| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Find tangent at given point (polynomial/algebraic) |
| Difficulty | Moderate -0.3 This is a standard C2 calculus question covering routine techniques: finding roots by factorization, differentiation for tangent equations and stationary points, and definite integration for area. All parts follow textbook procedures with no novel problem-solving required, though the multi-part structure and 11 total marks make it slightly more substantial than the most basic exercises. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
\includegraphics{figure_3}
Figure 3 shows part of the curve $C$ with equation
$$y = \frac{3}{5}x^2 - \frac{1}{4}x^3.$$
The curve $C$ touches the $x$-axis at the origin and passes through the point $A(p, 0)$.
\begin{enumerate}[label=(\alph*)]
\item Show that $p = 6$. [1]
\item Find an equation of the tangent to $C$ at $A$. [4]
\end{enumerate}
The curve $C$ has a maximum at the point $P$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the $x$-coordinate of $P$. [2]
\end{enumerate}
The shaded region $R$, in Fig. 3, is bounded by $C$ and the $x$-axis.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the area of $R$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q6 [11]}}