| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | Calculus with exponential models |
| Difficulty | Standard +0.3 This is a standard exponential modelling question requiring logarithmic manipulation to find constants (using two points on a line) and then differentiation of an exponential function. All techniques are routine for P3/C3 level with clear scaffolding through parts (a) and (b). Slightly easier than average due to straightforward application of well-practiced methods. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form1.06i Exponential growth/decay: in modelling context1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States or implies \(\log_{10} p = 0.32\) or \(\log_{10} q = \frac{0.56 - 0.32}{8}\) | M1 | Or equivalent equations |
| \(p =\) awrt \(2.089\) or \(q =\) awrt \(1.072\) | A1 | |
| States or implies \(\log_{10} p = 0.32\) and \(\log_{10} q = \frac{0.56 - 0.32}{8}\) | M1 | Or equivalent equations |
| \(p =\) awrt \(2.089\) and \(q =\) awrt \(1.072\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States or implies \(\frac{dA}{dt} = p \ln q \times q^t\) with their values for \(p\) and \(q\) | M1 A1 | Uses \(\frac{d}{dt}q^t \rightarrow kq^t\), \(k \neq 1\) |
| Rate of increase in pond weed after 6 days is \(0.22\) (m²/day) | A1 | Units not required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to differentiate and reaches \(\frac{1}{A}\frac{dA}{dt} = k\) | M1 | Or equivalent |
| \(\frac{1}{A \ln 10}\frac{dA}{dt} = 0.03\) | A1 | |
| awrt \(0.22\) | A1 | Units not required |
## Question 5:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States or implies $\log_{10} p = 0.32$ **or** $\log_{10} q = \frac{0.56 - 0.32}{8}$ | M1 | Or equivalent equations |
| $p =$ awrt $2.089$ **or** $q =$ awrt $1.072$ | A1 | |
| States or implies $\log_{10} p = 0.32$ **and** $\log_{10} q = \frac{0.56 - 0.32}{8}$ | M1 | Or equivalent equations |
| $p =$ awrt $2.089$ **and** $q =$ awrt $1.072$ | A1 | |
**(4 marks)**
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States or implies $\frac{dA}{dt} = p \ln q \times q^t$ with their values for $p$ and $q$ | M1 A1 | Uses $\frac{d}{dt}q^t \rightarrow kq^t$, $k \neq 1$ |
| Rate of increase in pond weed after 6 days is $0.22$ (m²/day) | A1 | Units not required |
**(3 marks)**
**Alt (b):** Using $\log_{10} A = 0.03t + 0.32$ as starting point:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to differentiate and reaches $\frac{1}{A}\frac{dA}{dt} = k$ | M1 | Or equivalent |
| $\frac{1}{A \ln 10}\frac{dA}{dt} = 0.03$ | A1 | |
| awrt $0.22$ | A1 | Units not required |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{76205772-5395-4ab2-96f9-ad9803b8388c-16_582_737_248_607}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The growth of duckweed on a pond is being studied.
The surface area of the pond covered by duckweed, $A \mathrm {~m} ^ { 2 }$, at a time $t$ days after the start of the study is modelled by the equation
$$A = p q ^ { t } \quad \text { where } p \text { and } q \text { are positive constants }$$
Figure 1 shows the linear relationship between $\log _ { 10 } A$ and $t$.\\
The points $( 0,0.32 )$ and $( 8,0.56 )$ lie on the line as shown.
\begin{enumerate}[label=(\alph*)]
\item Find, to 3 decimal places, the value of $p$ and the value of $q$.
Using the model with the values of $p$ and $q$ found in part (a),
\item find the rate of increase of the surface area of the pond covered by duckweed, in $\mathrm { m } ^ { 2 }$ / day, exactly 6 days after the start of the study.\\
Give your answer to 2 decimal places.\\
\includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-19_2649_1840_117_114}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2021 Q5 [7]}}