| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Differential equations |
| Type | Exponential growth/decay - direct proportionality (dN/dt = kN) |
| Difficulty | Moderate -0.3 This is a straightforward exponential growth question requiring standard techniques: forming dy/dt = ky, solving by separation of variables to get y = Ae^(kt), applying initial conditions, and substituting given values. All steps are routine A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.06i Exponential growth/decay: in modelling context1.07t Construct differential equations: in context1.08k Separable differential equations: dy/dx = f(x)g(y) |
6 Helga invests $\pounds 4000$ in a savings account.\\
After $t$ days, her investment is worth $\pounds y$.\\
The rate of increase of $y$ is $k y$, where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Write down a differential equation in terms of $t , y$ and $k$.
\item Solve your differential equation to find the value of Helga's investment after $t$ days. Give your answer in terms of $k$ and $t$.
It is given that $k = \frac { 1 } { 365 } \ln \left( 1 + \frac { r } { 100 } \right)$ where $r \%$ is the rate of interest per annum. During the first year the rate of interest is $6 \%$ per annum.
\item Find the value of Helga's investment after 90 days.
After one year (365 days), the rate of interest drops to 5\% per annum.
\item Find the total time that it will take for Helga's investment to double in value.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 Q6 [12]}}