| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Two Curves Intersection Area |
| Difficulty | Standard +0.3 This is a standard C2 integration question requiring finding intersection points by solving a quadratic equation, then integrating the difference of two functions. While it involves multiple steps (solving simultaneous equations, setting up and evaluating a definite integral), these are routine techniques with no conceptual challenges, making it slightly easier than average. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks |
|---|---|
| \(3x^2 - 9x - 12 = 0\) | M1 |
| \(3(x + 1)(x - 4) = 0\) | M1 |
| \(x = -1, 4\) | A1 |
| Therefore \((-1, 5), (4, 5)\) | A1 |
| (ii) \(\text{area} = \int_{-1}^{4} [(9 + 3x - x^2) - (2x^2 - 6x - 3)] \, dx\) | M1 |
| \(= \int_{-1}^{4} (12 + 9x - 3x^2) \, dx\) | A1 |
| \(= \left[12x + \frac{9}{2}x^2 - x^3\right]_{-1}^{4}\) | M1 A2 |
| \(= (48 + 72 - 64) - \left(-12 - \frac{9}{2} + 1\right) = 62\frac{1}{2}\) | M1 A1 (11) |
**(i)** $2x^2 - 6x - 3 = 9 + 3x - x^2$
$3x^2 - 9x - 12 = 0$ | M1 |
$3(x + 1)(x - 4) = 0$ | M1 |
$x = -1, 4$ | A1 |
Therefore $(-1, 5), (4, 5)$ | A1 |
**(ii)** $\text{area} = \int_{-1}^{4} [(9 + 3x - x^2) - (2x^2 - 6x - 3)] \, dx$ | M1 |
$= \int_{-1}^{4} (12 + 9x - 3x^2) \, dx$ | A1 |
$= \left[12x + \frac{9}{2}x^2 - x^3\right]_{-1}^{4}$ | M1 A2 |
$= (48 + 72 - 64) - \left(-12 - \frac{9}{2} + 1\right) = 62\frac{1}{2}$ | M1 A1 (11) |
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**Total (72)**9.\\
\includegraphics[max width=\textwidth, alt={}, center]{faa66f88-9bff-4dc9-955f-80cdab3fdd34-3_538_872_1790_447}
The diagram shows the curves $y = 2 x ^ { 2 } - 6 x - 3$ and $y = 9 + 3 x - x ^ { 2 }$.\\
(i) Find the coordinates of the points where the two curves intersect.\\
(ii) Find the area of the shaded region bounded by the two curves.
\hfill \mbox{\textit{OCR C2 Q9 [11]}}