| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find value where max/min occurs |
| Difficulty | Standard +0.3 This is a standard C4 differential equation question with routine techniques: separating variables (part i), verifying initial conditions (part ii), using R cos(θ-α) form and differentiation (part iii), and finding maxima (part iv). All parts follow textbook methods with no novel insight required, though the multi-part structure and algebraic manipulation make it slightly above average difficulty for typical A-level questions. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08k Separable differential equations: dy/dx = f(x)g(y)4.10a General/particular solutions: of differential equations4.10b Model with differential equations: kinematics and other contexts4.10f Simple harmonic motion: x'' = -omega^2 x |
The motion of a particle is modelled by the differential equation
$$v \frac{dv}{dt} + 4x = 0,$$
where $x$ is its displacement from a fixed point, and $v$ is its velocity.
Initially $x = 1$ and $v = 4$.
\begin{enumerate}[label=(\roman*)]
\item Solve the differential equation to show that $v^2 = 20 - 4x^2$. [4]
\end{enumerate}
Now consider motion for which $x = \cos 2t + 2 \sin 2t$, where $x$ is the displacement from a fixed point at time $t$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Verify that, when $t = 0$, $x = 1$. Use the fact that $v = \frac{dx}{dt}$ to verify that when $t = 0$, $v = 4$. [4]
\item Express $x$ in the form $R \cos(2t - \alpha)$, where $R$ and $\alpha$ are constants to be determined, and obtain the corresponding expression for $v$. Hence or otherwise verify that, for this motion too, $v^2 = 20 - 4x^2$. [7]
\item Use your answers to part (iii) to find the maximum value of $x$, and the earliest time at which $x$ reaches this maximum value. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 2013 Q6 [18]}}