| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2020 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.3 This is a standard second-order linear homogeneous differential equation with constant coefficients. Students need to find the auxiliary equation (m² - 2m - 15 = 0), solve for roots (m = 5, -3), write the general solution, apply initial conditions including the constraint that Q→finite limit as t→∞ (which forces the coefficient of e^(5t) to be zero), then evaluate at t=0.5. While it requires multiple steps and understanding of limiting behavior, it follows a completely standard procedure taught in FP2 with no novel insight required. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10e Second order non-homogeneous: complementary + particular integral |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (a) | AE: m2 – 2m – 15 = 0 => m = 5 or –3 |
| Answer | Marks |
|---|---|
| t = 0.5 ⇒Q=100e−1.5 =22.3 | M1 |
| Answer | Marks |
|---|---|
| [6] | 1.1 |
| Answer | Marks |
|---|---|
| 3.4 | Or dQ/dt tends to zero as t → ∞ |
| Answer | Marks | Guidance |
|---|---|---|
| soi | www | |
| (b) | (As t→∞ e−3t →0 so Q tends to) 0. | B1 |
| [1] | 3.4 | Only if from Q=ke−at, a > 0 |
Question 5:
5 | (a) | AE: m2 – 2m – 15 = 0 => m = 5 or –3
So GS is Q= Ae−3t +Be5t
Q tends to finite limit as t → ∞ => B = 0
t = 0, Q = 100 => A = 100
So Q=100e−3t
t = 0.5 ⇒Q=100e−1.5 =22.3 | M1
A1
B1
M1
A1
A1
[6] | 1.1
1.1
2.2a
3.4
1.1
3.4 | Or dQ/dt tends to zero as t → ∞
Using initial condition to find A
(or A + B)
soi | www
(b) | (As t→∞ e−3t →0 so Q tends to) 0. | B1
[1] | 3.4 | Only if from Q=ke−at, a > 0 | It must be clear that the limit is 0;
“Q is approximately 0” would not
be sufficient for B1.5 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 $t \geqslant 0 , Q$ can be modelled by the differential equation $\frac { d ^ { 2 } Q } { d t ^ { 2 } } - 2 \frac { d Q } { d t } - 15 Q = 0$.
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 $t = 0.5$.
\item Determine the finite limit of $Q$ as $t$ tends to infinity.
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 2 2020 Q5 [7]}}