OCR MEI C3 (Core Mathematics 3)

1

1. Show algebraically that the function \(\mathrm { f } ( x ) = \frac { 2 x } { 1 - x ^ { 2 } }\) is odd. Fig. 7 shows the curve \(y = \mathrm { f } ( x )\) for \(0 \leqslant x \leqslant 4\), together with the asymptote \(x = 1\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8350e810-3ceb-4876-a7a8-249e17616057-1_718_813_567_644} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
2. Use the copy of Fig. 7 to complete the curve for \(- 4 \leqslant x \leqslant 4\).

3 Each of the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\) below is obtained using a sequence of two transformations applied to the corresponding dashed graph. In each case, state suitable transformations, and hence find expressions for \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\).

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{8350e810-3ceb-4876-a7a8-249e17616057-2_433_716_569_710}
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{8350e810-3ceb-4876-a7a8-249e17616057-2_396_612_1130_761}

4 Fig. 4 shows the curve \(y = f ( x )\), where \(f ( x ) = \sqrt { 1 - 9 x ^ { 2 } } , - a \leqslant x \leqslant a\). \begin{figure}[h] \end{figure}

1. Find the value of \(a\).
2. Write down the range of \(\mathrm { f } ( x )\).
3. Sketch the curve \(y = \mathrm { f } \left( \frac { 1 } { 3 } x \right) - 1\).

5 You are given that \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) are odd functions, defined for \(x \in \mathbb { R }\).

1. Given that \(\mathrm { s } ( x ) = \mathrm { f } ( x ) + \mathrm { g } ( x )\), prove that \(\mathrm { s } ( x )\) is an odd function.
2. Given that \(\mathrm { p } ( x ) = \mathrm { f } ( x ) \mathrm { g } ( x )\), determine whether \(\mathrm { p } ( x )\) is odd, even or neither.

6

1. State the algebraic condition for the function \(\mathrm { f } ( x )\) to be an even function.
   What geometrical property does the graph of an even function have?
2. State whether the following functions are odd, even or neither.
   (A) \(\mathrm { f } ( x ) = x ^ { 2 } - 3\)
   (B) \(\mathrm { g } ( x ) = \sin x + \cos x\)
   (C) \(\mathrm { h } ( x ) = \frac { 1 } { x + x ^ { 3 } }\)

7 Fig. 8 shows part of the curve \(y = \mathrm { f } ( x )\), where \(\mathrm { f } ( x ) = \mathrm { e } ^ { - \frac { 1 } { 5 } x } \sin x\), for all \(x\). \begin{figure}[h] \end{figure}

1. Sketch the graphs of
   (A) \(y = \mathrm { f } ( 2 x )\),
   (B) \(y = \mathrm { f } ( x + \pi )\).
2. Show that the \(x\)-coordinate of the turning point P satisfies the equation \(\tan x = 5\). Hence find the coordinates of P .
3. Show that \(\mathrm { f } ( x + \pi ) = \mathrm { e } ^ { - \frac { 1 } { 5 } \pi } \mathrm { f } ( x )\). Hence, using the substitution \(u = x - \pi\), show that $$\int _ { \pi } ^ { 2 \pi } \mathrm { f } ( x ) \mathrm { d } x = \mathrm { e } ^ { - \frac { 1 } { 5 } \pi } \int _ { 0 } ^ { \pi } \mathrm { f } ( u ) \mathrm { d } u .$$ Interpret this result graphically. [You should not attempt to integrate \(\mathrm { f } ( x )\).]