| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Laws of Logarithms |
| Type | Model y=ab^x: linearise and find constants from graph/data |
| Difficulty | Moderate -0.3 This is a standard C2 logarithmic transformation question requiring routine application of log laws, plotting transformed data, finding a line of best fit, and interpreting the model. While multi-part with 13 marks total, each step follows a predictable template with no novel problem-solving required—slightly easier than average due to its formulaic nature. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form2.02c Scatter diagrams and regression lines |
| Year | 1965 | 1970 | 1975 | 1980 | 1985 | 1990 | 1995 | 2000 | 2005 | 2010 |
| Number of years since 1960 \((t)\) | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 |
| Reduction in thickness since 1960 \((h\) m\()\) | 0.7 | 1.0 | 1.7 | 2.3 | 3.6 | 4.7 | 6.0 | 8.2 | 12 | 15.9 |
The thickness of a glacier has been measured every five years from 1960 to 2010. The table shows the reduction in thickness from its measurement in 1960.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Year & 1965 & 1970 & 1975 & 1980 & 1985 & 1990 & 1995 & 2000 & 2005 & 2010 \\
\hline
Number of years since 1960 $(t)$ & 5 & 10 & 15 & 20 & 25 & 30 & 35 & 40 & 45 & 50 \\
\hline
Reduction in thickness since 1960 $(h$ m$)$ & 0.7 & 1.0 & 1.7 & 2.3 & 3.6 & 4.7 & 6.0 & 8.2 & 12 & 15.9 \\
\hline
\end{tabular}
\end{center}
An exponential model may be used for these data, assuming that the relationship between $h$ and $t$ is of the form $h = a \times 10^{bt}$, where $a$ and $b$ are constants to be determined.
\begin{enumerate}[label=(\roman*)]
\item Show that this relationship may be expressed in the form $\log_{10} h = mt + c$, stating the values of $m$ and $c$ in terms of $a$ and $b$. [2]
\item Complete the table of values in the answer book, giving your answers correct to 2 decimal places, and plot the graph of $\log_{10} h$ against $t$, drawing by eye a line of best fit. [4]
\item Use your graph to find $h$ in terms of $t$ for this model. [4]
\item Calculate by how much the glacier will reduce in thickness between 2010 and 2020, according to the model. [2]
\item Give one reason why this model will not be suitable in the long term. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 2014 Q13 [13]}}