| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Curve-Line Intersection Area |
| Difficulty | Moderate -0.3 This is a standard C2 integration question with routine steps: differentiation to find turning points, solving a cubic by factoring (given two roots), and finding area between curves using definite integration. All techniques are textbook exercises with no novel problem-solving required, though the multi-part structure and algebraic manipulation place it slightly below average difficulty. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(3x^2 - 6x - 22\) | M1 | Condone one incorrect term, but must be three terms; condone "\(y=\)" |
| Their \(y' = 0\) s.o.i. | M1 | At least one term correct in their \(y'\); may be implied by use of quadratic formula, completing square, attempt to factorise |
| \(3.89\) | A1 | If A0A0, SC1 for \(\frac{3 \pm 5\sqrt{3}}{3}\) or \(1 \pm \frac{5}{\sqrt{3}}\) or better, or both decimal answers given to a different accuracy or from truncation |
| \(-1.89\) | A1 | Exact values: \(3.886751346\) and \(-1.886751346\) |
| [4] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(x^3 - 3x^2 - 22x + 24 = 6x + 24\) | M1 | May be implied by \(x^3 - 3x^2 - 28x [= 0]\) |
| \(x^3 - 3x^2 - 28x [= 0]\) | M1 | May be implied by \(x^2 - 3x - 28 [= 0]\) |
| Other point when \(x = 7\) isw | A1 | Dependent on award of both M marks; ignore other values of \(x\) |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(F[x] = \frac{x^4}{4} - \frac{3x^3}{3} - \frac{22x^2}{2} + 24x\) | M1\* | Allow for three terms correct; condone \(+ c\). Alternative: M1 for \(\int((x^3 - 3x^2 - 22x + 24) - (6x + 24))\,dx\); M1\* for \(F[x] = \frac{x^4}{4} - \frac{3x^3}{3} - \frac{28x^2}{2}\); condone one error in integration |
| \(F[0] - F[-4]\) | M1dep | Allow \(0 - F[-4]\), condone \(-F[-4]\), but do not allow \(F[-4]\) only. M1dep for \(F[0] - F[-4]\) |
| Area of triangle \(= 48\) | B1 | |
| Area required \(= 96\) from fully correct working | A1 | A0 for \(-96\), ignore units; no marks for 96 unsupported |
| [4] |
## Question 2:
### Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $3x^2 - 6x - 22$ | **M1** | Condone one incorrect term, but must be three terms; condone "$y=$" |
| Their $y' = 0$ s.o.i. | **M1** | At least one term correct in their $y'$; may be implied by use of quadratic formula, completing square, attempt to factorise |
| $3.89$ | **A1** | If **A0A0**, **SC1** for $\frac{3 \pm 5\sqrt{3}}{3}$ or $1 \pm \frac{5}{\sqrt{3}}$ or better, or both decimal answers given to a different accuracy or from truncation |
| $-1.89$ | **A1** | Exact values: $3.886751346$ and $-1.886751346$ |
| **[4]** | | |
---
### Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $x^3 - 3x^2 - 22x + 24 = 6x + 24$ | **M1** | May be implied by $x^3 - 3x^2 - 28x [= 0]$ |
| $x^3 - 3x^2 - 28x [= 0]$ | **M1** | May be implied by $x^2 - 3x - 28 [= 0]$ |
| Other point when $x = 7$ isw | **A1** | Dependent on award of both **M** marks; ignore other values of $x$ |
| **[3]** | | |
---
### Part (iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $F[x] = \frac{x^4}{4} - \frac{3x^3}{3} - \frac{22x^2}{2} + 24x$ | **M1\*** | Allow for three terms correct; condone $+ c$. Alternative: **M1** for $\int((x^3 - 3x^2 - 22x + 24) - (6x + 24))\,dx$; **M1\*** for $F[x] = \frac{x^4}{4} - \frac{3x^3}{3} - \frac{28x^2}{2}$; condone one error in integration |
| $F[0] - F[-4]$ | **M1dep** | Allow $0 - F[-4]$, condone $-F[-4]$, but do not allow $F[-4]$ only. **M1dep** for $F[0] - F[-4]$ |
| Area of triangle $= 48$ | **B1** | |
| Area required $= 96$ from fully correct working | **A1** | **A0** for $-96$, ignore units; no marks for 96 unsupported |
| **[4]** | | |
---2 Fig. 9 shows a sketch of the curve $y = x ^ { 3 } - 3 x ^ { 2 } - 22 x + 24$ and the line $y = 6 x + 24$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{4a9ca68f-f980-4a8f-b387-80dbdca33dfe-2_782_1168_319_451}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{center}
\end{figure}
(i) Differentiate $y = x ^ { 3 } - 3 x ^ { 2 } - 22 x + 24$ and hence find the $x$-coordinates of the turning points of the curve. Give your answers to 2 decimal places.\\
(ii) You are given that the line and the curve intersect when $x = 0$ and when $x = - 4$. Find algebraically the $x$-coordinate of the other point of intersection.\\
(iii) Use calculus to find the area of the region bounded by the curve and the line $y = 6 x + 24$ for $- 4 \leqslant x \leqslant 0$, shown shaded on Fig. 9.
\hfill \mbox{\textit{OCR MEI C2 Q2 [11]}}