Edexcel C34 (Core Mathematics 3 & 4) 2014 January

1. $$\mathrm { f } ( x ) = \frac { 2 x } { x ^ { 2 } + 3 } , \quad x \in \mathbb { R }$$ Find the set of values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) > 0\) You must show your working.
(Solutions based entirely on graphical or numerical methods are not acceptable.)

2. Solve, for \(0 \leqslant x \leqslant 270 ^ { \circ }\), the equation $$\frac { \tan 2 x + \tan 50 ^ { \circ } } { 1 - \tan 2 x \tan 50 ^ { \circ } } = 2$$ Give your answers in degrees to 2 decimal places.
(6)
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3. Given that $$4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 \equiv ( A x + B ) \left( x ^ { 2 } + 4 \right) + C x + D$$

1. find the values of the constants \(A , B , C\) and \(D\).
2. Hence find $$\int _ { 1 } ^ { 4 } \frac { 4 x ^ { 3 } + 2 x ^ { 2 } + 17 x + 8 } { x ^ { 2 } + 4 } d x$$ giving your answer in the form \(p + \ln q\), where \(p\) and \(q\) are integers.

4. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 | 3 - x | + 5 , \quad x \geqslant 0$$ Figure 2 shows a sketch of part of the graph \(y = \mathrm { g } ( x )\), where $$\operatorname { g } ( x ) = \frac { x + 9 } { 2 x + 3 } , \quad x \geqslant 0$$

1. Find the value of \(\mathrm { fg } ( 1 )\)
2. State the range of g
3. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain. Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
4. state the range of possible values of \(k\).

1. (a) Prove, by using logarithms, that

$$\frac { \mathrm { d } } { \mathrm {~d} x } \left( 2 ^ { x } \right) = 2 ^ { x } \ln 2$$ The curve \(C\) has the equation $$2 x + 3 y ^ { 2 } + 3 x ^ { 2 } y + 12 = 4 \times 2 ^ { x }$$ The point \(P\), with coordinates \(( 2,0 )\), lies on \(C\).
(b) Find an equation of the tangent to \(C\) at \(P\).

6. Given that the binomial expansion, in ascending powers of \(x\), of $$\frac { 6 } { \sqrt { } \left( 9 + A x ^ { 2 } \right) } , \quad | x | < \frac { 3 } { \sqrt { } | A | }$$ is \(\quad B - \frac { 2 } { 3 } x ^ { 2 } + C x ^ { 4 } + \ldots\)

1. find the values of the constants \(A , B\) and \(C\).
2. Hence find the coefficient of \(x ^ { 6 }\)

7. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$f ( x ) = 2 x ( 1 + x ) \ln x , \quad x > 0$$ The curve has a minimum turning point at \(A\).

1. Find f'(x)
2. Hence show that the \(x\) coordinate of \(A\) is the solution of the equation $$x = \mathrm { e } ^ { - \frac { 1 + x } { 1 + 2 x } }$$
3. Use the iteration formula $$x _ { n + 1 } = \mathrm { e } ^ { - \frac { 1 + x _ { n } } { 1 + 2 x _ { n } } } , \quad x _ { 0 } = 0.46$$ to find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 4 decimal places.
4. Use your answer to part (c) to estimate the coordinates of \(A\) to 2 decimal places.

8. (a) Prove that $$\text { 2cosec } 2 A - \cot A \equiv \tan A , \quad A \neq \frac { n \pi } { 2 } , n \in \mathbb { Z }$$ (b) Hence solve, for \(0 \leqslant \theta \leqslant \frac { \pi } { 2 }\)

1. \(2 \operatorname { cosec } 4 \theta - \cot 2 \theta = \sqrt { } 3\)
2. \(\tan \theta + \cot \theta = 5\) Give your answers to 3 significant figures.

9. (a) Use the substitution \(u = 4 - \sqrt { } x\) to find $$\int \frac { \mathrm { d } x } { 4 - \sqrt { } x }$$ A team of scientists is studying a species of slow growing tree.
The rate of change in height of a tree in this species is modelled by the differential equation $$\frac { \mathrm { d } h } { \mathrm {~d} t } = \frac { 4 - \sqrt { } h } { 20 }$$ where \(h\) is the height in metres and \(t\) is the time measured in years after the tree is planted.
(b) Find the range in values of \(h\) for which the height of a tree in this species is increasing.
(c) Given that one of these trees is 1 metre high when it is planted, calculate the time it would take to reach a height of 10 metres. Write your answer to 3 significant figures.
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10. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 5 \mathbf { j } + 5 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } )
& l _ { 2 } : \mathbf { r } = ( 2 \mathbf { j } + 12 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) meet and find the position vector of their point of intersection.
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are perpendicular to each other. The point \(A\), with position vector \(5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }\), lies on \(l _ { 1 }\)
   The point \(B\) is the image of \(A\) after reflection in the line \(l _ { 2 }\)
3. Find the position vector of \(B\).
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11. The curve \(C\) has parametric equations $$x = 10 \cos 2 t , \quad y = 6 \sin t , \quad - \frac { \pi } { 2 } \leqslant t \leqslant \frac { \pi } { 2 }$$ The point \(A\) with coordinates \(( 5,3 )\) lies on \(C\).

1. Find the value of \(t\) at the point \(A\).
2. Show that an equation of the normal to \(C\) at \(A\) is $$3 y = 10 x - 41$$ The normal to \(C\) at \(A\) cuts \(C\) again at the point \(B\).
3. Find the exact coordinates of \(B\).

12. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve with equation $$y = x ( \sin x + \cos x ) , \quad 0 \leqslant x \leqslant \frac { \pi } { 4 }$$ The finite region \(R\), shown shaded in Figure 4, is bounded by the curve, the \(x\)-axis and the line \(x = \frac { \pi } { 4 }\). This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution, with volume \(V\).

1. Assuming the formula for volume of revolution show that \(V = \int _ { 0 } ^ { \frac { \pi } { 4 } } \pi x ^ { 2 } ( 1 + \sin 2 x ) \mathrm { d } x\)
2. Hence using calculus find the exact value of \(V\). You must show your working.
   (Solutions based entirely on graphical or numerical methods are not acceptable.)