| Exam Board | OCR |
|---|---|
| Module | AS Pure (AS Pure Mathematics) |
| Year | 2017 |
| Session | Specimen |
| Marks | 7 |
| Topic | Exponential Equations & Modelling |
| Type | log(y) vs x: convert and interpret |
| Difficulty | Moderate -0.8 This is a straightforward logarithmic transformation question requiring basic manipulation of exponential equations and reading values from a graph. Part (a) is routine algebra (taking log of both sides), part (b) involves reading gradient and intercept from a graph then converting back, and part (c) asks for a simple prediction with a standard reliability comment. All techniques are standard AS-level material with no problem-solving 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 form2.02c Scatter diagrams and regression lines |
| Number of weeks after the start (\(x\)) | 1 | 2 | 3 | 4 | 5 |
| Number of missed appointments (\(y\)) | 235 | 149 | 99 | 59 | 38 |
A doctors' surgery starts a campaign to reduce missed appointments.
The number of missed appointments for each of the first five weeks after the start of the campaign is shown below.
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Number of weeks after the start ($x$) & 1 & 2 & 3 & 4 & 5 \\
\hline
Number of missed appointments ($y$) & 235 & 149 & 99 & 59 & 38 \\
\hline
\end{tabular}
This data could be modelled by an equation of the form $y = pq^x$ where $p$ and $q$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Show that this relationship may be expressed in the form $\log_{10} y = mx + c$, expressing $m$ and $c$ in terms of $p$ and/or $q$. [2]
\end{enumerate}
The diagram below shows $\log_{10} y$ plotted against $x$, for the given data.
\includegraphics{figure_5}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Estimate the values of $p$ and $q$. [3]
\item Use the model to predict when the number of missed appointments will fall below 20.
Explain why this answer may not be reliable. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR AS Pure 2017 Q5 [7]}}