| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | log(y) vs x: convert and interpret |
| Difficulty | Standard +0.3 This is a standard exponential modelling question with routine calculations: part (a) involves basic substitution and solving e^t = 90 using logarithms; part (b) requires showing log₁₀P = log₁₀k + t·log₁₀a is linear (straightforward algebra), drawing a line of best fit from given data, and reading off intercept/gradient to find k and a. All techniques are textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| \(t\) | 1 | 2 | 3 | 4 | 5 |
| \(P\) | 100 | 500 | 1800 | 7000 | 19000 |
| \(\log _ { 10 } P\) | 2.00 | 2.70 | 3.26 | 3.85 | 4.28 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(5460\) (3 sf) | B1 | |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(9000 = 100e^t\), \(t = \ln 90\) | M1 | May be implied by answer |
| \(= 4.50\) (3 sf). Allow 4.5 ISW | A1 | Ignore units. Decimal answer needed |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\log_{10} P = \log_{10}(ka^t)\), \(\log_{10} P = \log_{10} k + \log_{10}(a^t)\) | M1 | At least two terms correct, may be implied by next line |
| \(\log_{10} P = \log_{10} k + t\log_{10} a\) | A1 | All correct, in this form |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Points plotted correctly \(\pm0.1\) | B1 | NB. May be implied by correct line of best fit |
| Line of best fit drawn, between \((1, 2.0)\) and \((1, 2.4)\) and between \((5, 4.2)\) and \((5, 4.5)\) | B1f | ft reasonable line through their points |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Read off \(c\) and attempt \(10^c\). May be implied by value of \(k\); \(k = 19.9\) to \(63.1\) | M1, A1 | ft their line. Probably \(c = 1.3\) to \(1.8\), \(k = 10^{1.3}\) to \(10^{1.8}\) |
| Attempt gradient of their graph AND correct ft equation in \(a\). May be implied by value of \(a\) | M1 | ft their line. Probably \(m = 0.5\) to \(0.7\) AND \(\log_{10} a = 0.5\) to \(0.7\) OR \(a = 10^{0.5}\) to \(10^{0.7}\) |
| \(a = 3.16\) to \(5.01\) (3 sf) | A1 | NB: Use of two points and simultaneous equations: no marks unless the two points used are on their line of best fit. |
| [4] |
## Question 6(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5460$ (3 sf) | B1 | |
| **[1]** | | |
---
## Question 6(a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $9000 = 100e^t$, $t = \ln 90$ | M1 | May be implied by answer |
| $= 4.50$ (3 sf). Allow 4.5 ISW | A1 | Ignore units. Decimal answer needed |
| **[2]** | | |
---
## Question 6(b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\log_{10} P = \log_{10}(ka^t)$, $\log_{10} P = \log_{10} k + \log_{10}(a^t)$ | M1 | At least two terms correct, may be implied by next line |
| $\log_{10} P = \log_{10} k + t\log_{10} a$ | A1 | All correct, in this form |
| **[2]** | | |
---
## Question 6(b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Points plotted correctly $\pm0.1$ | B1 | NB. May be implied by correct line of best fit |
| Line of best fit drawn, between $(1, 2.0)$ and $(1, 2.4)$ and between $(5, 4.2)$ and $(5, 4.5)$ | B1f | ft reasonable line through their points |
| **[2]** | | |
---
## Question 6(b)(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Read off $c$ and attempt $10^c$. May be implied by value of $k$; $k = 19.9$ to $63.1$ | M1, A1 | ft their line. Probably $c = 1.3$ to $1.8$, $k = 10^{1.3}$ to $10^{1.8}$ |
| Attempt gradient of their graph AND correct ft equation in $a$. May be implied by value of $a$ | M1 | ft their line. Probably $m = 0.5$ to $0.7$ AND $\log_{10} a = 0.5$ to $0.7$ OR $a = 10^{0.5}$ to $10^{0.7}$ |
| $a = 3.16$ to $5.01$ (3 sf) | A1 | NB: Use of two points and simultaneous equations: no marks unless the two points used are on their line of best fit. |
| **[4]** | | |
---6 During some research the size, $P$, of a population of insects, at time $t$ months after the start of the research, is modelled by the following formula.\\
$P = 100 \mathrm { e } ^ { t }$
\begin{enumerate}[label=(\alph*)]
\item Use this model to answer the following.
\begin{enumerate}[label=(\roman*)]
\item Find the value of $P$ when $t = 4$.
\item Find the value of $t$ when the population is 9000 .
\end{enumerate}\item It is suspected that a more appropriate model would be the following formula.\\
$P = k a ^ { t }$ where $k$ and $a$ are constants.
\begin{enumerate}[label=(\roman*)]
\item Show that, using this model, the graph of $\log _ { 10 } P$ against $t$ would be a straight line.
Some observations of $t$ and $P$ gave the following results.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$t$ & 1 & 2 & 3 & 4 & 5 \\
\hline
$P$ & 100 & 500 & 1800 & 7000 & 19000 \\
\hline
$\log _ { 10 } P$ & 2.00 & 2.70 & 3.26 & 3.85 & 4.28 \\
\hline
\end{tabular}
\end{center}
\item On the grid in the Printed Answer Booklet, draw a line of best fit for the data points $\left( t , \log _ { 10 } P \right)$ given in the table.
\item Hence estimate the values of $k$ and $a$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q6 [11]}}