| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2025 |
| Session | February |
| Marks | 6 |
| Topic | Exponential Equations & Modelling |
| Type | log(y) vs x: convert and interpret |
| Difficulty | Standard +0.3 This is a straightforward exponential modelling question requiring routine manipulation of logarithms (converting log-linear to exponential form), substitution to find initial conditions, and basic differentiation. All steps are standard textbook exercises with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06f Laws of logarithms: addition, subtraction, power rules1.06g Equations with exponentials: solve a^x = b1.06i Exponential growth/decay: in modelling context1.07k Differentiate trig: sin(kx), cos(kx), tan(kx) |
5. The number of bacteria on a surface is being monitored.
The number of bacteria, $N$, on the surface, $t$ hours after monitoring began is modelled by the equation
$$\log _ { 10 } N = 0.35 t + 2$$
Use the equation of the model to answer parts (a) to (c).
\begin{enumerate}[label=(\alph*)]
\item Find the initial number of bacteria on the surface.
\item Show that the equation of the model can be written in the form
$$N = a b ^ { t }$$
where $a$ and $b$ are constants to be found. Give the value of $b$ to 2 decimal places.
\item Hence find the rate of growth of bacteria on the surface exactly 5 hours after monitoring began.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2025 Q5 [6]}}