| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | First order differential equations (integrating factor) |
| Type | Separable variables |
| Difficulty | Standard +0.3 Part (a) is routine differentiation using chain rule. Part (b) is a standard separable differential equation requiring separation, integration, and applying initial conditions. Part (c) is a simple limit evaluation. All techniques are straightforward applications of core methods with no novel insight required, making this slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks |
|---|---|
| Show sufficient working to justify the given answer | B1 |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Correct separation of variables | B1 | e.g. \(-\int \frac{1}{t}dt = \int \frac{1}{x\ln x}dx\) |
| Obtain term \(\ln(\ln x)\) | B1 | |
| Obtain term \(-\ln t\) | B1 | |
| Evaluate a constant or use \(x = e\) and \(t = 2\) as limits in an expression involving \(\ln(\ln x)\) | M1 | |
| Obtain correct solution in any form, e.g. \(\ln(\ln x) = -\ln t + \ln 2\) | A1 | |
| Use log laws to enable removal of logarithms | M1 | |
| Obtain answer \(x = e^{\frac{2}{t}}\), or simplified equivalent | A1 | |
| Total: 7 |
| Answer | Marks |
|---|---|
| State that \(x\) tends to 1 coming from \(x = e^{\frac{k}{t}}\) | B1 |
| Total: 1 |
## Question 7(a):
| Show sufficient working to justify the given answer | B1 | |
| **Total: 1** | | |
## Question 7(b):
| Correct separation of variables | B1 | e.g. $-\int \frac{1}{t}dt = \int \frac{1}{x\ln x}dx$ |
| Obtain term $\ln(\ln x)$ | B1 | |
| Obtain term $-\ln t$ | B1 | |
| Evaluate a constant or use $x = e$ and $t = 2$ as limits in an expression involving $\ln(\ln x)$ | M1 | |
| Obtain correct solution in any form, e.g. $\ln(\ln x) = -\ln t + \ln 2$ | A1 | |
| Use log laws to enable removal of logarithms | M1 | |
| Obtain answer $x = e^{\frac{2}{t}}$, or simplified equivalent | A1 | |
| **Total: 7** | | |
## Question 7(c):
| State that $x$ tends to 1 coming from $x = e^{\frac{k}{t}}$ | B1 | |
| **Total: 1** | | |
---7
\begin{enumerate}[label=(\alph*)]
\item Given that $y = \ln ( \ln x )$, show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { x \ln x }$$
The variables $x$ and $t$ satisfy the differential equation
$$x \ln x + t \frac { \mathrm {~d} x } { \mathrm {~d} t } = 0$$
It is given that $x = \mathrm { e }$ when $t = 2$.
\item Solve the differential equation obtaining an expression for $x$ in terms of $t$, simplifying your answer.
\item Hence state what happens to the value of $x$ as $t$ tends to infinity.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q7 [8]}}