| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Area between curve and line |
| Difficulty | Moderate -0.3 This is a standard C2 curve sketching question with routine algebraic manipulation (factorising), differentiation to find turning points, and integration for area. All techniques are straightforward applications of core methods with no novel problem-solving required, making it slightly easier than average but not trivial due to the multi-step nature and 12 total marks. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
\includegraphics{figure_5}
Figure 2 shows part of the curve with equation
$$y = x^3 - 6x^2 + 9x.$$
The curve touches the $x$-axis at $A$ and has a maximum turning point at $B$.
\begin{enumerate}[label=(\alph*)]
\item Show that the equation of the curve may be written as
$$y = x(x - 3)^2,$$
and hence write down the coordinates of $A$. [2]
\item Find the coordinates of $B$. [5]
\end{enumerate}
The shaded region $R$ is bounded by the curve and the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the area of $R$. [5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q8 [12]}}