| 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 | 1.02q Use intersection points: of graphs to solve equations1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(x + 1 = 6x - x^2 - 3\) | M1 | |
| \(x^2 - 5x + 4 = 0\) with \((x - 1)(x - 4)\) | M1 | |
| \(x = 1\) and \(x = 4\) | A1 | |
| \(y = 2\) and \(y = 5\) | M1 A1 | (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\int (6x - x^2 - 3)dx = 3x^2 - \frac{x^3}{3} - 3x\) | M1 A1 | |
| Limits \(x_A\) and \(x_B\): \((48 - \frac{64}{3} - 12) - (3 - \frac{1}{3} - 3)\) (= 15) | M1 A1 | |
| Trapezium: \(\frac{1}{2}(2 + 5) \times 3 = 10.5\) | B1ft | |
| Area of \(R = 15 - 10.5 = 4.5\) | M1 A1 | (7 marks) |
**Part (a)**
$x + 1 = 6x - x^2 - 3$ | M1 |
$x^2 - 5x + 4 = 0$ with $(x - 1)(x - 4)$ | M1 |
$x = 1$ and $x = 4$ | A1 |
$y = 2$ and $y = 5$ | M1 A1 | (5 marks)
**Part (b)**
$\int (6x - x^2 - 3)dx = 3x^2 - \frac{x^3}{3} - 3x$ | M1 A1 |
Limits $x_A$ and $x_B$: $(48 - \frac{64}{3} - 12) - (3 - \frac{1}{3} - 3)$ (= 15) | M1 A1 |
Trapezium: $\frac{1}{2}(2 + 5) \times 3 = 10.5$ | B1ft |
Area of $R = 15 - 10.5 = 4.5$ | M1 A1 | (7 marks)
**Total for Question 7: 12 marks**
---\includegraphics{figure_2}
Fig. 2 shows the line with equation $y = x + 1$ and the curve with equation $y = 6x - x^2 - 3$.
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 Q7 [12]}}