\) | A1 | accept 28 or greater accuracy |
| OR: Attempt calculation involving multiplication of power of \(\frac{31.4}{40}\) | M1 | |
| Obtain \(31.4 \times \left(\frac{31.4}{40}\right)^{0.5}\) or \(40 \times \left(\frac{31.4}{40}\right)^{1.5}\) | A1 | |
| Obtain \(27.8\) | A1 | accept 28 or greater accuracy |
## Question 7:
### Part (i)(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $e^{-0.132t} = 0.25$ | B1 | or equiv such as $40e^{-0.132t} = 10$ |
| Attempt solution of eqn of form $e^{-0.132t} = k$ | M1 | using sound process; implied by correct ans; allow trial and improvement attempt |
| Obtain $10.5$ | A1 | or greater accuracy |
### Part (i)(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Differentiate to obtain $ke^{-0.132t}$ | M1 | where $k$ is a constant not equal to 40 (allow even if process looks like integration) |
| Obtain $5.28e^{-0.132t}$ or $-5.28e^{-0.132t}$ | A1 | or (unsimplified) equiv |
| Substitute 5 to obtain $2.73$ or $-2.73$ | A1 | accept $2.7$ or $-2.7$ or greater accuracy; allow $2.73$ or $-2.73$ whatever it is claimed to be |
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| **EITHER:** Attempt to solve $40e^{2\lambda} = 31.4$ or $40e^{-2\lambda} = 31.4$ | M1 | using sound process; method implied by correct formula for mass of $B$ obtained |
| Obtain or imply $40e^{-0.121t}$ | A1 | or greater accuracy $(-0.12103...)$ or $0.5\ln 0.785$ |
| Substitute 3 to obtain $27.8$ | A1 | accept 28 or greater accuracy |
| **OR:** Attempt calculation involving multiplication of power of $\frac{31.4}{40}$ | M1 | |
| Obtain $31.4 \times \left(\frac{31.4}{40}\right)^{0.5}$ or $40 \times \left(\frac{31.4}{40}\right)^{1.5}$ | A1 | |
| Obtain $27.8$ | A1 | accept 28 or greater accuracy |
---7 (i) Substance $A$ is decaying exponentially and its mass is recorded at regular intervals. At time $t$ years, the mass, $M$ grams, of substance $A$ is given by
$$M = 40 \mathrm { e } ^ { - 0.132 t }$$
\begin{enumerate}[label=(\alph*)]
\item Find the time taken for the mass of substance $A$ to decrease to $25 \%$ of its value when $t = 0$.
\item Find the rate at which the mass of substance $A$ is decreasing when $t = 5$.\\
(ii) Substance $B$ is also decaying exponentially. Initially its mass was 40 grams and, two years later, its mass is 31.4 grams. Find the mass of substance $B$ after a further year.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2012 Q7 [9]}}