| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Linear transformation to find constants |
| Difficulty | Moderate -0.8 This is a straightforward application of logarithm laws to convert between linear and exponential forms. Part (a) requires basic log manipulation (recognizing log₁₀A = log₁₀p + t·log₁₀q), parts (b-c) are routine substitution and interpretation, and part (d) asks for a standard modeling limitation. All steps are procedural with no problem-solving insight required, making this easier than average. |
| Spec | 1.02z Models in context: use functions in modelling1.06c Logarithm definition: log_a(x) as inverse of a^x1.06f Laws of logarithms: addition, subtraction, power rules1.06i Exponential growth/decay: in modelling context |
| Answer | Marks | Guidance |
|---|---|---|
| \(p = 10^{0.5}\) (or \(\log_{10} p = 0.5\)) or \(q = 10^{0.03}\) (or \(\log_{10} q = 0.03\)) | M1 | Correct equation in \(p\) or \(q\); may be implied by correct value |
| \(p =\) awrt \(3.162\) or \(q =\) awrt \(1.072\) | A1 | Either correct value |
| \(p = 10^{0.5}\) and \(q = 10^{0.03}\) (both conditions met) | dM1 | Correct equations in both \(p\) and \(q\) |
| \(A = 3.162 \times 1.072^t\) | A1 | Complete equation with both values embedded |
| Answer | Marks | Guidance |
|---|---|---|
| The initial mass (in kg) of algae (in the pond) | B1 | Must reference mass of algae and relate to initially/at the start; e.g. "The mass of algae originally in the pond" or "\(p\) is the mass of algae when \(t=0\)" |
| Answer | Marks | Guidance |
|---|---|---|
| The ratio of algae from one week to the next | B1 | Must reference rate of change/multiplier and time frame per week; e.g. "\(q\) is the rate at which the mass of algae increases every week" or "The amount of algae increases by 7.2% each week" |
| Answer | Marks | Guidance |
|---|---|---|
| \(5.5\) kg | B1 | cao including units |
| Answer | Marks | Guidance |
|---|---|---|
| \(4 = \text{"3.162"} \times \text{"1.072"}^t\) or \(\log_{10} 4 = 0.03t + 0.5\) | M1 | Setting up correct equation to find \(t\); substitution of \(A=4\) |
| awrt \(3.4\) (weeks) | A1 | Accept any acceptable method including trial and improvement; condone lack of units |
| Answer | Marks | Guidance |
|---|---|---|
| - The weather may affect the rate of growth | B1 | Any reason why rate of change/growth or mass might change or why model is not realistic; e.g. seasonal changes, overcrowding, algae may die/be removed/eaten |
## Question 5(a):
$p = 10^{0.5}$ (or $\log_{10} p = 0.5$) or $q = 10^{0.03}$ (or $\log_{10} q = 0.03$) | M1 | Correct equation in $p$ or $q$; may be implied by correct value
$p =$ awrt $3.162$ or $q =$ awrt $1.072$ | A1 | Either correct value
$p = 10^{0.5}$ and $q = 10^{0.03}$ (both conditions met) | dM1 | Correct equations in both $p$ and $q$
$A = 3.162 \times 1.072^t$ | A1 | Complete equation with both values embedded
**Total: 4 marks**
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## Question 5(b)(i):
The initial mass (in kg) of algae (in the pond) | B1 | Must reference mass of algae and relate to initially/at the start; e.g. "The mass of algae originally in the pond" or "$p$ is the mass of algae when $t=0$"
---
## Question 5(b)(ii):
The ratio of algae from one week to the next | B1 | Must reference rate of change/multiplier and time frame per week; e.g. "$q$ is the rate at which the mass of algae increases every week" or "The amount of algae increases by 7.2% each week"
**Total: 2 marks**
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## Question 5(c)(i):
$5.5$ kg | B1 | cao including units
---
## Question 5(c)(ii):
$4 = \text{"3.162"} \times \text{"1.072"}^t$ or $\log_{10} 4 = 0.03t + 0.5$ | M1 | Setting up correct equation to find $t$; substitution of $A=4$
awrt $3.4$ (weeks) | A1 | Accept any acceptable method including trial and improvement; condone lack of units
**Total: 3 marks**
---
## Question 5(d):
- The model predicts unlimited growth
- The weather may affect the rate of growth | B1 | Any reason why rate of change/growth or mass might change or why model is not realistic; e.g. seasonal changes, overcrowding, algae may die/be removed/eaten
**Total: 1 mark**
---\begin{enumerate}
\item The mass, $A$ kg, of algae in a small pond, is modelled by the equation
\end{enumerate}
$$A = p q ^ { t }$$
where $p$ and $q$ are constants and $t$ is the number of weeks after the mass of algae was first recorded.
Data recorded indicates that there is a linear relationship between $t$ and $\log _ { 10 } A$ given by the equation
$$\log _ { 10 } A = 0.03 t + 0.5$$
(a) Use this relationship to find a complete equation for the model in the form
$$A = p q ^ { t }$$
giving the value of $p$ and the value of $q$ each to 4 significant figures.\\
(b) With reference to the model, interpret\\
(i) the value of the constant $p$,\\
(ii) the value of the constant $q$.\\
(c) Find, according to the model,\\
(i) the mass of algae in the pond when $t = 8$, giving your answer to the nearest 0.5 kg ,\\
(ii) the number of weeks it takes for the mass of algae in the pond to reach 4 kg .\\
(d) State one reason why this may not be a realistic model in the long term.
\hfill \mbox{\textit{Edexcel AS Paper 1 2022 Q5 [10]}}