| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2005 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Optimise perimeter or area of 2D region |
| Difficulty | Challenging +1.8 This AEA optimization problem requires setting up the area function with symmetry considerations, differentiating a product involving cos(p), solving the transcendental equation tan(p) = p for critical points, then proving inequalities about the solution. While multi-step and requiring careful algebraic manipulation, the techniques are standard calculus with some inequality verification—challenging but not requiring deep novel insight. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.07n Stationary points: find maxima, minima using derivatives |
4.A rectangle $A B C D$ is drawn so that $A$ and $B$ lie on the $x$-axis,and $C$ and $D$ lie on the curve with equation $y = \cos x , - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ .The point $A$ has coordinates $( p , 0 )$ ,where $0 < p < \frac { \pi } { 2 }$ .
\begin{enumerate}[label=(\alph*)]
\item Find an expression,in terms of $p$ ,for the area of this rectangle.
The maximum area of $A B C D$ is $S$ and occurs when $p = \alpha$ .Show that
\item $\frac { \pi } { 4 } < \alpha < 1$ ,
\item $S = \frac { 2 \alpha ^ { 2 } } { \sqrt { } \left( 1 + \alpha ^ { 2 } \right) }$ ,
\item $\frac { \pi ^ { 2 } } { 2 \sqrt { } \left( 16 + \pi ^ { 2 } \right) } < S < \sqrt { } 2$ .
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2005 Q4 [13]}}