| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Time to reach target in exponential model |
| Difficulty | Moderate -0.3 This is a straightforward application of exponential functions requiring (i) taking natural logarithms to solve for t, and (ii) differentiating and substituting a value. Both parts are standard C3 techniques with no conceptual challenges beyond routine manipulation, making it slightly easier than average. |
| Spec | 1.06g Equations with exponentials: solve a^x = b1.06i Exponential growth/decay: in modelling context1.07j Differentiate exponentials: e^(kx) and a^(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt solution involving (natural) logarithm | M1 | |
| Obtain \(-0.017t = \ln \frac{25}{180}\) | A1 | [or equiv] |
| Obtain 116 | A1 | Total: 3 marks [or greater accuracy rounding to 116] |
| (ii) Differentiate to obtain \(ke^{-0.017t}\) | M1 | [any constant \(k\) different from 180; solution must involve differentiation] |
| Obtain correct \(-3.06e^{-0.017t}\) | A1 | [or unsimplified equiv; accept + or –] |
| Obtain 1.2 | A1 | Total: 3 marks [or greater accuracy; accept + or – answer] |
**(i)** Attempt solution involving (natural) logarithm | M1 |
Obtain $-0.017t = \ln \frac{25}{180}$ | A1 | [or equiv]
Obtain 116 | A1 | Total: 3 marks [or greater accuracy rounding to 116]
**(ii)** Differentiate to obtain $ke^{-0.017t}$ | M1 | [any constant $k$ different from 180; solution must involve differentiation]
Obtain correct $-3.06e^{-0.017t}$ | A1 | [or unsimplified equiv; accept + or –]
Obtain 1.2 | A1 | Total: 3 marks [or greater accuracy; accept + or – answer]3 The mass, $m$ grams, of a substance at time $t$ years is given by the formula
$$m = 180 \mathrm { e } ^ { - 0.017 t } .$$
(i) Find the value of $t$ for which the mass is 25 grams.\\
(ii) Find the rate at which the mass is decreasing when $t = 55$.
\hfill \mbox{\textit{OCR C3 2005 Q3 [6]}}