Edexcel C3 (Core Mathematics 3) 2009 June

1. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = - x ^ { 3 } + 2 x ^ { 2 } + 2\), which intersects the \(x\)-axis at the point \(A\) where \(x = \alpha\). To find an approximation to \(\alpha\), the iterative formula $$x _ { n + 1 } = \frac { 2 } { \left( x _ { n } \right) ^ { 2 } } + 2$$ is used.

1. Taking \(x _ { 0 } = 2.5\), find the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\). Give your answers to 3 decimal places where appropriate.
2. Show that \(\alpha = 2.359\) correct to 3 decimal places.

2. (a) Use the identity \(\cos ^ { 2 } \theta + \sin ^ { 2 } \theta = 1\) to prove that \(\tan ^ { 2 } \theta = \sec ^ { 2 } \theta - 1\).
(b) Solve, for \(0 \leqslant \theta < 360 ^ { \circ }\), the equation $$2 \tan ^ { 2 } \theta + 4 \sec \theta + \sec ^ { 2 } \theta = 2$$

1. Rabbits were introduced onto an island. The number of rabbits, \(P , t\) years after they were introduced is modelled by the equation

$$P = 80 \mathrm { e } ^ { \frac { 1 } { 5 } t } , \quad t \in \mathbb { R } , t \geqslant 0$$

1. Write down the number of rabbits that were introduced to the island.
2. Find the number of years it would take for the number of rabbits to first exceed 1000.
3. Find \(\frac { \mathrm { d } P } { \mathrm {~d} t }\).
4. Find \(P\) when \(\frac { \mathrm { d } P } { \mathrm {~d} t } = 50\).

4. (i) Differentiate with respect to \(x\)

1. \(x ^ { 2 } \cos 3 x\)
2. \(\frac { \ln \left( x ^ { 2 } + 1 \right) } { x ^ { 2 } + 1 }\)
   (ii) A curve \(C\) has the equation $$y = \sqrt { } ( 4 x + 1 ) , \quad x > - \frac { 1 } { 4 } , \quad y > 0$$ The point \(P\) on the curve has \(x\)-coordinate 2 . Find an equation of the tangent to \(C\) at \(P\) in the form \(a x + b y + c = 0\), where \(a , b\) and \(c\) are integers.

5. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\).
The curve meets the coordinate axes at the points \(A ( 0,1 - k )\) and \(B \left( \frac { 1 } { 2 } \ln k , 0 \right)\), where \(k\) is a constant and \(k > 1\), as shown in Figure 2. On separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ^ { - 1 } ( x )\). Show on each sketch the coordinates, in terms of \(k\), of each point at which the curve meets or cuts the axes. Given that \(\mathrm { f } ( x ) = \mathrm { e } ^ { 2 x } - k\),
3. state the range of f ,
4. find \(\mathrm { f } ^ { - 1 } ( x )\),
5. write down the domain of \(\mathrm { f } ^ { - 1 }\).

1. (a) Use the identity \(\cos ( A + B ) = \cos A \cos B - \sin A \sin B\), to show that

$$\cos 2 A = 1 - 2 \sin ^ { 2 } A$$ The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$\begin{aligned} & C _ { 1 } : \quad y = 3 \sin 2 x
& C _ { 2 } : \quad y = 4 \sin ^ { 2 } x - 2 \cos 2 x \end{aligned}$$ (b) Show that the \(x\)-coordinates of the points where \(C _ { 1 }\) and \(C _ { 2 }\) intersect satisfy the equation $$4 \cos 2 x + 3 \sin 2 x = 2$$ (c) Express \(4 \cos 2 x + 3 \sin 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\), giving the value of \(\alpha\) to 2 decimal places.
(d) Hence find, for \(0 \leqslant x < 180 ^ { \circ }\), all the solutions of $$4 \cos 2 x + 3 \sin 2 x = 2$$ giving your answers to 1 decimal place.

7. The function f is defined by $$\mathrm { f } ( x ) = 1 - \frac { 2 } { ( x + 4 ) } + \frac { x - 8 } { ( x - 2 ) ( x + 4 ) } , \quad x \in \mathbb { R } , x \neq - 4 , x \neq 2$$

1. Show that \(\mathrm { f } ( x ) = \frac { x - 3 } { x - 2 }\) The function g is defined by $$\mathrm { g } ( x ) = \frac { \mathrm { e } ^ { x } - 3 } { \mathrm { e } ^ { x } - 2 } , \quad x \in \mathbb { R } , x \neq \ln 2$$
2. Differentiate \(\mathrm { g } ( x )\) to show that \(\mathrm { g } ^ { \prime } ( x ) = \frac { \mathrm { e } ^ { x } } { \left( \mathrm { e } ^ { x } - 2 \right) ^ { 2 } }\)
3. Find the exact values of \(x\) for which \(\mathrm { g } ^ { \prime } ( x ) = 1\)

8. (a) Write down \(\sin 2 x\) in terms of \(\sin x\) and \(\cos x\).
(b) Find, for \(0 < x < \pi\), all the solutions of the equation $$\operatorname { cosec } x - 8 \cos x = 0$$ giving your answers to 2 decimal places.