| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 12 |
| 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 C2 logarithmic transformation question requiring routine application of log laws to linearise exponential data, followed by drawing a graph and reading off gradient/intercept. Part (iii) is direct substitution. While it involves multiple steps (8 marks for part ii), each step follows a well-practiced procedure with no novel problem-solving required, making it slightly easier than average. |
| 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 |
| Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Number of flu viruses | 7 | 10 | 24 | 32 | 40 | 38 | 63 | 96 | 234 | 480 |
There are many different flu viruses. The numbers of flu viruses detected in the first few weeks of the 2012–2013 flu epidemic in the UK were as follows.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Week & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Number of flu viruses & 7 & 10 & 24 & 32 & 40 & 38 & 63 & 96 & 234 & 480 \\
\hline
\end{tabular}
These data may be modelled by an equation of the form $y = a \times 10^{bt}$, where $y$ is the number of flu viruses detected in week $t$ of the epidemic, and $a$ and $b$ are constants to be determined.
\begin{enumerate}[label=(\roman*)]
\item Explain why this model leads to a straight-line graph of $\log_{10} y$ against $t$. State the gradient and intercept of this graph in terms of $a$ and $b$. [3]
\item Complete the values of $\log_{10} y$ in the table, draw the graph of $\log_{10} y$ against $t$, and draw by eye a line of best fit for the data.
Hence determine the values of $a$ and $b$ and the equation for $y$ in terms of $t$ for this model. [8]
\end{enumerate}
During the decline of the epidemic, an appropriate model was
$$y = 921 \times 10^{-0.137w},$$
where $y$ is the number of flu viruses detected in week $w$ of the decline.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Use this to find the number of viruses detected in week 4 of the decline. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 2016 Q11 [12]}}