Edexcel C3 (Core Mathematics 3) 2010 January

1. Express

$$\frac { x + 1 } { 3 x ^ { 2 } - 3 } - \frac { 1 } { 3 x + 1 }$$ as a single fraction in its simplest form.

2. $$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 3 x - 11$$

1. Show that \(\mathrm { f } ( x ) = 0\) can be rearranged as $$x = \sqrt { } \left( \frac { 3 x + 11 } { x + 2 } \right) , \quad x \neq - 2 .$$ The equation \(\mathrm { f } ( x ) = 0\) has one positive root \(\alpha\). The iterative formula \(x _ { n + 1 } = \sqrt { } \left( \frac { 3 x _ { n } + 11 } { x _ { n } + 2 } \right)\) is used to find an approximation to \(\alpha\).
2. Taking \(x _ { 1 } = 0\), find, to 3 decimal places, the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
3. Show that \(\alpha = 2.057\) correct to 3 decimal places.

3. (a) Express \(5 \cos x - 3 \sin x\) in the form \(R \cos ( x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
(b) Hence, or otherwise, solve the equation $$5 \cos x - 3 \sin x = 4$$ for \(0 \leqslant x < 2 \pi\), giving your answers to 2 decimal places.

4. (i) Given that \(y = \frac { \ln \left( x ^ { 2 } + 1 \right) } { x }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(ii) Given that \(x = \tan y\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).

5. Sketch the graph of \(y = \ln | x |\), stating the coordinates of any points of intersection with the axes.

6. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the graph of \(y = \mathrm { f } ( x )\).
The graph intersects the \(y\)-axis at the point \(( 0,1 )\) and the point \(A ( 2,3 )\) is the maximum turning point. Sketch, on separate axes, the graphs of

1. \(y = \mathrm { f } ( - x ) + 1\),
2. \(y = \mathrm { f } ( x + 2 ) + 3\),
3. \(y = 2 \mathrm { f } ( 2 x )\). On each sketch, show the coordinates of the point at which your graph intersects the \(y\)-axis and the coordinates of the point to which \(A\) is transformed.

1. (a) By writing \(\sec x\) as \(\frac { 1 } { \cos x }\), show that \(\frac { \mathrm { d } ( \sec x ) } { \mathrm { d } x } = \sec x \tan x\).

Given that \(y = \mathrm { e } ^ { 2 x } \sec 3 x\),
(b) find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). The curve with equation \(y = \mathrm { e } ^ { 2 x } \sec 3 x , - \frac { \pi } { 6 } < x < \frac { \pi } { 6 }\), has a minimum turning point at \(( a , b )\).
(c) Find the values of the constants \(a\) and \(b\), giving your answers to 3 significant figures.

8. Solve $$\operatorname { cosec } ^ { 2 } 2 x - \cot 2 x = 1$$ for \(0 \leqslant x \leqslant 180 ^ { \circ }\).

9. (i) Find the exact solutions to the equations

1. \(\ln ( 3 x - 7 ) = 5\)
2. \(3 ^ { x } \mathrm { e } ^ { 7 x + 2 } = 15\)
   (ii) The functions f and g are defined by $$\begin{array} { l l } \mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } + 3 , & x \in \mathbb { R }
   \mathrm {~g} ( x ) = \ln ( x - 1 ) , & x \in \mathbb { R } , x > 1 \end{array}$$
3. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
4. Find fg and state its range.