| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Linear transformation to find constants |
| Difficulty | Moderate -0.5 This is a standard C2 exponential modelling question requiring plotting points, taking logarithms to linearise data, and finding constants from a graph. While it involves multiple steps, each step follows a routine procedure taught explicitly in the syllabus with no novel problem-solving required. The logarithmic transformation method is a textbook technique, making this slightly easier than average. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form1.06i Exponential growth/decay: in modelling context |
| Time \(( t\) hours \()\) | 1 | 2 | 3 | 4 | 5 |
| Number of bacteria \(( N )\) | 120 | 170 | 250 | 370 | 530 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) 5 points plotted accurately; Smooth curve drawn | B1, B1 | Total: 2 marks |
| (ii) exponential shape or increasing curve | B1 | Total: 1 mark |
| (iii) \(\log N = t \log b + \log a\) | B1 | |
| slope \(= \log b\) | B1 | |
| intercept \(= \log a\) | B1 | Total: 3 marks |
| (iv) \(\log N\) values \(= 2.08, 2.23, 2.40, 2.57, 2.72\) | B1 | |
| 5 points plotted and single best fit line drawn | B1 | |
| \(\log b = 0.16\) approximately | M1, A1 | |
| \(b = 1.45\) | A1 | |
| \(\log a = 1.92\) approximately | M1 | |
| \(a = 83\) | A1 | Total: 6 marks |
**(i)** 5 points plotted accurately; Smooth curve drawn | B1, B1 | **Total: 2 marks**
**(ii)** exponential shape or increasing curve | B1 | **Total: 1 mark**
**(iii)** $\log N = t \log b + \log a$ | B1 |
slope $= \log b$ | B1 |
intercept $= \log a$ | B1 | **Total: 3 marks**
**(iv)** $\log N$ values $= 2.08, 2.23, 2.40, 2.57, 2.72$ | B1 |
5 points plotted and single best fit line drawn | B1 |
$\log b = 0.16$ approximately | M1, A1 |
$b = 1.45$ | A1 |
$\log a = 1.92$ approximately | M1 |
$a = 83$ | A1 | **Total: 6 marks**10 A culture of bacteria is observed during an experiment. The number of bacteria is denoted by $N$ and the time in hours after the start of the experiment by $t$.\\
The table gives observations of $t$ and $N$.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Time $( t$ hours $)$ & 1 & 2 & 3 & 4 & 5 \\
\hline
Number of bacteria $( N )$ & 120 & 170 & 250 & 370 & 530 \\
\hline
\end{tabular}
\end{center}
(i) Plot the points $( t , N )$ on graph paper and join them with a smooth curve.\\
(ii) Explain why the curve suggests why the relationship connecting $t$ and $N$ may be of the form $N = a b ^ { t }$.\\
(iii) Explain how, by using logarithms, the curve given by plotting $N$ against $t$ can be transformed into a straight line.\\
State the gradient of this straight line and its intercept with the vertical axis in terms of $a$ and $b$.\\
(iv) Complete a table of values for $\log _ { 10 } N$ and plot the points $\left( t , \log _ { 10 } N \right)$ on graph paper. Draw the best fit line through the points and use it to estimate the values of $a$ and $b$.
\hfill \mbox{\textit{OCR MEI C2 Q10 [12]}}