Edexcel C3 (Core Mathematics 3) 2007 January

1. (a) By writing \(\sin 3 \theta\) as \(\sin ( 2 \theta + \theta )\), show that

$$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$ (b) Given that \(\sin \theta = \frac { \sqrt { } 3 } { 4 }\), find the exact value of \(\sin 3 \theta\).

2. $$f ( x ) = 1 - \frac { 3 } { x + 2 } + \frac { 3 } { ( x + 2 ) ^ { 2 } } , x \neq - 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + x + 1 } { ( x + 2 ) ^ { 2 } } , x \neq - 2\).
2. Show that \(x ^ { 2 } + x + 1 > 0\) for all values of \(x\).
3. Show that \(\mathrm { f } ( x ) > 0\) for all values of \(x , x \neq - 2\).

3. The curve \(C\) has equation $$x = 2 \sin y .$$

1. Show that the point \(P \left( \sqrt { } 2 , \frac { \pi } { 4 } \right)\) lies on \(C\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 2 } }\) at \(P\).
3. Find an equation of the normal to \(C\) at \(P\). Give your answer in the form \(y = m x + c\), where \(m\) and \(c\) are exact constants.

4. (i) The curve \(C\) has equation $$y = \frac { x } { 9 + x ^ { 2 } }$$ Use calculus to find the coordinates of the turning points of \(C\).
(ii) Given that $$y = \left( 1 + \mathrm { e } ^ { 2 x } \right) ^ { \frac { 3 } { 2 } }$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { 1 } { 2 } \ln 3\).

5. \begin{figure}[h] \end{figure} Figure 1 shows an oscilloscope screen. The curve shown on the screen satisfies the equation $$y = \sqrt { 3 } \cos x + \sin x$$

1. Express the equation of the curve in the form \(y = R \sin ( x + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\).
2. Find the values of \(x , 0 \leqslant x < 2 \pi\), for which \(y = 1\).

1. The function \(f\) is defined by

$$\mathrm { f } : x \mapsto \ln ( 4 - 2 x ) , \quad x < 2 \quad \text { and } \quad x \in \mathbb { R } .$$

1. Show that the inverse function of f is defined by $$\mathrm { f } ^ { - 1 } : x \mapsto 2 - \frac { 1 } { 2 } \mathrm { e } ^ { x }$$ and write down the domain of \(\mathrm { f } ^ { - 1 }\).
2. Write down the range of \(\mathrm { f } ^ { - 1 }\).
3. In the space provided on page 16, sketch the graph of \(y = f ^ { - 1 } ( x )\). State the coordinates of the points of intersection with the \(x\) and \(y\) axes. The graph of \(y = x + 2\) crosses the graph of \(y = f ^ { - 1 } ( x )\) at \(x = k\). The iterative formula $$x _ { n + 1 } = - \frac { 1 } { 2 } e ^ { x _ { n } } , x _ { 0 } = - 0.3$$ is used to find an approximate value for \(k\).
4. Calculate the values of \(x _ { 1 }\) and \(x _ { 2 }\), giving your answers to 4 decimal places.
5. Find the value of \(k\) to 3 decimal places.

7. $$f ( x ) = x ^ { 4 } - 4 x - 8$$

1. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \([ - 2 , - 1 ]\).
2. Find the coordinates of the turning point on the graph of \(y = \mathrm { f } ( x )\).
3. Given that \(\mathrm { f } ( x ) = ( x - 2 ) \left( x ^ { 3 } + a x ^ { 2 } + b x + c \right)\), find the values of the constants, \(a , b\) and \(c\).
4. In the space provided on page 21, sketch the graph of \(y = \mathrm { f } ( x )\).
5. Hence sketch the graph of \(y = | \mathrm { f } ( x ) |\).

1. 1. Prove that

$$\sec ^ { 2 } x - \operatorname { cosec } ^ { 2 } x \equiv \tan ^ { 2 } x - \cot ^ { 2 } x$$ (ii) Given that $$y = \arccos x , \quad - 1 \leqslant x \leqslant 1 \text { and } 0 \leqslant y \leqslant \pi ,$$

1. express arcsin \(x\) in terms of \(y\).
2. Hence evaluate \(\arccos x + \arcsin x\). Give your answer in terms of \(\pi\).