| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | ln(y) vs ln(x) linear graph |
| Difficulty | Moderate -0.5 This is a standard logarithmic linearization question requiring routine manipulation of logs (taking log of both sides), reading a graph to find gradient and intercept, and substituting back. Part (c) requires understanding extrapolation limitations, which is straightforward. The 'show that' in part (a) is purely algebraic manipulation with no problem-solving required. |
| Spec | 1.02z Models in context: use functions in modelling1.06h 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 logs of both sides and shows the addition law |
| \(\Rightarrow \log_{10} N = \log_{10} a + b\log_{10} T\) so \(m = b\) and \(c = \log_{10} a\) | A1 | Uses power law, writes full expression 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 | Implied by \(b = 2.3\) or \(a = 10^{1.8} \approx 63\) |
| Uses graph to find both \(a\) and \(b\): \(a = 10^{\text{intercept}}\) and \(b =\) gradient | M1 | Implied by \(b = 2.3\) and \(a = 10^{1.8} \approx 63\) |
| Uses \(T = 3\) in \(N = aT^b\) with their \(a\) and \(b\) | M1 | Implied by attempt at \(63 \times 3^{2.3}\) |
| Number of microbes \(\approx 800\) | A1 | Allow \(800 \pm 150\) following correct work |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(N = 1000000 \Rightarrow \log_{10} N = 6\) | M1 | For using \(N = 1000000\) and stating \(\log_{10} N = 6\) |
| We cannot 'extrapolate' the graph and assume the model still holds | A1 | Statement that we only have information for \(\log N\) between 1.8 and 4.5, so cannot be certain relationship still holds |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States that \(a\) is the number of microbes 1 day after the start of the experiment | B1 | Allow numerical explanation: \(T=1 \Rightarrow N = a1^b \Rightarrow N = a\) giving \(a\) is value of \(N\) at \(T=1\) |
## Question 12:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $N = aT^b \Rightarrow \log_{10} N = \log_{10} a + \log_{10} T^b$ | M1 | Takes logs of both sides and shows the addition law |
| $\Rightarrow \log_{10} N = \log_{10} a + b\log_{10} T$ so $m = b$ **and** $c = \log_{10} a$ | A1 | Uses power law, writes full expression 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 | Implied by $b = 2.3$ or $a = 10^{1.8} \approx 63$ |
| Uses graph to find both $a$ and $b$: $a = 10^{\text{intercept}}$ **and** $b =$ gradient | M1 | Implied by $b = 2.3$ **and** $a = 10^{1.8} \approx 63$ |
| Uses $T = 3$ in $N = aT^b$ with their $a$ and $b$ | M1 | Implied by attempt at $63 \times 3^{2.3}$ |
| Number of microbes $\approx 800$ | A1 | Allow $800 \pm 150$ following correct work |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $N = 1000000 \Rightarrow \log_{10} N = 6$ | M1 | For using $N = 1000000$ and stating $\log_{10} N = 6$ |
| We cannot 'extrapolate' the graph and assume the model still holds | A1 | Statement that we only have information for $\log N$ between 1.8 and 4.5, so cannot be certain relationship still holds |
### Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States that $a$ is the number of microbes 1 day after the start of the experiment | B1 | Allow numerical explanation: $T=1 \Rightarrow N = a1^b \Rightarrow N = a$ giving $a$ is value of $N$ at $T=1$ |
---\begin{enumerate}
\item 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
\end{enumerate}
$$N = a T ^ { b } , \quad \text { where } a \text { and } b \text { are constants }$$
(a) 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]{f7994129-07ee-4f6d-9531-08a15a38b794-18_1232_1046_804_513}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows the line of best fit for values of $\log _ { 10 } N$ plotted against values of $\log _ { 10 } T$\\
(b) Use the information provided to estimate the number of microbes present in the culture 3 days after the start of the experiment.\\
(c) Explain why the information provided could not reliably be used to estimate the day when the number of microbes in the culture first exceeds 1000000 .\\
(d) With reference to the model, interpret the value of the constant $a$.
\hfill \mbox{\textit{Edexcel Paper 1 Q12 [9]}}