| Exam Board | OCR |
|---|---|
| Module | H240/03 (Pure Mathematics and Mechanics) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Parametric integration |
| Type | Emblem or applied region area |
| Difficulty | Standard +0.8 This is a challenging parametric integration problem requiring students to derive the area integral formula (part a), then evaluate a non-trivial integral involving trigonometric products and apply multiple integration techniques including double angle formulas and substitution (part b). The multi-step nature, need for careful algebraic manipulation, and integration of sin t cos² t (requiring substitution or product-to-sum formulas) place this above average difficulty, though it remains within standard A-level techniques. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.08d Evaluate definite integrals: between limits |
A mathematics department is designing a new emblem to place on the walls outside its classrooms. The design for the emblem is shown in the diagram below.
\includegraphics{figure_5}
The emblem is modelled by the region between the $x$-axis and the curve with parametric equations
$$x = 1 + 0.2t - \cos t, \quad y = k \sin^2 t,$$
where $k$ is a positive constant and $0 \leq t \leq \pi$.
Lengths are in metres and the area of the emblem must be $1 \text{m}^2$.
\begin{enumerate}[label=(\alph*)]
\item Show that $k \int_0^\pi (0.2 + \sin t - 0.2 \cos^2 t - \sin t \cos^2 t) dt = 1$. [3]
\item Determine the exact value of $k$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/03 2023 Q5 [9]}}