| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Integration by Substitution |
| Type | Area under curve using substitution |
| Difficulty | Standard +0.8 This is a substantial multi-part question requiring quotient rule differentiation (with chain rule for the denominator), finding a tangent equation, identifying an asymptote, and performing integration by substitution with algebraic manipulation. The integration part (9 marks) likely involves non-trivial substitution and algebraic simplification to find exact area. While the techniques are all C3 standard, the combination of steps, the need to handle the asymptote geometry, and the extended integration with exact form pushes this moderately above average difficulty. |
| Spec | 1.07m Tangents and normals: gradient and equations1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08h Integration by substitution |
Fig. 8 shows the curve $y = \frac{x}{\sqrt{x+4}}$ and the line $x = 5$. The curve has an asymptote $l$.
The tangent to the curve at the origin O crosses the line $l$ at P and the line $x = 5$ at Q.
\includegraphics{figure_8}
\begin{enumerate}[label=(\roman*)]
\item Show that for this curve $\frac{dy}{dx} = \frac{x + 8}{2(x + 4)^{\frac{3}{2}}}$. [5]
\item Find the coordinates of the point P. [4]
\item Using integration by substitution, find the exact area of the region enclosed by the curve, the tangent OQ and the line $x = 5$. [9]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 2016 Q8 [18]}}