| Exam Board | WJEC |
|---|---|
| Module | Unit 3 (Unit 3) |
| Year | 2024 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Separable variables - standard (polynomial/exponential x-side) |
| Difficulty | Standard +0.3 Part (a) is a routine quotient rule differentiation that students can verify mechanically. Part (b) requires recognizing that the given result from (a) can be used to integrate, then separating variables and applying the initial condition. While it requires some insight to connect parts (a) and (b), the technique is standard for A-level and the algebra is straightforward once the substitution is spotted. |
| Spec | 1.07l Derivative of ln(x): and related functions1.08k Separable differential equations: dy/dx = f(x)g(y) |
\begin{enumerate}[label=(\alph*)]
\item Given that $y = \frac{1 + \ln x}{x}$, show that $\frac{dy}{dx} = \frac{-\ln x}{x^2}$. [2]
\item Hence, solve the differential equation
$$\frac{dx}{dt} = \frac{x^2 t}{\ln x},$$
given that $t = 3$ when $x = 1$.
Give your answer in the form $t^2 = g(x)$, where $g$ is a function of $x$. [5]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 3 2024 Q14 [7]}}