| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area under polynomial curve |
| Difficulty | Moderate -0.3 This is a standard C2 calculus question covering routine techniques: finding roots, differentiation for tangents and stationary points, and definite integration. All parts follow textbook procedures with no novel problem-solving required, though the multi-part structure and integration calculation make it slightly more substantial than the most basic exercises. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Solve \(\frac{3}{2}x^2 - \frac{1}{4}x^3 = 0\) to find \(p = 6\), or verify: \(\frac{3}{2} \times 6^2 - \frac{1}{4} \times 6^4 = 0\) (*) | B1 | (1 mark) |
| (b) \(\frac{dy}{dx} = 3x - \frac{3x^2}{4}\) | M1, A1 | |
| \(m = -9\), \(y - 0 = -9(x - 6)\) (Any correct form) | M1, A1 | (4 marks) |
| (c) \(3x - \frac{3x^2}{4} = 0\), \(x = 4\) | M1, A1ft | (2 marks) |
| (d) \(\int(\frac{3x^2}{2} - \frac{x^4}{4})dx = \frac{x^3}{2} - \frac{x^4}{16}\) (Allow unsimplified versions) | M1, A1 | |
| \([.........]_b^6 = \frac{6^3}{2} - \frac{6^4}{16} = 27\) | M1, A1 | M: Need 6 and 0 as limits. |
**(a)** Solve $\frac{3}{2}x^2 - \frac{1}{4}x^3 = 0$ to find $p = 6$, or verify: $\frac{3}{2} \times 6^2 - \frac{1}{4} \times 6^4 = 0$ (*) | B1 | (1 mark)
**(b)** $\frac{dy}{dx} = 3x - \frac{3x^2}{4}$ | M1, A1 |
$m = -9$, $y - 0 = -9(x - 6)$ (Any correct form) | M1, A1 | (4 marks)
**(c)** $3x - \frac{3x^2}{4} = 0$, $x = 4$ | M1, A1ft | (2 marks)
**(d)** $\int(\frac{3x^2}{2} - \frac{x^4}{4})dx = \frac{x^3}{2} - \frac{x^4}{16}$ (Allow unsimplified versions) | M1, A1 |
$[.........]_b^6 = \frac{6^3}{2} - \frac{6^4}{16} = 27$ | M1, A1 | M: Need 6 and 0 as limits. | (4 marks)\includegraphics{figure_1}
Fig. 1 shows part of the curve $C$ with equation $y = \frac{1}{3}x^2 - \frac{1}{4}x^3$.
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. 1, 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 Q7 [11]}}