| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Topic | Second order differential equations |
| Type | Modeling context with interpretation |
| Difficulty | Standard +0.3 This is a standard second-order linear homogeneous differential equation with constant coefficients. Students solve the auxiliary equation (m-5)(m+3)=0, apply the boundary conditions Q(0)=100 and the finite limit condition (which forces the coefficient of e^{5t} to be zero), then substitute t=0.5. While it requires multiple steps, it follows a completely routine procedure taught explicitly in FP2 with no novel problem-solving required. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method |
A capacitor is an electrical component which stores charge. The value of the charge stored by the capacitor, in suitable units, is denoted by $Q$. The capacitor is placed in an electrical circuit.
At any time $t$ seconds, where $t \geq 0$, $Q$ can be modelled by the differential equation
$$\frac{d^2Q}{dt^2} - 2\frac{dQ}{dt} - 15Q = 0.$$
Initially the charge is 100 units and it is given that $Q$ tends to a finite limit as $t$ tends to infinity.
\begin{enumerate}[label=(\alph*)]
\item Determine the charge on the capacitor when $t = 0.5$. [6]
\item Determine the finite limit of $Q$ as $t$ tends to infinity. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2021 Q3 [7]}}