| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core (Further Pure Core) |
| Year | 2021 |
| Session | November |
| Marks | 20 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Tank/reservoir mixing problems |
| Difficulty | Challenging +1.2 This is a structured mixing problem with clear scaffolding through multiple parts. While it requires setting up and solving a first-order linear DE with integrating factor (a Further Maths topic), the question guides students through the model derivation, provides the specific cases to solve, and asks for interpretation. The mathematical techniques are standard for this topic, though the physical reasoning in parts (c)(i) and (c)(ii) adds modest challenge beyond routine calculation. |
| Spec | 4.10b Model with differential equations: kinematics and other contexts4.10c Integrating factor: first order equations |
| Answer | Marks | Guidance |
|---|---|---|
| 17 | (a) | (i) |
| Answer | Marks |
|---|---|
| 1 + ( a − b ) t | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 3.1b | |
| 2.4 | AG | |
| 17 | (a) | (ii) |
| Answer | Marks |
|---|---|
| d t 1 + ( a − b ) t | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| [2] | 3.3 | |
| 3.3 | AG | |
| 17 | (b) | (i) |
| Answer | Marks |
|---|---|
| x=1−e −at | M1 |
| Answer | Marks |
|---|---|
| [4] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 3.4 | separating variables | or IF eat |
| Answer | Marks | Guidance |
|---|---|---|
| 17 | (b) | (ii) |
| Answer | Marks |
|---|---|
| rate of inflow = 0.693 l/min | M1 |
| Answer | Marks |
|---|---|
| [2] | 3.3 |
| Answer | Marks | Guidance |
|---|---|---|
| 17 | (c) | (i) |
| Answer | Marks | Guidance |
|---|---|---|
| a | B1 | |
| [1] | 3.5b | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| 17 | (c) | (ii) |
| Answer | Marks |
|---|---|
| so maximum amount of chemical is 0.25 l | B1 |
| Answer | Marks |
|---|---|
| [9] | 1.1 |
| Answer | Marks |
|---|---|
| 3.2a | or completing the square |
Question 17:
17 | (a) | (i) | at litres of chemical in, bt litres of mixture out
amount of liquid in container = 1 + ( a − b ) t litres
x
proportion of chemical =
1 + ( a − b ) t | M1
A1
[2] | 3.1b
2.4 | AG
17 | (a) | (ii) | rate of chemical in = a litres/hr
b x
rate of chemical out = litres/hr
1 + ( a − b ) t
d x b x
= a −
d t 1 + ( a − b ) t
d x b x
+ = a
d t 1 + ( a − b ) t | M1
A1
[2] | 3.3
3.3 | AG
17 | (b) | (i) | dx
+ax=a
dt
1
dx=adt
1−x
− l n (1 − x ) = a t + c
when t=0, x=0c=0
1−x=e −at
x=1−e −at | M1
A1
B1
A1
[4] | 1.1
1.1
3.3
3.4 | separating variables | or IF eat
x e a t = e a t + c
c = −1
17 | (b) | (ii) | 12 = 1 − e − a
a = ln 2= 0.693
rate of inflow = 0.693 l/min | M1
A1
[2] | 3.3
3.4
17 | (c) | (i) | 1
when t= the container has no liquid left
a | B1
[1] | 3.5b | 1
oe e.g. volume negative when t
a
17 | (c) | (ii) | d x 2 a x
+ = a
d t 1 − a t
2− aa d t
I F = e 1 t
= e − 2 ln (1 − a t ) = (1 − a t ) − 2
d
( x (1 − a t ) − 2 ) = a (1 − a t ) − 2
d t
x (1 − a t ) − 2 = (1 − a t ) − 1 + c
when t = 0, x = 0 c = −1
x = (1 − a t ) − (1 − a t ) 2 = a t (1 − a t )
d x 1
= a − 2 a 2 t = 0 w h e n t =
d t 2 a
x = 12 − 14 = 14
so maximum amount of chemical is 0.25 l | B1
M1
A1
M1
A1
M1
A1
M1
A1
[9] | 1.1
3.1a
1.1
1.1
1.1
3.4
1.1
3.4
3.2a | or completing the square
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recorded or monitored17 In a chemical process, a vessel contains 1 litre of pure water. A liquid chemical is then passed into the top of the vessel at a constant rate of $a$ litres per minute and thoroughly mixed with the water. At the same time, the resulting mixture is drawn from the bottom of the vessel at a constant rate of $b$ litres per minute. You may assume that the chemical mixes instantly and uniformly with the water. After $t$ minutes, the mixture in the vessel contains $x$ litres of the chemical.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that the proportion of chemical present in the vessel after $t$ minutes is
$$\frac { x } { 1 + ( a - b ) t } .$$
\item Hence show that $\frac { d x } { d t } + \frac { b x } { 1 + ( a - b ) t } = a$.
\end{enumerate}\item First, consider the case where $\mathbf { b } = \mathbf { a }$.
\begin{enumerate}[label=(\roman*)]
\item Solve the differential equation to find $x$ in terms of $a$ and $t$.
\item Given that after 1 minute the vessel contains equal amounts of water and chemical, find the rate of inflow of chemical.
\end{enumerate}\item Now consider the case where $\mathrm { b } = 2 \mathrm { a }$.
\begin{enumerate}[label=(\roman*)]
\item Explain why the differential equation in part (a)(ii) is now invalid for $\mathrm { t } \geqslant \frac { 1 } { \mathrm { a } }$.
\item Find the maximum amount of chemical in the vessel.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core 2021 Q17 [20]}}