| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2004 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Completing the square and sketching |
| Type | Discriminant for real roots condition |
| Difficulty | Challenging +1.8 This AEA question requires multiple sophisticated techniques: verifying a fixed point (trivial), finding conditions for repeated roots using discriminant analysis of a cubic, then determining the range of a horizontal line intersecting a cubic at three points using calculus to find local extrema. The final part demands careful analysis of the cubic's shape and critical values, which is non-routine but systematic for strong students. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07n Stationary points: find maxima, minima using derivatives |
$$f(x) = x^3 - (k+4)x + 2k,$$ where $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Show that, for all values of $k$, the curve with equation $y = f(x)$ passes through the point $(2, 0)$. [1]
\item Find the values of $k$ for which the equation $f(x) = 0$ has exactly two distinct roots. [5]
\end{enumerate}
Given that $k > 0$, that the $x$-axis is a tangent to the curve with equation $y = f(x)$, and that the line $y = p$ intersects the curve in three distinct points,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the set of values that $p$ can take. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2004 Q3 [11]}}