| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Chemical reaction kinetics |
| Difficulty | Standard +0.3 This is a straightforward separable differential equation requiring substitution of the given expression for x, separation of variables, integration (including ln), and application of initial conditions. Part (ii) involves simple limit evaluation. While it requires multiple steps and careful algebraic manipulation, it follows standard A-level techniques without requiring novel insight or particularly challenging integration. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks |
|---|---|
| 5(i) | dy 2y |
| Answer | Marks |
|---|---|
| dt (1+t)2 | B1 |
| Separate variables correctly and attempt integration of one side | M1 |
| Obtain term lny, or equivalent | A1 |
| Answer | Marks |
|---|---|
| (1+t) | A1 |
| Answer | Marks |
|---|---|
| 1+t | M1 |
| Answer | Marks |
|---|---|
| 1+t | A1 |
| Total: | 6 |
| Answer | Marks |
|---|---|
| 5(ii) | 100 |
| Answer | Marks | Guidance |
|---|---|---|
| e2 | B1 | |
| State or imply that the mass of A tends to zero | B1 | |
| Total: | 2 | |
| Question | Answer | Marks |
Question 5:
--- 5(i) ---
5(i) | dy 2y
State =− , or equivalent
dt (1+t)2 | B1
Separate variables correctly and attempt integration of one side | M1
Obtain term lny, or equivalent | A1
2
Obtain term , or equivalent
(1+t) | A1
Use y = 100 and t = 0 to evaluate a constant, or as limits in an expression containing terms of
b
the form alny and
1+t | M1
2
Obtain correct solution in any form, e.g. lny= −2+ln100
1+t | A1
Total: | 6
--- 5(ii) ---
5(ii) | 100
State that the mass of B approaches , or exact equivalent
e2 | B1
State or imply that the mass of A tends to zero | B1
Total: | 2
Question | Answer | MarksIn a certain chemical process a substance $A$ reacts with and reduces a substance $B$. The masses of $A$ and $B$ at time $t$ after the start of the process are $x$ and $y$ respectively. It is given that $\frac{dy}{dt} = -0.2xy$ and $x = \frac{10}{(1 + t)^2}$. At the beginning of the process $y = 100$.
\begin{enumerate}[label=(\roman*)]
\item Form a differential equation in $y$ and $t$, and solve this differential equation. [6]
\item Find the exact value approached by the mass of $B$ as $t$ becomes large. State what happens to the mass of $A$ as $t$ becomes large. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q5 [8]}}