Edexcel C3 (Core Mathematics 3) 2009 January

1. (a) Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at the point where \(x = 2\) on the curve with equation

$$y = x ^ { 2 } \sqrt { } ( 5 x - 1 )$$ (b) Differentiate \(\frac { \sin 2 x } { x ^ { 2 } }\) with respect to \(x\).

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

1. Express \(\mathrm { f } ( x )\) as a single fraction in its simplest form.
2. Hence show that \(\mathrm { f } ^ { \prime } ( x ) = \frac { 2 } { ( x - 3 ) ^ { 2 } }\)

3. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x ) , \quad 1 < x < 9\).
The points \(T ( 3,5 )\) and \(S ( 7,2 )\) are turning points on the graph.
Sketch, on separate diagrams, the graphs of

1. \(y = 2 \mathrm { f } ( x ) - 4\),
2. \(y = | \mathrm { f } ( x ) |\). Indicate on each diagram the coordinates of any turning points on your sketch.

4. Find the equation of the tangent to the curve \(x = \cos ( 2 y + \pi )\) at \(\left( 0 , \frac { \pi } { 4 } \right)\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be found.

5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \mapsto 3 x + \ln x , \quad x > 0 , \quad x \in \mathbb { R }
& \mathrm {~g} : x \mapsto \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } \end{aligned}$$

1. Write down the range of g.
2. Show that the composite function fg is defined by $$\mathrm { fg } : x \mapsto x ^ { 2 } + 3 \mathrm { e } ^ { x ^ { 2 } } , \quad x \in \mathbb { R } .$$
3. Write down the range of fg.
4. Solve the equation \(\frac { \mathrm { d } } { \mathrm { d } x } [ \mathrm { fg } ( x ) ] = x \left( x \mathrm { e } ^ { x ^ { 2 } } + 2 \right)\).

6. (a) (i) By writing \(3 \theta = ( 2 \theta + \theta )\), show that $$\sin 3 \theta = 3 \sin \theta - 4 \sin ^ { 3 } \theta$$ (ii) Hence, or otherwise, for \(0 < \theta < \frac { \pi } { 3 }\), solve $$8 \sin ^ { 3 } \theta - 6 \sin \theta + 1 = 0 .$$ Give your answers in terms of \(\pi\).
(b) Using \(\sin ( \theta - \alpha ) = \sin \theta \cos \alpha - \cos \theta \sin \alpha\), or otherwise, show that $$\sin 15 ^ { \circ } = \frac { 1 } { 4 } ( \sqrt { } 6 - \sqrt { } 2 )$$

7. $$f ( x ) = 3 x e ^ { x } - 1$$ The curve with equation \(y = \mathrm { f } ( x )\) has a turning point \(P\).

1. Find the exact coordinates of \(P\). The equation \(\mathrm { f } ( x ) = 0\) has a root between \(x = 0.25\) and \(x = 0.3\)
2. Use the iterative formula $$x _ { n + 1 } = \frac { 1 } { 3 } \mathrm { e } ^ { - x _ { n } }$$ with \(x _ { 0 } = 0.25\) to find, to 4 decimal places, the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\).
3. By choosing a suitable interval, show that a root of \(\mathrm { f } ( x ) = 0\) is \(x = 0.2576\) correct to 4 decimal places.

8. (a) Express \(3 \cos \theta + 4 \sin \theta\) in the form \(R \cos ( \theta - \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\).
(b) Hence find the maximum value of \(3 \cos \theta + 4 \sin \theta\) and the smallest positive value of \(\theta\) for which this maximum occurs. The temperature, \(\mathrm { f } ( t )\), of a warehouse is modelled using the equation $$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$ where \(t\) is the time in hours from midday and \(0 \leqslant t < 24\).
(c) Calculate the minimum temperature of the warehouse as given by this model.
(d) Find the value of \(t\) when this minimum temperature occurs.