| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | ln(y) vs ln(x) linear graph |
| Difficulty | Standard +0.3 This is a standard logarithmic linearization question requiring students to convert between exponential and linear forms, read values from a graph, and interpret constants. The techniques are routine (taking logs, identifying gradient and intercept) with straightforward application to find p and q. Parts (b) and (c) require basic interpretation rather than complex reasoning. Slightly above average difficulty due to the modelling context and multi-part nature, but all steps are textbook-standard. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\log_{10} h = 2.25 - 0.235\log_{10} m \Rightarrow h = 10^{2.25 - 0.235\log_{10} m} \Rightarrow h = 10^{2.25} \times m^{-0.235}\) | M1 | Establishes link between \(h = pm^q\) and \(\log_{10} h = 2.25 - 0.235\log_{10} m\); may be implied by correct equation in \(p\) or \(q\) |
| Either: \(p = 10^{2.25}\), \(q = -0.235\) OR: \(\log_{10} p = 2.25\), \(q = -0.235\) | A1 | For a correct equation in \(p\) or \(q\) |
| \(p = 178\) and \(q = -0.235\) | A1 | Both values correct |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(h = \text{"178"} \times 5^{"-0.235"}\) OR \(\log_{10} h = \text{"2.25"} - \text{"0.235"}\log_{10} 5\) | M1 | Uses either model to set up equation in \(h\) (or \(m\)) |
| \(h = 122\) | A1 | \(h =\) awrt 122; condone \(h =\) awrt 122 bpm |
| Reasonably accurate (to 2 sf) so suitable | A1ft | Comment consistent with their answer; e.g. only "3" bpm away is suitable ✓; "122" bpm \(\neq\) 119 bpm is unsuitable ✗ |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| "\(p\)" would be the (resting) heart rate (in bpm) of a mammal with a mass of 1 kg | B1 | — |
## Question 13:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\log_{10} h = 2.25 - 0.235\log_{10} m \Rightarrow h = 10^{2.25 - 0.235\log_{10} m} \Rightarrow h = 10^{2.25} \times m^{-0.235}$ | M1 | Establishes link between $h = pm^q$ and $\log_{10} h = 2.25 - 0.235\log_{10} m$; may be implied by correct equation in $p$ or $q$ |
| Either: $p = 10^{2.25}$, $q = -0.235$ OR: $\log_{10} p = 2.25$, $q = -0.235$ | A1 | For a correct equation in $p$ or $q$ |
| $p = 178$ and $q = -0.235$ | A1 | Both values correct |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $h = \text{"178"} \times 5^{"-0.235"}$ OR $\log_{10} h = \text{"2.25"} - \text{"0.235"}\log_{10} 5$ | M1 | Uses either model to set up equation in $h$ (or $m$) |
| $h = 122$ | A1 | $h =$ awrt 122; condone $h =$ awrt 122 bpm |
| Reasonably accurate (to 2 sf) so suitable | A1ft | Comment consistent with their answer; e.g. only "3" bpm away is suitable ✓; "122" bpm $\neq$ 119 bpm is unsuitable ✗ |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| "$p$" would be the (resting) heart rate (in bpm) of a mammal with a mass of 1 kg | B1 | — |
---13.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{235cd1dc-a3ab-473a-bf77-3e41b274dfd8-30_549_709_251_621}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
The resting heart rate, $h$, of a mammal, measured in beats per minute, is modelled by the equation
$$h = p m ^ { q }$$
where $p$ and $q$ are constants and $m$ is the mass of the mammal measured in kg .\\
Figure 2 illustrates the linear relationship between $\log _ { 10 } h$ and $\log _ { 10 } m$\\
The line meets the vertical $\log _ { 10 } h$ axis at 2.25 and has a gradient of - 0.235
\begin{enumerate}[label=(\alph*)]
\item Find, to 3 significant figures, the value of $p$ and the value of $q$.
A particular mammal has a mass of 5 kg and a resting heart rate of 119 beats per minute.
\item Comment on the suitability of the model for this mammal.
\item With reference to the model, interpret the value of the constant $p$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 2021 Q13 [7]}}