| 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 -0.8 This is a straightforward C2 question involving standard techniques: finding a turning point by differentiation, sketching a quadratic, computing a definite integral, and finding an area between curves using subtraction. All steps are routine with no problem-solving insight required, making it easier than average but not trivial due to the multi-part nature. |
| Spec | 1.07n 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 | Guidance |
|---|---|---|
| \(y'=6-2x\) | M1 | condone one error |
| \(y'=0\) used | M1 | |
| \(x=3\) | A1 | |
| \(y=16\) | A1 | |
| \((0,7)\), \((-1,0)\) and \((7,0)\) found or marked on graph | 3 | 1 each |
| sketch of correct shape | 1 | must reach pos. \(y\)-axis |
| Answer | Marks | Guidance |
|---|---|---|
| \(58.6\) to \(58.7\) | 3 | B1 for \(7x+3x^2-\dfrac{x^3}{3}\) |
| M1 | [their value at 5] \(-\) [their value at 1] dependent on integration attempted | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| using his (ii) and 48 | 1 |
## Question 5:
**Part i:**
$y'=6-2x$ | M1 | condone one error
$y'=0$ used | M1 |
$x=3$ | A1 |
$y=16$ | A1 |
$(0,7)$, $(-1,0)$ and $(7,0)$ found or marked on graph | 3 | 1 each
sketch of correct shape | 1 | must reach pos. $y$-axis | 8
**Part ii:**
$58.6$ to $58.7$ | 3 | B1 for $7x+3x^2-\dfrac{x^3}{3}$
| M1 | [their value at 5] $-$ [their value at 1] dependent on integration attempted | 3
**Part iii:**
using his (ii) and 48 | 1 | | 15 The equation of a curve is $\quad y = 7 + 6 x - x ^ { 2 }$.\\
(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 Q5 [12]}}