| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2018 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Critique single model appropriateness |
| Difficulty | Standard +0.3 This is a straightforward multi-part question involving substitution into given formulas and basic interpretation. Parts (i)-(iii) require simple quadratic manipulation with clear conditions. Parts (iv)-(v) involve direct substitution into an exponential model and evaluating a limit. All steps are routine with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02z Models in context: use functions in modelling1.06a Exponential function: a^x and e^x graphs and properties |
| \(t\) | 1 | 3 | 5 |
| \(p\) | 59 | 84 | 89 |
| Answer | Marks | Guidance |
|---|---|---|
| \(C = 2\); \(A = 62\); \(B = 10\) | B1, B1, B1 [3] | AO 3.3, 3.3, 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| substitution of 0.75 in \(p = 62 - 10(t-2)^2\); \(46\) | M1, A1 [2] | AO 3.4, 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| their \(62 - 10(t-2)^2 = 0\); \([t=]\) 4 hours 29 minutes or 4 hours 30 minutes | M1, A1 [2] | AO 3.4, 2.4 |
| Answer | Marks | Guidance |
|---|---|---|
| substitution of \(t = 1, 3\) and \(5\); awrt \(59.4 \approx 59\), awrt \(83.8 \approx 84\), awrt \(88.8 \approx 89\) | M1, A1 [2] | AO 3.4, 3.5a |
| Answer | Marks | Guidance |
|---|---|---|
| \(p \to 90\) as \(t \to\) large or when \(t=12\), \(p = 89.99539\)...rounded to 2 or more sf | B1 | AO 3.5a |
| comparison with value of \(p\) for \(t=5\) eg model predicts \(p=89\) for \(t=5\) and \(p=90\) for \(t=12\) so not good advice | B1 [2] | AO 3.5a |
## Question 16(i):
$C = 2$; $A = 62$; $B = 10$ | B1, B1, B1 [3] | AO 3.3, 3.3, 1.1 | since max when $t=2$; since max when $(t-2)^2=0$; from substitution of 22, 62 and 2
---
## Question 16(ii):
substitution of 0.75 in $p = 62 - 10(t-2)^2$; $46$ | M1, A1 [2] | AO 3.4, 1.1 | FT their 2, 62, 10; allow 46.375 rounded to 2 or more sf
---
## Question 16(iii):
their $62 - 10(t-2)^2 = 0$; $[t=]$ 4 hours 29 minutes or 4 hours 30 minutes | M1, A1 [2] | AO 3.4, 2.4 | or $\geq 0$ or $> 0$ for M1; NB $t = 2 + \sqrt{6.2}$; allow 4.49 or 4.5 [hours]
---
## Question 16(iv):
substitution of $t = 1, 3$ and $5$; awrt $59.4 \approx 59$, awrt $83.8 \approx 84$, awrt $88.8 \approx 89$ | M1, A1 [2] | AO 3.4, 3.5a | or awrt 59.4, 83.8 and 88.8 found and supporting comment made eg they are approximately the same as the values in the table; if **M0** allow **SC1** for two values correctly found and shown to be consistent or supporting comment made
---
## Question 16(v):
$p \to 90$ as $t \to$ large or when $t=12$, $p = 89.99539$...rounded to 2 or more sf | B1 | AO 3.5a |
comparison with value of $p$ for $t=5$ eg model predicts $p=89$ for $t=5$ and $p=90$ for $t=12$ so not good advice | B1 [2] | AO 3.5a | allow equivalent comment on 7 hours work for one extra mark; or model predicts $p=90$ for (any) $t \geq 7$ so not good advice
---16 In the first year of a course, an A-level student, Aaishah, has a mathematics test each week. The night before each test she revises for $t$ hours. Over the course of the year she realises that her percentage mark for a test, $p$, may be modelled by the following formula, where $A , B$ and $C$ are constants.
$$p = A - B ( t - C ) ^ { 2 }$$
\begin{itemize}
\item Aaishah finds that, however much she revises, her maximum mark is achieved when she does 2 hours revision. This maximum mark is 62 .
\item Aaishah had a mark of 22 when she didn't spend any time revising.\\
(i) Find the values of $A , B$ and $C$.\\
(ii) According to the model, if Aaishah revises for 45 minutes on the night before the test, what mark will she achieve?\\
(iii) What is the maximum amount of time that Aaishah could have spent revising for the model to work?
\end{itemize}
In an attempt to improve her marks Aaishah now works through problems for a total of $t$ hours over the three nights before the test. After taking a number of tests, she proposes the following new formula for $p$.
$$p = 22 + 68 \left( 1 - \mathrm { e } ^ { - 0.8 t } \right)$$
For the next three tests she recorded the data in Fig. 16.
\begin{table}[h]
\begin{center}
\begin{tabular}{ | c | c | c | c | }
\hline
$t$ & 1 & 3 & 5 \\
\hline
$p$ & 59 & 84 & 89 \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 16}
\end{center}
\end{table}
(iv) Verify that the data is consistent with the new formula.\\
(v) Aaishah's tutor advises her to spend a minimum of twelve hours working through problems in future. Determine whether or not this is good advice.
\hfill \mbox{\textit{OCR MEI Paper 2 2018 Q16 [11]}}