| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Half-life and doubling time |
| Difficulty | Moderate -0.3 This is a straightforward exponential decay question requiring standard techniques: (i) solving 120 = 240e^(-0.04t) using logarithms, and (ii) differentiating to get dm/dt = -9.6e^(-0.04t), then solving -9.6e^(-0.04t) = -2.1. Both parts are routine applications of C3 material with no conceptual challenges, making it slightly easier than average. |
| Spec | 1.06i Exponential growth/decay: in modelling context1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State \(e^{-0.04t} = 0.5\) | B1 | or equiv |
| Attempt solution of equation of form \(e^{-0.04t} = k\) | M1 | using sound process; maybe implied |
| Obtain \(17\) | A1 | 3 or greater accuracy (17.328…) |
| (ii) Differentiate to obtain form \(ke^{-0.04t}\) | *M1 | constant \(k\) different from 240 |
| Obtain \((\pm) 9.6e^{-0.04t}\) | A1 | or (unsimplified) equiv |
| Equate attempt at first derivative to \((\pm) 2.1\) and attempt solution | M1 | dep *M; method may be implied |
| Obtain \(38\) | A1 | 4 or greater accuracy (37.9956…) |
(i) State $e^{-0.04t} = 0.5$ | B1 | or equiv
Attempt solution of equation of form $e^{-0.04t} = k$ | M1 | using sound process; maybe implied
Obtain $17$ | A1 | 3 or greater accuracy (17.328…)
(ii) Differentiate to obtain form $ke^{-0.04t}$ | *M1 | constant $k$ different from 240
Obtain $(\pm) 9.6e^{-0.04t}$ | A1 | or (unsimplified) equiv
Equate attempt at first derivative to $(\pm) 2.1$ and attempt solution | M1 | dep *M; method may be implied
Obtain $38$ | A1 | 4 or greater accuracy (37.9956…)
---5 A substance is decaying in such a way that its mass, m kg , at a time t years from now is given by the formula
$$\mathrm { m } = 240 \mathrm { e } ^ { - 0.04 \mathrm { t } }$$
(i) Find the time taken for the substance to halve its mass.\\
(ii) Find the value of $t$ for which the mass is decreasing at a rate of 2.1 kg per year.
\hfill \mbox{\textit{OCR C3 2007 Q5 [7]}}