| Exam Board | AQA |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential model with shifted asymptote |
| Difficulty | Moderate -0.3 This is a standard exponential modeling question with a shifted asymptote (Newton's Law of Cooling). Part (a) requires simple substitution at t=0, part (b) involves solving for k using logarithms (routine AS-level technique), part (c) is another logarithmic equation, and part (d) tests understanding of model limitations. While it requires multiple steps, all techniques are standard AS-level material with no novel insight needed—slightly easier than average due to its predictable structure and clear scaffolding. |
| Spec | 1.06i Exponential growth/decay: in modelling context |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(a = 14\) | B1 | AO3.3 – Obtains 14 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(12 = 20 - 14e^{-10k}\) | M1 | AO3.4 – Substitutes 12 and 10 into given equation |
| Forms a fully correct equation | A1F | AO1.1a – FT their value for \(a\) |
| \(k = 0.056\) | A1 | AO1.1b – Obtains correct value of \(k\); AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(18 = 20 - 14e^{-0.056t}\) | B1 | AO3.4 – Substitutes their \(k\) and other values correctly into given model |
| Solves model equation to obtain a value for \(t\), where \(t > 10\) | M1 | AO1.1a |
| \(t = 35\) minutes | A1 | AO3.2a – Obtains correct value of \(t\) including units; AWRT 35 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| It is not likely that the room temperature will stay at 20°C over such a long period | E1 | AO3.5b – Explains that conditions may change, e.g. someone may have drunk the water; OR states that after 3 hours the water will be effectively at room temperature |
## Question 10(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $a = 14$ | B1 | AO3.3 – Obtains 14 |
---
## Question 10(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $12 = 20 - 14e^{-10k}$ | M1 | AO3.4 – Substitutes 12 and 10 into given equation |
| Forms a fully correct equation | A1F | AO1.1a – FT their value for $a$ |
| $k = 0.056$ | A1 | AO1.1b – Obtains correct value of $k$; AWRT |
---
## Question 10(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $18 = 20 - 14e^{-0.056t}$ | B1 | AO3.4 – Substitutes their $k$ and other values correctly into given model |
| Solves model equation to obtain a value for $t$, where $t > 10$ | M1 | AO1.1a |
| $t = 35$ minutes | A1 | AO3.2a – Obtains correct value of $t$ including units; AWRT 35 |
---
## Question 10(d):
| Answer | Mark | Guidance |
|--------|------|----------|
| It is not likely that the room temperature will stay at 20°C over such a long period | E1 | AO3.5b – Explains that conditions may change, e.g. someone may have drunk the water; OR states that after 3 hours the water will be effectively at room temperature |
---10 A bottle of water has a temperature of $6 ^ { \circ } \mathrm { C }$ when it is removed from a refrigerator.
It is placed in a room where the temperature is $20 ^ { \circ } \mathrm { C }$\\
10 minutes later, the temperature of the water is $12 ^ { \circ } \mathrm { C }$\\
The temperature of the water, $T ^ { \circ } \mathrm { C }$, at time $t$ minutes after it is removed from the refrigerator, may be modelled by the equation
$$T = 20 - a \mathrm { e } ^ { - k t }$$
10
\begin{enumerate}[label=(\alph*)]
\item Find the value of $a$.
10
\item Calculate the value of $k$, giving your answer to two significant figures.\\
10
\item Using this model, estimate how long it takes the water to reach a temperature of\\
$18 ^ { \circ } \mathrm { C }$ after it is taken out of the refrigerator. $18 ^ { \circ } \mathrm { C }$ after it is taken out of the refrigerator.
10
\item Explain why the model may not be appropriate to predict the temperature of the water three hours after it is taken out of the refrigerator.
\end{enumerate}
\hfill \mbox{\textit{AQA AS Paper 2 2022 Q10 [8]}}