| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | October |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | log(y) vs x: convert and interpret |
| Difficulty | Moderate -0.3 This is a straightforward logarithm manipulation question requiring conversion from log-linear to exponential form using standard index laws (10^(A+B) = 10^A × 10^B), followed by a simple substitution. While it involves multiple steps and understanding of exponential models, the techniques are routine for P3 students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
(a) Sight of $101.478$ or $10^{0.0646}$ or $10^{0.0646t+1.478}$ M1
$a = $ awrt $30$ or awrt $1.16$ A1
$\log_{10} N = 0.0646t + 1.478 \Rightarrow N = 10^{0.0646t+1.478} = 10^{0.0646t} \cdot 10^{1.478}$ dM1
$N = 30 \times 1.16^t$ A1
(4 marks)
(b) Attempts $N = 30 \times 1.16^t = $ awrt $2600$ M1 A1
(2 marks)
(6 marks)\begin{enumerate}
\item A scientist monitored the growth of bacteria on a dish over a 30 -day period.
\end{enumerate}
The area, $N \mathrm {~mm} ^ { 2 }$, of the dish covered by bacteria, $t$ days after monitoring began, is modelled by the equation
$$\log _ { 10 } N = 0.0646 t + 1.478 \quad 0 \leqslant t \leqslant 30$$
(a) Show that this equation may be written in the form
$$N = a b ^ { t }$$
where $a$ and $b$ are constants to be found. Give the value of $a$ to the nearest integer and give the value of $b$ to 3 significant figures.\\
(b) Use the model to find the area of the dish covered by bacteria 30 days after monitoring began. Give your answer, in $\mathrm { mm } ^ { 2 }$, to 2 significant figures.
\hfill \mbox{\textit{Edexcel P3 2020 Q2 [6]}}