| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Curve-Line Intersection Area |
| Difficulty | Moderate -0.8 This is a straightforward C2 question with routine tasks: finding intersection points by solving x^4 = 8x, computing a simple definite integral for area between curves, and completing a standard first principles differentiation exercise. All parts require only direct application of basic techniques with no problem-solving insight needed. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.04a Binomial expansion: (a+b)^n for positive integer n1.07g Differentiation from first principles: for small positive integer powers of x1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| \(x^4 = 8x\) | M1 | |
| \((2, 16)\) c.a.o. | A1 | |
| \(PQ = 16\) and completion to show \(\frac{1}{2} \times 2 \times 16 = 16\) | A1 | NB answer 16 given |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{x^5}{5}\) | M1 | |
| evaluating their integral at their co-ord of P and zero [or 32/5 o.e.] | M1 | ft only if integral attempted, not for \(x^4\) or differentiation |
| \(9.6\) o.e. | A1 | c.a.o. |
| Answer | Marks | Guidance |
|---|---|---|
| \(6x^2h^2 + 4xh^3 + h^4\) | 2 | B1 for two terms correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(4x^3 + 6x^2h + 4xh^2 + h^3\) | 2 | B1 for three terms correct |
| Answer | Marks | Guidance |
|---|---|---|
| \(4x^3\) | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| gradient of [tangent to] curve | 1 |
## Question 1:
**Part iA:**
$x^4 = 8x$ | M1 |
$(2, 16)$ c.a.o. | A1 |
$PQ = 16$ and completion to show $\frac{1}{2} \times 2 \times 16 = 16$ | A1 | NB answer 16 given | **[3]**
**Part iB:**
$\frac{x^5}{5}$ | M1 |
evaluating their integral at their co-ord of P and zero [or 32/5 o.e.] | M1 | ft only if integral attempted, not for $x^4$ or differentiation |
$9.6$ o.e. | A1 | c.a.o. | **[3]**
**Part iiA:**
$6x^2h^2 + 4xh^3 + h^4$ | 2 | B1 for two terms correct | **[2]**
**Part iiB:**
$4x^3 + 6x^2h + 4xh^2 + h^3$ | 2 | B1 for three terms correct | **[2]**
**Part iiC:**
$4x^3$ | 1 | | **[1]**
**Part iiD:**
gradient of [tangent to] curve | 1 | | **[1]**
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\begin{enumerate}[label=(\roman*)]
\item \begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{1a6d059d-8ab8-41e0-8bf3-54e248f820e4-1_650_759_252_762}
\captionsetup{labelformat=empty}
\caption{Fig. 12}
\end{center}
\end{figure}
Fig. 12 shows part of the curve $y = x ^ { 4 }$ and the line $y = 8 x$, which intersect at the origin and the point P .\\
(A) Find the coordinates of P , and show that the area of triangle OPQ is 16 square units.\\
(B) Find the area of the region bounded by the line and the curve.
\item You are given that $\mathrm { f } ( x ) = x ^ { 4 }$.\\
(A) Complete this identity for $\mathrm { f } ( x + h )$.
$$\mathrm { f } ( x + h ) = ( x + h ) ^ { 4 } = x ^ { 4 } + 4 x ^ { 3 } h + \ldots$$
(B) Simplify $\frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }$.\\
(C) Find $\lim _ { h \rightarrow 0 } \frac { \mathrm { f } ( x + h ) - \mathrm { f } ( x ) } { h }$.\\
(D) State what this limit represents.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 Q1 [12]}}