| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Function Transformations |
| Type | Sequence of transformations order |
| Difficulty | Standard +0.3 This is a multi-part question covering standard C3 transformations, integration, differentiation, and inverse functions. Part (i) requires identifying two transformations (horizontal stretch by factor 1/2, then vertical translation up 1 unit), which is routine. Parts (ii)-(v) involve standard techniques with no novel insights required. The most challenging aspect is correctly ordering transformations and finding the inverse function, but these are well-practiced skills at this level. Overall slightly easier than average due to the structured, guided nature of the parts. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08d Evaluate definite integrals: between limits |
8 Fig. 8 shows the curve $y = \mathrm { f } ( x )$, where $\mathrm { f } ( x ) = 1 + \sin 2 x$ for $- \frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 4 } \pi$.
\begin{tikzpicture}[>=latex, x=3.5cm, y=1.6cm]
% Draw horizontal x-axis
\draw[->] (-1.1, 0) -- (1.1, 0) node[right] {$x$};
% Draw vertical y-axis
\draw[->] (0, -0.4) -- (0, 2.5) node[above] {$y$};
% Origin label
\node[below left] at (0, 0) {O};
% Plot the curve: f(x) = 1 + sin(2x)
\draw[domain=-0.785398:0.785398, smooth, samples=100, thick]
plot (\x, {1 + sin(deg(2*\x))});
% Ticks on x-axis
\draw ({-pi/4}, 0) -- ({-pi/4}, -0.05) node[below] {$-\frac{1}{4}\pi$};
\draw ({pi/4}, 0) -- ({pi/4}, -0.05) node[below] {$\frac{1}{4}\pi$};
% Tick on y-axis for y = 2
\draw (0, 2) -- (-0.03, 2) node[left] {$2$};
\end{tikzpicture}
(i) State a sequence of two transformations that would map part of the curve $y = \sin x$ onto the curve $y = \mathrm { f } ( x )$.\\
(ii) Find the area of the region enclosed by the curve $y = \mathrm { f } ( x )$, the $x$-axis and the line $x = \frac { 1 } { 4 } \pi$.\\
(iii) Find the gradient of the curve $y = \mathrm { f } ( x )$ at the point $( 0,1 )$. Hence write down the gradient of the curve $y = \mathrm { f } ^ { - 1 } ( x )$ at the point $( 1,0 )$.\\
(iv) State the domain of $\mathrm { f } ^ { - 1 } ( x )$. Add a sketch of $y = \mathrm { f } ^ { - 1 } ( x )$ to a copy of Fig. 8.\\
(v) Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{OCR MEI C3 2008 Q8 [17]}}