| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2020 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Show integral then evaluate area |
| Difficulty | Standard +0.8 This question requires finding a maximum using parametric differentiation (chain rule with dy/dx = (dy/dt)/(dx/dt)), converting area integral limits from x to t (requiring algebraic manipulation), and integration by parts twice. While each technique is A-level standard, the combination across three parts with parametric equations and the exponential function creates moderate challenge above typical questions. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation1.08e Area between curve and x-axis: using definite integrals1.08i Integration by parts |
\includegraphics{figure_5}
The diagram shows the curve $C$ with parametric equations
$x = \frac{3}{t}$, $y = t^2 e^{-2t}$, where $t > 0$.
The maximum point on $C$ is denoted by $P$.
\begin{enumerate}[label=(\alph*)]
\item Determine the exact coordinates of $P$. [4]
The shaded region $R$ is enclosed by the curve, the $x$-axis and the lines $x = 1$ and $x = 6$.
\item Show that the area of $R$ is given by
$$\int_a^b 3te^{-2t} dt,$$
where $a$ and $b$ are constants to be determined. [3]
\item Hence determine the exact area of $R$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2020 Q5 [12]}}