| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2017 |
| Session | Specimen |
| Marks | 11 |
| Topic | Areas by integration |
| Type | Area of sector/segment problems |
| Difficulty | Challenging +1.2 This is a standard integration problem requiring verification of intersection points and finding area between curves. Part (a) is routine substitution. Part (b) requires setting up integrals for circular arc and parabola areas, using symmetry, and applying standard techniques (trigonometric substitution for circle). While it has multiple steps and requires careful setup, all techniques are standard A-level material with no novel insight needed. The 8 marks reflect computational length rather than conceptual difficulty. |
| Spec | 1.03d Circles: equation (x-a)^2+(y-b)^2=r^21.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
The diagram shows the circle with centre O and radius 2, and the parabola $y = \frac{1}{\sqrt{3}}(4 - x^2)$.
\includegraphics{figure_5}
The circle meets the parabola at points $P$ and $Q$, as shown in the diagram.
\begin{enumerate}[label=(\alph*)]
\item Verify that the coordinates of $Q$ are $(1, \sqrt{3})$. [3]
\item Find the exact area of the shaded region enclosed by the arc $PQ$ of the circle and the parabola. [8]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2017 Q5 [11]}}