| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Linear transformation to find constants |
| Difficulty | Moderate -0.3 This is a standard logarithmic linearization question requiring routine application of log laws (part i), plotting points (part ii), reading gradient and intercept from a graph (part iii), substituting t=0 (part iv), and solving a logarithmic equation (part v). While multi-part with several marks, each step follows a well-practiced procedure with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06h Logarithmic graphs: reduce y=ax^n and y=kb^x to linear form2.02c Scatter diagrams and regression lines |
| Years after proposal accepted \(( t )\) | 1 | 2 | 3 | 4 | 5 |
| Cost \(( \pounds y\) million \()\) | 250 | 300 | 360 | 440 | 530 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\log a + \log(b^t)\) www | B1 | condone omission of base throughout question |
| clear use of \(\log(b^t) = t\log b\) dep | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \((2.398),\ 2.477,\ 2.556,\ 2.643,\ 2.724\) | T1 | |
| points plotted correctly f.t. | P1 | On correct square |
| ruled line of best fit f.t. | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\log a = 2.31\) to \(2.33\) | M1 | ft their intercept |
| \(a = 204\) to \(214\) | A1 | |
| \(\log b = 0.08\) approx | M1 | ft their gradient |
| \(b = 1.195\) to \(1.215\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| e.g. £210 million dep | 1 | their £\(a\) million |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{\log 1000 - \text{their intercept}}{\text{their gradient}} \approx \frac{3 - 2.32}{0.08}\) | M1 | |
| \(= 8.15\) to \(8.85\) | A1 | or B2 from trials |
## Question 12:
### Part i:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\log a + \log(b^t)$ www | B1 | condone omission of base throughout question |
| clear use of $\log(b^t) = t\log b$ dep | B1 | | **[2]** |
### Part ii:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $(2.398),\ 2.477,\ 2.556,\ 2.643,\ 2.724$ | T1 | |
| points plotted correctly f.t. | P1 | On correct square |
| ruled line of best fit f.t. | 1 | | **[3]** |
### Part iii:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\log a = 2.31$ to $2.33$ | M1 | ft their intercept |
| $a = 204$ to $214$ | A1 | |
| $\log b = 0.08$ approx | M1 | ft their gradient |
| $b = 1.195$ to $1.215$ | A1 | | **[4]** |
### Part iv:
| Answer/Working | Marks | Guidance |
|---|---|---|
| e.g. £210 million dep | 1 | their £$a$ million | **[1]** |
### Part v:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{\log 1000 - \text{their intercept}}{\text{their gradient}} \approx \frac{3 - 2.32}{0.08}$ | M1 | |
| $= 8.15$ to $8.85$ | A1 | or B2 from trials | **[2]** |12 Answer part (ii) of this question on the insert provided.
The proposal for a major building project was accepted, but actual construction was delayed. Each year a new estimate of the cost was made. The table shows the estimated cost, $\pounds y$ million, of the project $t$ years after the project was first accepted.
\begin{center}
\begin{tabular}{ | l | c | c | c | c | c | }
\hline
Years after proposal accepted $( t )$ & 1 & 2 & 3 & 4 & 5 \\
\hline
Cost $( \pounds y$ million $)$ & 250 & 300 & 360 & 440 & 530 \\
\hline
\end{tabular}
\end{center}
The relationship between $y$ and $t$ is modelled by $y = a b ^ { t }$, where $a$ and $b$ are constants.\\
(i) Show that $y = a b ^ { t }$ may be written as
$$\log _ { 10 } y = \log _ { 10 } a + t \log _ { 10 } b$$
(ii) On the insert, complete the table and plot $\log _ { 10 } y$ against $t$, drawing by eye a line of best fit.\\
(iii) Use your graph and the results of part (i) to find the values of $\log _ { 10 } a$ and $\log _ { 10 } b$ and hence $a$ and $b$.\\
(iv) According to this model, what was the estimated cost of the project when it was first accepted?\\
(v) Find the value of $t$ given by this model when the estimated cost is $\pounds 1000$ million. Give your answer rounded to 1 decimal place.
\hfill \mbox{\textit{OCR MEI C2 2009 Q12 [12]}}