| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2018 |
| Session | Specimen |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Model y=ax^b: linearise and find constants from graph/data |
| Difficulty | Moderate -0.8 This is a standard logarithmic modelling question requiring routine application of log laws to linearise a power relationship, reading values from a graph, and basic interpretation. Part (a) is pure algebraic manipulation (take log of both sides), part (b) involves reading from a graph and converting back, and part (c) tests understanding of what a represents when T=1. All techniques are textbook exercises with no novel problem-solving required, making it 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(N = aT^b \Rightarrow \log_{10} N = \log_{10} a + \log_{10} T^b\) | M1 | Takes \(\log_{10}\)'s of both sides and attempts addition law. Condone \(\log = \log_{10}\) |
| \(\Rightarrow \log_{10} N = \log_{10} a + b\log_{10} T\) so \(m = b\) and \(c = \log_{10} a\) | A1 | Proceeds correctly and states both \(m = b\) and \(c = \log_{10} a\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Uses graph to find either \(a\) or \(b\): \(a = 10^{\text{intercept}}\) or \(b =\) gradient | M1 | May be implied by \(a = 10^{1.75\text{ to }1.85}\) or \(b = 2.27\) to \(2.33\) |
| Uses graph to find both \(a\) and \(b\) | M1 | |
| Uses \(T = 3\) in \(N = aT^b\) with their \(a\) and \(b\) | M1 | |
| Number of microbes \(\approx 800\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_{10} 3 \approx 0.48\) and using graph to find \(\log_{10} N\) | M1 | |
| Using graph to find \(\log_{10} N\) (FYI \(\log_{10} N \approx 2.9\)) | M1 | |
| \(\log_{10} N = k \Rightarrow N = 10^k\) | M1 | |
| Number of microbes \(\approx 800\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States that \(a\) is the number of microbes 1 day after the start of the experiment | B1 |
## Question 8:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $N = aT^b \Rightarrow \log_{10} N = \log_{10} a + \log_{10} T^b$ | M1 | Takes $\log_{10}$'s of both sides and attempts addition law. Condone $\log = \log_{10}$ |
| $\Rightarrow \log_{10} N = \log_{10} a + b\log_{10} T$ so $m = b$ and $c = \log_{10} a$ | A1 | Proceeds correctly and states both $m = b$ and $c = \log_{10} a$ |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses graph to find either $a$ or $b$: $a = 10^{\text{intercept}}$ **or** $b =$ gradient | M1 | May be implied by $a = 10^{1.75\text{ to }1.85}$ or $b = 2.27$ to $2.33$ |
| Uses graph to find both $a$ and $b$ | M1 | |
| Uses $T = 3$ in $N = aT^b$ with their $a$ and $b$ | M1 | |
| Number of microbes $\approx 800$ | A1 | |
**Way Two (line of best fit):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_{10} 3 \approx 0.48$ and using graph to find $\log_{10} N$ | M1 | |
| Using graph to find $\log_{10} N$ (FYI $\log_{10} N \approx 2.9$) | M1 | |
| $\log_{10} N = k \Rightarrow N = 10^k$ | M1 | |
| Number of microbes $\approx 800$ | A1 | |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| States that $a$ is the number of microbes 1 day after the start of the experiment | B1 | |
---8. In a controlled experiment, the number of microbes, $N$, present in a culture $T$ days after the start of the experiment were counted.\\
$N$ and $T$ are expected to satisfy a relationship of the form
$$N = a T ^ { b } \quad \text { where } a \text { and } b \text { are constants }$$
\begin{enumerate}[label=(\alph*)]
\item Show that this relationship can be expressed in the form
$$\log _ { 10 } N = m \log _ { 10 } T + c$$
giving $m$ and $c$ in terms of the constants $a$ and/or $b$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{d8e25332-3a45-43ca-a5b8-0a16f47f13b9-24_1223_1043_895_461}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
Figure 2 shows the line of best fit for values of $\log _ { 10 } N$ plotted against values of $\log _ { 10 } T$
\item Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.
\item With reference to the model, interpret the value of the constant $a$.
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2018 Q8 [7]}}