| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 2 (AS Paper 2) |
| Year | 2020 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Exponential Functions |
| Type | Exponential model with shifted asymptote |
| Difficulty | Standard +0.3 This is a straightforward exponential modelling question requiring substitution to find parameters, evaluation at given points, differentiation for acceleration, and limit analysis. All techniques are standard AS-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06i Exponential growth/decay: in modelling context1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| \(A = 11\) | B1 (1.1) | from \(t=0\) and \(V=11\) |
| Substitution of \(t=3\) and \(V=13.8\) | M1 (3.3) | \(13.8 = A + B(1 - e^{-0.17 \times 3})\); implied by 7.0086 |
| \(B = 7.0\) | A1 (1.1) [3] | Allow 7, 7.01, 7.009, etc |
| Answer | Marks | Guidance |
|---|---|---|
| \(t=4\) and \(V=14.453681053\)... to 1 or more dp good fit | B1 (3.4) | allow SC1 for two correct values with no comment or incorrect comment(s); allow full marks if more accurate value of \(B\) used: 14.458 and 15.013 |
| \(t=5\) and \(V=15.008095476\).. to 1 or more dp good fit | B1 (1.1) [2] | If B0B0, allow sc M1 for subst with their values |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dV}{dt} =\) their \(7.0 \times 0.17\, e^{-0.85}\) | M1 (3.1a) | Allow for their \(7.0 \times 0.17\, e^{-0.17t}\); condone use of 7.0086 for \(B\) |
| \(0.509\ \text{ms}^{-2}\) | A1 (1.1) [2] |
| Answer | Marks | Guidance |
|---|---|---|
| \(t \to \infty,\quad V \to 11 + 7\) oe | M1 (3.3) | Allow ft from (a) for M1M1 |
| their \(18 \times 3.6\) | M1 (1.1) | Allow for any attempt to change to km/hr, accept use of 'their 18' |
| \(64.8 > 60\) so the motorist is fined | A1 (3.5a) [3] | Need to show \(64.8 > 60\) for full marks |
## Question 11(a):
$A = 11$ | B1 (1.1) | from $t=0$ and $V=11$
Substitution of $t=3$ and $V=13.8$ | M1 (3.3) | $13.8 = A + B(1 - e^{-0.17 \times 3})$; implied by 7.0086
$B = 7.0$ | A1 (1.1) **[3]** | Allow 7, 7.01, 7.009, etc
---
## Question 11(b):
$t=4$ and $V=14.453681053$... to 1 or more dp good fit | B1 (3.4) | allow SC1 for two correct values with no comment or incorrect comment(s); allow full marks if more accurate value of $B$ used: 14.458 and 15.013
$t=5$ and $V=15.008095476$.. to 1 or more dp good fit | B1 (1.1) **[2]** | If B0B0, allow sc M1 for subst with their values
---
## Question 11(c):
$\frac{dV}{dt} =$ their $7.0 \times 0.17\, e^{-0.85}$ | M1 (3.1a) | Allow for their $7.0 \times 0.17\, e^{-0.17t}$; condone use of 7.0086 for $B$
$0.509\ \text{ms}^{-2}$ | A1 (1.1) **[2]** |
---
## Question 11(d):
$t \to \infty,\quad V \to 11 + 7$ oe | M1 (3.3) | Allow ft from (a) for M1M1
their $18 \times 3.6$ | M1 (1.1) | Allow for any attempt to change to km/hr, accept use of 'their 18'
$64.8 > 60$ so the motorist is fined | A1 (3.5a) **[3]** | Need to show $64.8 > 60$ for full marks11 A car is travelling along a stretch of road at a steady speed of $11 \mathrm {~ms} ^ { - 1 }$.\\
The driver accelerates, and $t$ seconds after starting to accelerate the speed of the car, $V$, is modelled by the formula\\
$\mathrm { V } = \mathrm { A } + \mathrm { B } \left( 1 - \mathrm { e } ^ { - 0.17 \mathrm { t } } \right)$.\\
When $t = 3 , V = 13.8$.
\begin{enumerate}[label=(\alph*)]
\item Find the values of $A$ and $B$, giving your answers correct to 2 significant figures.
When $t = 4 , V = 14.5$ and when $t = 5 , V = 14.9$.
\item Determine whether the model is a good fit for these data.
\item Determine the acceleration of the car according to the model when $t = 5$, giving your answer correct to 3 decimal places.
The car continues to accelerate until it reaches its maximum speed.\\
The speed limit on this road is $60 \mathrm { kmh } ^ { - 1 }$. All drivers who exceed this speed limit are recorded by a speed camera and automatically fined $\pounds 100$.
\item Determine whether, according to the model, the driver of this car is fined $\pounds 100$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 2 2020 Q11 [10]}}