| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | January |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Area with parametric/substitution |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard C3 techniques: verification by substitution, quotient rule differentiation, integration by substitution, and area calculation. While it requires several steps and careful algebra, each part follows routine procedures with clear guidance (e.g., 'show that', 'using the substitution'). The symmetry argument is straightforward once the gradient is found. Slightly above average due to the algebraic manipulation required, but no novel insights needed. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08f Area between two curves: using integration1.08h Integration by substitution |
Fig. 8 shows the curve $y = \frac{x}{\sqrt{x-2}}$, together with the lines $y = x$ and $x = 11$.
The curve meets these lines at P and Q respectively. R is the point $(11, 11)$.
\includegraphics{figure_8}
\begin{enumerate}[label=(\roman*)]
\item Verify that the $x$-coordinate of P is 3. [2]
\item Show that, for the curve, $\frac{dy}{dx} = \frac{x-4}{2(x-2)^{\frac{3}{2}}}$.
Hence find the gradient of the curve at P. Use the result to show that the curve is \textbf{not} symmetrical about $y = x$. [7]
\item Using the substitution $u = x - 2$, show that $\int_3^{11} \frac{x}{\sqrt{x-2}} \, dx = 25\frac{1}{3}$.
Hence find the area of the region PQR bounded by the curve and the lines $y = x$ and $x = 11$. [9]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 2012 Q8 [18]}}