| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area between curve and line |
| Difficulty | Standard +0.3 This is a standard C2 integration question requiring finding intersection points by solving a quadratic equation, then calculating the area between a line and curve using definite integration. While it involves multiple steps (12 marks total), each step follows routine procedures taught in C2 with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(x^2 - 2x + 3 = 9 - x\) | M1 | |
| \(x^2 - x - 6 = 0\); \((x+2)(x-3) = 0\) | M1 A1 | \(x = -2, 3\) |
| \(y = 11, 6\) | M1 A1 ft | (5) |
| (b) \(\int(x^2 - 2x + 3)dx = \frac{x^3}{3} - x^2 + 3x\) | M1 A1 | |
| \(\left[\frac{x^3}{3} - x^2 + 3x\right]_{-2}^{3} = (9 - 9 + 9) - \left(\frac{-8}{3} - 4 - 6\right)\) \((= 21\frac{2}{3})\) | M1 A1 | |
| Trapezium: \(\frac{1}{2}(11 + 6) \times 5\) | B1 ft | \((= 42\frac{1}{2})\) |
| Area \(= 42\frac{1}{2} - 21\frac{2}{3} = 20\frac{5}{6}\) | M1 A1 | (7) |
**(a)** $x^2 - 2x + 3 = 9 - x$ | M1 |
$x^2 - x - 6 = 0$; $(x+2)(x-3) = 0$ | M1 A1 | $x = -2, 3$
$y = 11, 6$ | M1 A1 ft | (5)
**(b)** $\int(x^2 - 2x + 3)dx = \frac{x^3}{3} - x^2 + 3x$ | M1 A1 |
$\left[\frac{x^3}{3} - x^2 + 3x\right]_{-2}^{3} = (9 - 9 + 9) - \left(\frac{-8}{3} - 4 - 6\right)$ $(= 21\frac{2}{3})$ | M1 A1 |
Trapezium: $\frac{1}{2}(11 + 6) \times 5$ | B1 ft | $(= 42\frac{1}{2})$
Area $= 42\frac{1}{2} - 21\frac{2}{3} = 20\frac{5}{6}$ | M1 A1 | (7)\includegraphics{figure_3}
Fig. 3 shows the line with equation $y = 9 - x$ and the curve with equation $y = x^2 - 2x + 3$. The line and the curve intersect at the points $A$ and $B$, and $O$ is the origin.
\begin{enumerate}[label=(\alph*)]
\item Calculate the coordinates of $A$ and the coordinates of $B$. [5]
\end{enumerate}
The shaded region $R$ is bounded by the line and the curve.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Calculate the area of $R$. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q8 [12]}}