| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2018 |
| Session | September |
| Marks | 8 |
| Topic | Exponential Functions |
| Type | Critique single model appropriateness |
| Difficulty | Standard +0.3 This is a straightforward exponential modeling question requiring students to form an equation from given information (sales and rate), test a data point, and make a qualitative comment about extrapolation. The mathematics is routine (solving dS/dt = kS, substituting values), but the multi-part structure and need to interpret the model's reliability elevates it slightly above average difficulty. |
| Spec | 1.02z Models in context: use functions in modelling1.06i Exponential growth/decay: in modelling context |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \(S = Ae^{kt}\) | B1 |
| \(S = 3.1e^{kt}\) | B1 | Identify correct initial value (May still be A and not 3.1) |
| \(\frac{dS}{dt} = 3.1ke^{kt}\) | M1 | Attempt differentiation (OR \(\frac{dS}{dt} = 3.1(\ln b)b^t\) OR \(a = 3.1\)) |
| \(0.8 = 3.1ke^0\) hence \(k = 0.258\) | M1 | Substitute into derivative and attempt to find \(k\) |
| \(S = 3.1e^{0.258t}\), where \(S\) is the annual sales in millions of devices and \(t\) is the number of years after 2015 | A1 | Correct equation with variables clearly defined |
Total: [5]
| Answer | Marks | Guidance |
|---|---|---|
| (ii) | when \(t = 2\), \(S = 3.1e^{0.516} = 5.19\) (millions) | M1 |
| E.g. so observed value was 5.2 (millions) so model appears to be reliable | E1 | Comment on reliability of model (Must have correct 5.2 million, from correct model) |
Total: [2]
| Answer | Marks | Guidance |
|---|---|---|
| (iii) | E.g. unlikely to be a reliable prediction as market will become saturated so sales unlikely to increase at same rate | E1 |
Total: [1]
(i) | $S = Ae^{kt}$ | B1 | State or imply appropriate exponential model (Other models are possible eg using $t$ as number of years after a year other than 2015 OR $S = ab^t$ OR $a = 3.1$)
| $S = 3.1e^{kt}$ | B1 | Identify correct initial value (May still be A and not 3.1)
| $\frac{dS}{dt} = 3.1ke^{kt}$ | M1 | Attempt differentiation (OR $\frac{dS}{dt} = 3.1(\ln b)b^t$ OR $a = 3.1$)
| $0.8 = 3.1ke^0$ hence $k = 0.258$ | M1 | Substitute into derivative and attempt to find $k$
| $S = 3.1e^{0.258t}$, where $S$ is the annual sales in millions of devices and $t$ is the number of years after 2015 | A1 | Correct equation with variables clearly defined
Total: [5]
(ii) | when $t = 2$, $S = 3.1e^{0.516} = 5.19$ (millions) | M1 | Find value of $S$ when $t = 2$ (Using their model which must be of the form $Ae^{kt}$ or $ab^t$, with numerical parameters)
| E.g. so observed value was 5.2 (millions) so model appears to be reliable | E1 | Comment on reliability of model (Must have correct 5.2 million, from correct model)
Total: [2]
(iii) | E.g. unlikely to be a reliable prediction as market will become saturated so sales unlikely to increase at same rate | E1 | Comment about trend unlikely to continue, or device becoming obsolete or extrapolation may not be reliable
Total: [1]9 An analyst believes that the sales of a particular electronic device are growing exponentially. In 2015 the sales were 3.1 million devices and the rate of increase in the annual sales is 0.8 million devices per year.\\
(i) Find a model to represent the annual sales, defining any variables used.\\
(ii) In 2017 the sales were 5.2 million devices. Determine whether this is consistent with the model in part (i).\\
(iii) The analyst uses the model in part (i) to predict the sales for 2025. Comment on the reliability of this prediction.
\hfill \mbox{\textit{OCR H240/01 2018 Q9 [8]}}