| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Equations & Modelling |
| Type | log(y) vs x: convert and interpret |
| Difficulty | Moderate -0.3 This is a standard logarithmic transformation question requiring students to linearize exponential data and find a line of best fit. While it involves multiple steps (taking logs, plotting, finding gradient/intercept, back-transforming), each step follows a routine procedure taught in C2. The mathematical techniques are straightforward applications of log laws and graph work, making it slightly easier than average for A-level but not trivial due to the multi-part nature and data handling requirements. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| Year | 1955 | 1965 | 1975 | 1985 | 1995 | 2005 |
| Population (millions) | 131 | 161 | 209 | 277 | 372 | 492 |
Answer part (ii) of this question on the insert provided.
Since 1945 the populations of many countries have been growing. The table shows the estimated population of 15- to 59-year-olds in Africa during the period 1955 to 2005.
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Year & 1955 & 1965 & 1975 & 1985 & 1995 & 2005 \\
\hline
Population (millions) & 131 & 161 & 209 & 277 & 372 & 492 \\
\hline
\end{tabular}
\end{center}
Source: United Nations
Such estimates are used to model future population growth and world needs of resources. One model is $P = a10^{bt}$, where the population is $P$ millions, $t$ is the number of years after 1945 and $a$ and $b$ are constants.
\begin{enumerate}[label=(\roman*)]
\item Show that, using this model, the graph of $\log_{10} P$ against $t$ is a straight line of gradient $b$. State the intercept of this line on the vertical axis. [3]
\item On the insert, complete the table, giving values correct to 2 decimal places, and plot the graph of $\log_{10} P$ against $t$. Draw, by eye, a line of best fit on your graph. [3]
\item Use your graph to find the equation for $P$ in terms of $t$. [4]
\item Use your results to estimate the population of 15- to 59-year-olds in Africa in 2050. Comment, with a reason, on the reliability of this estimate. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 2010 Q12 [13]}}