| Exam Board | SPS |
|---|---|
| Module | SPS SM (SPS SM) |
| Year | 2025 |
| Session | November |
| Marks | 11 |
| Topic | Exponential Equations & Modelling |
| Type | log(y) vs x: convert and interpret |
| Difficulty | Moderate -0.8 This is a standard logarithmic transformation question requiring students to take logs of an exponential model, complete a table, draw a graph, and find constants from the line of best fit. While it involves multiple steps (11 marks total), each step is routine: explaining why log y vs t is linear is textbook material, calculating logs is straightforward, and finding a and b from gradient/intercept is a standard technique taught in all A-level statistics courses. No novel problem-solving or insight required. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form |
| 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 |
| \(t\) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| \(\log_{10} y\) | 1.51 | 1.58 | 1.98 | 2.68 |
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{center}
\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}
\end{center}
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}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$t$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
$\log_{10} y$ & & & & 1.51 & & 1.58 & & 1.98 & & 2.68 \\
\hline
\end{tabular}
\end{center}
\hfill \mbox{\textit{SPS SPS SM 2025 Q7 [11]}}