| Exam Board | Edexcel |
|---|---|
| Module | AEA (Advanced Extension Award) |
| Year | 2017 |
| Session | June |
| Marks | 21 |
| Paper | Download PDF ↗ |
| Topic | Standard Integrals and Reverse Chain Rule |
| Type | Area under curve using integration |
| Difficulty | Challenging +1.8 This is a sophisticated multi-part question requiring understanding of tangency conditions (double roots), integration by parts with a specific target form, algebraic expansion and coefficient matching. While each individual technique is A-level standard, the combination—especially recognizing why the tangency condition gives the factored form and executing the integration by parts to achieve the elegant result—requires above-average problem-solving insight and algebraic fluency typical of AEA questions. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07a Derivative as gradient: of tangent to curve1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration1.08i Integration by parts |
7.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{15e3f7f2-a77c-4ee4-8f0a-ac739e9fede5-7_583_1198_217_440}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
Figure 3 shows part of the curve $C$ with equation $y = x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - 34 x$ and the line $L$ with equation $y = m x + c$ .
The line $L$ touches $C$ at the points $P$ and $Q$ with $x$ coordinates $p$ and $q$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Explain why
$$x ^ { 4 } - 10 x ^ { 3 } + 33 x ^ { 2 } - ( 34 + m ) x - c = ( x - p ) ^ { 2 } ( x - q ) ^ { 2 }$$
The finite region $R$ ,shown shaded in Figure 3,is bounded by $C$ and $L$ .
\item Use integration by parts to show that the area of $R$ is $\frac { ( q - p ) ^ { 5 } } { 30 }$
\item Show that
$$( x - p ) ^ { 2 } ( x - q ) ^ { 2 } = x ^ { 4 } - 2 ( p + q ) x ^ { 3 } + S x ^ { 2 } - T x + U$$
where $S , T$ and $U$ are expressions to be found in terms of $p$ and $q$ .
\item Using part(a)and part(c)find the value of $p$ ,the value of $q$ and the equation of $L$ .
\end{enumerate}
\hfill \mbox{\textit{Edexcel AEA 2017 Q7 [21]}}