| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2002 |
| Session | Specimen |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Polynomial with rational/modulus curves |
| Difficulty | Challenging +1.8 This AEA question requires proving existence of a tangency condition between piecewise and quadratic functions, finding the specific k value through simultaneous equations, sketching both curves, and computing an area integral. While conceptually accessible, it demands careful handling of the piecewise definition, algebraic manipulation to establish tangency (equal values and derivatives), and integration across the domain—representing substantial multi-step reasoning beyond standard A-level. |
| Spec | 1.02o Sketch reciprocal curves: y=a/x and y=a/x^21.02w Graph transformations: simple transformations of f(x)1.08e Area between curve and x-axis: using definite integrals |
5.The function f is defined on the domain $[ - 2,2 ]$ by:
$$f ( x ) = \left\{ \begin{array} { r l r }
- k x ( 2 + x ) & \text { if } & - 2 \leq x < 0 , \\
k x ( 2 - x ) & \text { if } & 0 \leq x \leq 2 ,
\end{array} \right.$$
where $k$ is a positive constant.\\
The function g is defined on the domain $[ - 2,2 ]$ by $\mathrm { g } ( x ) = ( 2.5 ) ^ { 2 } - x ^ { 2 }$ .
\begin{enumerate}[label=(\alph*)]
\item Prove that there is a value of $k$ such that the graph of f touches the graph of g .
\item For this value of $k$ sketch the graphs of the functions f and g on the same axes,stating clearly where the graphs touch.
\item Find the exact area of the region bounded by the two graphs.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2002 Q5 [17]}}