| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Topic | Exponential Functions |
| Type | Time to reach target in exponential model |
| Difficulty | Moderate -0.8 This is a straightforward applied exponential decay problem requiring only basic manipulation of exponential functions (solving ae^{-kt} = 1 for t) and understanding superposition of doses. All parts are routine A-level techniques with no conceptual challenges or novel problem-solving required. |
| Spec | 1.06a Exponential function: a^x and e^x graphs and properties1.06g Equations with exponentials: solve a^x = b1.06i Exponential growth/decay: in modelling context |
| Answer | Marks | Guidance |
|---|---|---|
| Attempt to solve \(c = 1\) (or \(c < 1\)) for at least one drug, and obtain a solution | M1 | |
| Obtain 54.9 (hours) for Antiflu | A1 | |
| Obtain 23.0 (hours) for Coldcure | A1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Two decaying exponentials in the first quadrant showing | M1 | |
| correct intercepts on the \(c\)-axis and crossing for some \(t > 0\) | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Assume additive nature of the concentrations | M1 | |
| \(5e^{-0.07 \times 30} + 5e^{-0.07 \times 10} = 3.10\) | A1 | [2] |
**Part (i)**
Attempt to solve $c = 1$ (or $c < 1$) for at least one drug, and obtain a solution | M1
Obtain 54.9 (hours) for Antiflu | A1
Obtain 23.0 (hours) for Coldcure | A1 | [3]
**Part (ii)**
Two decaying exponentials in the first quadrant showing | M1
correct intercepts on the $c$-axis and crossing for some $t > 0$ | A1 | [2]
**Part (iii)**
Assume additive nature of the concentrations | M1
$5e^{-0.07 \times 30} + 5e^{-0.07 \times 10} = 3.10$ | A1 | [2]Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time $t$ hours after the injection, the concentration of Antiflu in Diane's bloodstream is $3e^{-0.02t}$ units and the concentration of Coldcure is $5e^{-0.07t}$ units. Each drug becomes ineffective when its concentration falls below 1 unit.
\begin{enumerate}[label=(\roman*)]
\item Show that Coldcure becomes ineffective before Antiflu. [3]
\item Sketch, on the same diagram, the graphs of concentration against time for each drug. [2]
\item 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later. [2]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2011 Q5 [7]}}