| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2002 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Area between curve and line |
| Difficulty | Hard +2.3 This AEA question requires systematic analysis of curve properties (symmetry, intercepts), solving simultaneous Diophantine constraints, computing areas via integration with fractional powers, and finding gradient equality points. The multi-part structure demands sustained problem-solving across algebra, calculus, and optimization—significantly beyond standard A-level but accessible with careful reasoning. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07i Differentiate x^n: for rational n and sums1.08e Area between curve and x-axis: using definite integrals |
\includegraphics{figure_2}
Figure 2 shows a sketch of part of two curves $C_1$ and $C_2$ for $y \geq 0$.
The equation of $C_1$ is $y = m_1 - x^{n_1}$ and the equation of $C_2$ is $y = m_2 - x^{n_2}$, where $m_1$, $m_2$, $n_1$ and $n_2$ are positive integers with $m_2 > m_1$.
Both $C_1$ and $C_2$ are symmetric about the line $x = 0$ and they both pass through the points $(3, 0)$ and $(-3, 0)$.
Given that $n_1 + n_2 = 12$, find
\begin{enumerate}[label=(\alph*)]
\item the possible values of $n_1$ and $n_2$,
[4]
\item the exact value of the smallest possible area between $C_1$ and $C_2$, simplifying your answer,
[8]
\item the largest value of $x$ for which the gradients of the two curves can be the same. Leave your answer in surd form.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2002 Q6 [17]}}