| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 18 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard C3 techniques (finding constants from graphs, differentiation, inverse functions, integration). Part (i) requires reading values from a graph; part (ii) involves routine differentiation and showing a maximum using calculus; part (iii) requires finding an inverse function with domain/range and using the inverse function gradient relationship; part (iv) is straightforward integration. All parts follow textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.05f Trigonometric function graphs: symmetries and periodicities1.08d Evaluate definite integrals: between limits |
Fig. 9 shows the curve $y = f(x)$. The endpoints of the curve are P $(-\pi, 1)$ and Q $(\pi, 3)$, and $f(x) = a + \sin bx$, where $a$ and $b$ are constants.
\includegraphics{figure_9}
\begin{enumerate}[label=(\roman*)]
\item Using Fig. 9, show that $a = 2$ and $b = \frac{1}{2}$. [3]
\item Find the gradient of the curve $y = f(x)$ at the point $(0, 2)$.
Show that there is no point on the curve at which the gradient is greater than this. [5]
\item Find $f^{-1}(x)$, and state its domain and range.
Write down the gradient of $y = f^{-1}(x)$ at the point $(2, 0)$. [6]
\item Find the area enclosed by the curve $y = f(x)$, the $x$-axis, the $y$-axis and the line $x = \pi$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 2011 Q9 [18]}}