Edexcel C3 (Core Mathematics 3) 2008 January

1. Given that

$$\frac { 2 x ^ { 4 } - 3 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } - 1 \right) } \equiv \left( a x ^ { 2 } + b x + c \right) + \frac { d x + e } { \left( x ^ { 2 } - 1 \right) }$$ find the values of the constants \(a , b , c , d\) and \(e\).
(4)

2. A curve \(C\) has equation $$y = \mathrm { e } ^ { 2 x } \tan x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$

1. Show that the turning points on \(C\) occur where \(\tan x = - 1\).
2. Find an equation of the tangent to \(C\) at the point where \(x = 0\).

3. $$\mathrm { f } ( x ) = \ln ( x + 2 ) - x + 1 , \quad x > - 2 , x \in \mathbb { R } .$$

1. Show that there is a root of \(\mathrm { f } ( x ) = 0\) in the interval \(2 < x < 3\).
2. Use the iterative formula $$x _ { n + 1 } = \ln \left( x _ { n } + 2 \right) + 1 , x _ { 0 } = 2.5$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) giving your answers to 5 decimal places.
3. Show that \(x = 2.505\) is a root of \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\).
The curve passes through the origin \(O\) and the points \(A ( 5,4 )\) and \(B ( - 5 , - 4 )\).
In separate diagrams, sketch the graph with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ( | x | )\),
3. \(y = 2 f ( x + 1 )\). On each sketch, show the coordinates of the points corresponding to \(A\) and \(B\).

5. The radioactive decay of a substance is given by $$R = 1000 \mathrm { e } ^ { - c t } , \quad t \geqslant 0 .$$ where \(R\) is the number of atoms at time \(t\) years and \(c\) is a positive constant.

1. Find the number of atoms when the substance started to decay. It takes 5730 years for half of the substance to decay.
2. Find the value of \(c\) to 3 significant figures.
3. Calculate the number of atoms that will be left when \(t = 22920\).
4. In the space provided on page 13, sketch the graph of \(R\) against \(t\).

6. (a) Use the double angle formulae and the identity $$\cos ( A + B ) \equiv \cos A \cos B - \sin A \sin B$$ to obtain an expression for \(\cos 3 x\) in terms of powers of \(\cos x\) only.
(b) (i) Prove that $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } \equiv 2 \sec x , \quad x \neq ( 2 n + 1 ) \frac { \pi } { 2 }$$ (ii) Hence find, for \(0 < x < 2 \pi\), all the solutions of $$\frac { \cos x } { 1 + \sin x } + \frac { 1 + \sin x } { \cos x } = 4$$

1. A curve \(C\) has equation

$$y = 3 \sin 2 x + 4 \cos 2 x , - \pi \leqslant x \leqslant \pi$$ The point \(A ( 0,4 )\) lies on \(C\).

1. Find an equation of the normal to the curve \(C\) at \(A\).
2. Express \(y\) in the form \(R \sin ( 2 x + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\). Give the value of \(\alpha\) to 3 significant figures.
3. Find the coordinates of the points of intersection of the curve \(C\) with the \(x\)-axis. Give your answers to 2 decimal places.

1. The functions \(f\) and \(g\) are defined by

$$\begin{aligned} & \mathrm { f } : x \mapsto 1 - 2 x ^ { 3 } , x \in \mathbb { R }
& \mathrm {~g} : x \mapsto \frac { 3 } { x } - 4 , x > 0 , x \in \mathbb { R } \end{aligned}$$

1. Find the inverse function \(\mathrm { f } ^ { - 1 }\).
2. Show that the composite function gf is $$\text { gf } : x \mapsto \frac { 8 x ^ { 3 } - 1 } { 1 - 2 x ^ { 3 } }$$
3. Solve \(\operatorname { gf } ( x ) = 0\).
4. Use calculus to find the coordinates of the stationary point on the graph of \(y = \operatorname { gf } ( x )\).