| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 10 |
| 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 modeling question requiring standard techniques: differentiation for rate of change, finding exponential model parameters from data, and solving exponential equations using logarithms. All steps are routine C3 procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06i Exponential growth/decay: in modelling context1.07b Gradient as rate of change: dy/dx notation |
| \(t\) (hours) | 0 | 10 | 20 |
| \(M _ { 2 }\) (grams) | 75 | 120 | 192 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Differentiate to obtain form \(ke^{-0.014t}\) | M1 | any constant \(k\) different from 400 |
| Obtain \(5.6e^{-0.014t}\) or \(-5.6e^{-0.014t}\) | A1 | or (unsimplified) equiv |
| Obtain 4.9 or -4.9 or 4.87 or -4.87 | A1 | 3 marks: but not greater accuracy; allow if final statement seems contradictory; answer only earns 0/3 – differentiation is needed |
| (ii) Either: State or imply \(M_1 = 75e^{0t}\) | B1 | or equiv |
| Attempt to find formula for \(M_2\) | M1 | |
| Obtain \(M_2 = 75e^{0.0477}\) | A1 | or equiv such as \(75e^{(\ln 1.5) / t}\) |
| Equate masses and attempt rearrangement | M1 | as far as equation with e appearing once |
| Obtain \(e^{0.0677} = \frac{16}{3}\) | A1 | 5 marks: or equiv of required form which might involve 5.33 or greater accuracy on RHS; final two marks might be earned in part iii |
| Or: State or imply \(M_2 = 75 \times r^{0 \cdot t}\) | B1 | for positive value \(r\) |
| Obtain \(75 \times 1.6^{0 \cdot t}\) | B1 | |
| Attempt to find \(M_2\) in terms of e | M1 | |
| Equate masses and attempt rearrangement | M1 | |
| Obtain \(e^{0.061t} = \frac{16}{3}\) | A1 | 5 marks: or equiv of required form which might involve 5.33 or greater accuracy on RHS; final two marks might be earned in part iii |
| (iii) Attempt solution involving logarithm of any equation of form \(e^{m} = c_1\) | M1 | whether the conclusion of part ii or not |
| Obtain 27.4 | A1 | 2 marks: or greater accuracy 27.4422...; correct answer only earns both marks |
**(i)** Differentiate to obtain form $ke^{-0.014t}$ | M1 | any constant $k$ different from 400
Obtain $5.6e^{-0.014t}$ or $-5.6e^{-0.014t}$ | A1 | or (unsimplified) equiv
Obtain 4.9 or -4.9 or 4.87 or -4.87 | A1 | 3 marks: but not greater accuracy; allow if final statement seems contradictory; answer only earns 0/3 – differentiation is needed
**(ii)** Either: State or imply $M_1 = 75e^{0t}$ | B1 | or equiv
Attempt to find formula for $M_2$ | M1 |
Obtain $M_2 = 75e^{0.0477}$ | A1 | or equiv such as $75e^{(\ln 1.5) / t}$
Equate masses and attempt rearrangement | M1 | as far as equation with e appearing once
Obtain $e^{0.0677} = \frac{16}{3}$ | A1 | 5 marks: or equiv of required form which might involve 5.33 or greater accuracy on RHS; final two marks might be earned in part iii
Or: State or imply $M_2 = 75 \times r^{0 \cdot t}$ | B1 | for positive value $r$
Obtain $75 \times 1.6^{0 \cdot t}$ | B1 |
Attempt to find $M_2$ in terms of e | M1 |
Equate masses and attempt rearrangement | M1 |
Obtain $e^{0.061t} = \frac{16}{3}$ | A1 | 5 marks: or equiv of required form which might involve 5.33 or greater accuracy on RHS; final two marks might be earned in part iii
**(iii)** Attempt solution involving logarithm of any equation of form $e^{m} = c_1$ | M1 | whether the conclusion of part ii or not
Obtain 27.4 | A1 | 2 marks: or greater accuracy 27.4422...; correct answer only earns both marks
---8 An experiment involves two substances, Substance 1 and Substance 2, whose masses are changing. The mass, $M _ { 1 }$ grams, of Substance 1 at time $t$ hours is given by
$$M _ { 1 } = 400 \mathrm { e } ^ { - 0.014 t } .$$
The mass, $M _ { 2 }$ grams, of Substance 2 is increasing exponentially and the mass at certain times is shown in the following table.
\begin{center}
\begin{tabular}{ | l | c | c | c | }
\hline
$t$ (hours) & 0 & 10 & 20 \\
\hline
$M _ { 2 }$ (grams) & 75 & 120 & 192 \\
\hline
\end{tabular}
\end{center}
A critical stage in the experiment is reached at time $T$ hours when the masses of the two substances are equal.\\
(i) Find the rate at which the mass of Substance 1 is decreasing when $t = 10$, giving your answer in grams per hour correct to 2 significant figures.\\
(ii) Show that $T$ is the root of an equation of the form $\mathrm { e } ^ { k t } = c$, where the values of the constants $k$ and $c$ are to be stated.\\
(iii) Hence find the value of $T$ correct to 3 significant figures.
\hfill \mbox{\textit{OCR C3 2011 Q8 [10]}}