| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2002 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Calculate intersection coordinates algebraically |
| Difficulty | Hard +2.3 This AEA question requires composition of trigonometric functions, implicit differentiation, geometric reasoning about convexity to establish inequalities, and integration bounds. Part (c) demands genuine insight to connect convexity with the linear/curved bounds, while part (d) requires careful application of these inequalities to integration—all significantly beyond standard A-level. |
| Spec | 1.05a Sine, cosine, tangent: definitions for all arguments1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
\includegraphics{figure_1}
Figure 1 shows a sketch of part of the curve with equation
$$y = \sin (\cos x).$$
The curve cuts the $x$-axis at the points $A$ and $C$ and the $y$-axis at the point $B$.
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the points $A$, $B$ and $C$.
[3]
\item Prove that $B$ is a stationary point.
[2]
\end{enumerate}
Given that the region $OCB$ is convex,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item show that, for $0 \leq x \leq \frac{\pi}{2}$,
$$\sin (\cos x) \leq \cos x$$
and
$$(1 - \frac{2}{\pi} x) \sin 1 \leq \sin (\cos x)$$
and state in each case the value or values of $x$ for which equality is achieved.
[6]
\item Hence show that
$$\frac{\pi}{4} \sin 1 < \int_0^{\frac{\pi}{2}} \sin(\cos x) \, dx < 1.$$
[4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2002 Q5 [15]}}