| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Session | Specimen |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area between two curves |
| Difficulty | Moderate -0.8 This is a routine C2 integration question with straightforward steps: substituting given points to verify intersection, setting up the integral by subtracting functions, and evaluating a polynomial integral. All techniques are standard with no problem-solving insight required, making it easier than average but not trivial due to the arithmetic involved. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.08e Area between curve and x-axis: using definite integrals1.08f Area between two curves: using integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(-75 + 45 + 30 = 0, \quad 25 - 15 - 10 = 0\) | B1 | For checking one point in both equations |
| B1 | 2 | For checking the other point in both equations |
| \(-12 - 18 + 30 = 0, \quad 4 + 6 - 10 = 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| Area is \(\int_{-5}^{2} \left[(-3x^2 - 9x + 30) - (x^2 + 3x - 10)\right] dx\) | M1 | For use of \(\int(y_1 - y_2) dx\) |
| A1 | 2 | For showing given answer correctly |
| i.e. \(\int_{-5}^{2} (-4x^2 - 12x + 40) dx\), as required |
| Answer | Marks | Guidance |
|---|---|---|
| Area is \(\left[-\frac{4}{3}x^3 - 6x^2 + 40x\right]_{-5}^{2}\) | M1 | For integration attempt with one term OK |
| A1 | For at least two terms correct | |
| A1 | For completely correct indefinite integral | |
| \(= \left(-\frac{32}{3} - 24 + 80\right) - \left(\frac{500}{3} - 150 - 200\right)\) | M1 | For correct use of limits |
| A1 | For showing given answer correctly | |
| \(= 228\frac{2}{3}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Area under top curve is | M1 | For complete evaluation attempt |
| A1 | For correct indefinite integration (allow for other curve if not earned here) | |
| \(\left[-x^3 - \frac{9}{2}x^2 + 30x\right]_{-5}^{2} = 171\frac{1}{2}\) | A1 | For correct value |
| Area above lower curve is | M1 | For evaluation and sign change |
| \(-\left[\frac{1}{4}x^4 + \frac{3}{2}x^3 - 10x\right]_{-5}^{2} = 57\frac{1}{6}\) | ||
| So area between is \(171\frac{1}{2} + 57\frac{1}{6} = 228\frac{2}{3}\) | A1 | 5 |
| Total: 9 |
## Part (i)
$-75 + 45 + 30 = 0, \quad 25 - 15 - 10 = 0$ | B1 | For checking one point in both equations
| B1 | 2 | For checking the other point in both equations
$-12 - 18 + 30 = 0, \quad 4 + 6 - 10 = 0$ | |
## Part (ii)
Area is $\int_{-5}^{2} \left[(-3x^2 - 9x + 30) - (x^2 + 3x - 10)\right] dx$ | M1 | For use of $\int(y_1 - y_2) dx$
| A1 | 2 | For showing given answer correctly
i.e. $\int_{-5}^{2} (-4x^2 - 12x + 40) dx$, as required | |
## Part (iii) - EITHER
Area is $\left[-\frac{4}{3}x^3 - 6x^2 + 40x\right]_{-5}^{2}$ | M1 | For integration attempt with one term OK
| A1 | | For at least two terms correct
| A1 | | For completely correct indefinite integral
$= \left(-\frac{32}{3} - 24 + 80\right) - \left(\frac{500}{3} - 150 - 200\right)$ | M1 | For correct use of limits
| A1 | | For showing given answer correctly
$= 228\frac{2}{3}$ | |
## Part (iii) - OR
Area under top curve is | M1 | For complete evaluation attempt
| A1 | | For correct indefinite integration (allow for other curve if not earned here)
$\left[-x^3 - \frac{9}{2}x^2 + 30x\right]_{-5}^{2} = 171\frac{1}{2}$ | A1 | For correct value
Area above lower curve is | M1 | For evaluation and sign change
$-\left[\frac{1}{4}x^4 + \frac{3}{2}x^3 - 10x\right]_{-5}^{2} = 57\frac{1}{6}$ | |
So area between is $171\frac{1}{2} + 57\frac{1}{6} = 228\frac{2}{3}$ | A1 | 5 | For showing given answer correctly
| **Total: 9** | |\includegraphics{figure_7}
The diagram shows the curves $y = -3x^2 - 9x + 30$ and $y = x^2 + 3x - 10$.
\begin{enumerate}[label=(\roman*)]
\item Verify that the curves intersect at the points $A(-5, 0)$ and $B(2, 0)$. [2]
\item Show that the area of the shaded region between the curves is given by $\int_{-5}^{2} (-4x^2 - 12x + 40) dx$. [2]
\item Hence or otherwise show that the area of the shaded region between the curves is $228\frac{2}{3}$. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q7 [9]}}