| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2005 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas Between Curves |
| Type | Area Between Curve and Both Axes |
| Difficulty | Standard +0.3 This is a standard C2 integration question with routine differentiation, tangent finding, and area calculation. While it has multiple parts and requires careful handling of the area between -2 and 6 (splitting at the x-axis crossing), all techniques are straightforward applications of core calculus methods with no novel problem-solving required. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07i Differentiate x^n: for rational n and sums1.07m Tangents and normals: gradient and equations1.08e Area between curve and x-axis: using definite integrals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(3x^2 - 20x + 12\) | B1 if one error "\(+c\)" is an error | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| eg \(y = -16x + 96\) | M1 for subst \(x = 2\) in their \(y'\) A1 for \(y' = -16\) and B1 for \(y = 64\) | 4 |
| (iii) Factorising \(f(x) \equiv (x+2)(x-6)^2\) OR Expanding \((x+2)(x-6)^2\) | B3 or B1 for \(f(-2) = -8-40-24+72 = 0\) and B1 for \(f'(6) = 0\) and B1dep for \(f(6)=0\) | 3 |
| (iv) \(\frac{x^4}{4} - \frac{10x^3}{3} + 6x^2 + 72x\) value at \((x = 6)\) ~ value at \((x = -2)\) \(341(3...)\) cao | B2 M1 Must have integrated \(f(x)\) A1 | 4 |
(i) $3x^2 - 20x + 12$ | B1 if one error "$+c$" is an error | 2
(ii) $y - 64 = -16(x - 2)$ o.e.
eg $y = -16x + 96$ | M1 for subst $x = 2$ in their $y'$ A1 for $y' = -16$ and B1 for $y = 64$ | 4
(iii) Factorising $f(x) \equiv (x+2)(x-6)^2$ OR Expanding $(x+2)(x-6)^2$ | B3 or B1 for $f(-2) = -8-40-24+72 = 0$ and B1 for $f'(6) = 0$ and B1dep for $f(6)=0$ | 3
(iv) $\frac{x^4}{4} - \frac{10x^3}{3} + 6x^2 + 72x$ value at $(x = 6)$ ~ value at $(x = -2)$ $341(3...)$ cao | B2 M1 Must have integrated $f(x)$ A1 | 49
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{faeaf2aa-ed4e-4926-b402-40c4c9aad479-3_535_790_450_630}
\captionsetup{labelformat=empty}
\caption{Fig. 9}
\end{center}
\end{figure}
Fig. 9 shows a sketch of the graph of $y = x ^ { 3 } - 10 x ^ { 2 } + 12 x + 72$.\\
(i) Write down $\frac { \mathrm { d } y } { \mathrm {~d} x }$.\\
(ii) Find the equation of the tangent to the curve at the point on the curve where $x = 2$.\\
(iii) Show that the curve crosses the $x$-axis at $x = - 2$. Show also that the curve touches the $x$-axis at $x = 6$.\\
(iv) Find the area of the finite region bounded by the curve and the $x$-axis, shown shaded in Fig. 9 . [4]
\hfill \mbox{\textit{OCR MEI C2 2005 Q9 [13]}}