| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Curve with Horizontal Line |
| Difficulty | Moderate -0.3 This is a straightforward C2 question testing standard techniques: finding a turning point by differentiation, sketching a quadratic, integrating a polynomial, and finding an area between curves. All steps are routine with no problem-solving insight required, though it involves multiple parts and careful arithmetic, making it slightly easier than average overall. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks |
|---|---|
| \(y' = 6 - 2x\) | M1 |
| Answer | Marks |
|---|---|
| \(y' = 0\) used | M1 |
| \(x = 3\) | A1 |
| \(y = 16\) | A1 |
| Answer | Marks |
|---|---|
| \((0, 7)\) \((-1, 0)\) and \((7, 0)\) found or marked on graph | B1 (condone one error; 1 each) |
| sketch of correct shape | B1 (must reach pos. y-axis) |
| Answer | Marks |
|---|---|
| \(58.6\) to \(58.7\) | M1 |
| using his (ii) and 48 | M1 |
| [their value at 5] − [their value at 1] | A1 (dependent on integration attempted) |
# Question 1
**i**
$y' = 6 - 2x$ | M1
**ii**
$y' = 0$ used | M1
$x = 3$ | A1
$y = 16$ | A1
**iii**
$(0, 7)$ $(-1, 0)$ and $(7, 0)$ found or marked on graph | B1 (condone one error; 1 each)
sketch of correct shape | B1 (must reach pos. y-axis)
**Integration**
B1 for $7x + 3x^2 - \frac{x^3}{3}$
$58.6$ to $58.7$ | M1
using his (ii) and 48 | M1
[their value at 5] − [their value at 1] | A1 (dependent on integration attempted)1 The equation of a curve is $y = 7 + 6 x - x ^ { 2 }$.\\
(i) Use calculus to find the coordinates of the turning point on this curve.
Find also the coordinates of the points of intersection of this curve with the axes, and sketch the curve.\\
(ii) Find $\int _ { 1 } ^ { 5 } \left( 7 + 6 x - x ^ { 2 } \right) \mathrm { d } x$, showing your working.\\
(iii) The curve and the line $y = 12$ intersect at $( 1,12 )$ and $( 5,12 )$. Using your answer to part (ii), find the area of the finite region between the curve and the line $y = 12$.
\hfill \mbox{\textit{OCR MEI C2 Q1 [12]}}