| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2003 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Solutions from graphical analysis |
| Difficulty | Hard +2.3 This AEA question requires sketching a quartic function, finding its range through calculus (locating turning points), and most challengingly, part (c) demands sophisticated reasoning about how the number of intersections between y=k and y=|f(x)| relates to k itself—a self-referential condition requiring careful analysis of the graph's structure and systematic case-work with parameters. |
| Spec | 1.02l Modulus function: notation, relations, equations and inequalities1.02n Sketch curves: simple equations including polynomials |
5.The function $f$ is given by
$$f ( x ) = \frac { 1 } { \lambda } \left( x ^ { 2 } - 4 \right) \left( x ^ { 2 } - 25 \right)$$
where $x$ is real and $\lambda$ is a positive integer.
\begin{enumerate}[label=(\alph*)]
\item Sketch the graph of $y = \mathrm { f } ( x )$ showing clearly where the graph crosses the coordinate axes.
\item Find,in terms of $\lambda$ ,the range of f .
\item Find the sets of positive integers $k$ and $\lambda$ such that the equation
$$k = | \mathrm { f } ( x ) |$$
has exactly $k$ distinct real roots.
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2003 Q5 [17]}}