| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | June |
| Marks | 9 |
| 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 growth question requiring standard techniques: substituting initial conditions to find constants, solving exponential equations using logarithms, and differentiating to find rate of change. All steps are routine C3 material with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06i Exponential growth/decay: in modelling context1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | State \(A = 42\) | B1 |
| State \(k = \frac{1}{5}\) | B1 | or 0.11 or greater accuracy |
| Attempt correct process for finding \(m\) | M1 | involving logarithms or equiv |
| Obtain \(\frac{1}{5}\ln 2\) or 0.077 | A1 | or 0.08 or greater accuracy |
| (ii) | Attempt solution for \(t\) using either formula | M1 |
| Obtain 11.3 | A1 | or greater accuracy; allow 11.3 ≥ 0.1 |
| (iii) | Differentiate to obtain form \(Be^{mt}\) | M1 |
| Obtain 3.235\(e^{0.077t}\) | A1√ | or equiv; following their \(A\) and \(m\) |
| Obtain 47.9 | A1 | allow 48 or greater accuracy |
(i) | State $A = 42$ | B1 | |
| State $k = \frac{1}{5}$ | B1 | or 0.11 or greater accuracy |
| Attempt correct process for finding $m$ | M1 | involving logarithms or equiv |
| Obtain $\frac{1}{5}\ln 2$ or 0.077 | A1 | or 0.08 or greater accuracy |
(ii) | Attempt solution for $t$ using either formula | M1 | using correct process (log'ms or T&I or …) |
| Obtain 11.3 | A1 | or greater accuracy; allow 11.3 ≥ 0.1 |
(iii) | Differentiate to obtain form $Be^{mt}$ | M1 | where $B$ is different from $A$ |
| Obtain 3.235$e^{0.077t}$ | A1√ | or equiv; following their $A$ and $m$ |
| Obtain 47.9 | A1 | allow 48 or greater accuracy |7 It is claimed that the number of plants of a certain species in a particular locality is doubling every 9 years. The number of plants now is 42 . The number of plants is treated as a continuous variable and is denoted by $N$. The number of years from now is denoted by $t$.\\
(i) Two equivalent expressions giving $N$ in terms of $t$ are
$$N = A \times 2 ^ { k t } \quad \text { and } \quad N = A \mathrm { e } ^ { m t } .$$
Determine the value of each of the constants $A , k$ and $m$.\\
(ii) Find the value of $t$ for which $N = 100$, giving your answer correct to 3 significant figures.\\
(iii) Find the rate at which the number of plants will be increasing at a time 35 years from now.
\hfill \mbox{\textit{OCR C3 2008 Q7 [9]}}