Edexcel C4 (Core Mathematics 4) 2013 January

1. Given

$$f ( x ) = ( 2 + 3 x ) ^ { - 3 } , \quad | x | < \frac { 2 } { 3 }$$ find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\). Give each coefficient as a simplified fraction.

2. (a) Use integration to find $$\int \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$$ (b) Hence calculate $$\int _ { 1 } ^ { 2 } \frac { 1 } { x ^ { 3 } } \ln x \mathrm {~d} x$$

3. Express \(\frac { 9 x ^ { 2 } + 20 x - 10 } { ( x + 2 ) ( 3 x - 1 ) }\) in partial fractions.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \frac { x } { 1 + \sqrt { } x }\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, the line with equation \(x = 1\) and the line with equation \(x = 4\).

1. Complete the table with the value of \(y\) corresponding to \(x = 3\), giving your answer to 4 decimal places.
   (1)

   \(x\)	1	2	3	4
   \(y\)	0.5	0.8284		1.3333

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate of the area of the region \(R\), giving your answer to 3 decimal places.
3. Use the substitution \(u = 1 + \sqrt { } x\), to find, by integrating, the exact area of \(R\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a98d4a7f-1e6d-4294-9b5c-c945e8fbe83e-07_743_1568_219_182} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with parametric equations $$x = 1 - \frac { 1 } { 2 } t , \quad y = 2 ^ { t } - 1$$ The curve crosses the \(y\)-axis at the point \(A\) and crosses the \(x\)-axis at the point \(B\).

1. Show that \(A\) has coordinates \(( 0,3 )\).
2. Find the \(x\) coordinate of the point \(B\).
3. Find an equation of the normal to \(C\) at the point \(A\). The region \(R\), as shown shaded in Figure 2, is bounded by the curve \(C\), the line \(x = - 1\) and the \(x\)-axis.
4. Use integration to find the exact area of \(R\).

6. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of part of the curve with equation \(y = 1 - 2 \cos x\), where \(x\) is measured in radians. The curve crosses the \(x\)-axis at the point \(A\) and at the point \(B\).

1. Find, in terms of \(\pi\), the \(x\) coordinate of the point \(A\) and the \(x\) coordinate of the point \(B\). The finite region \(S\) enclosed by the curve and the \(x\)-axis is shown shaded in Figure 3. The region \(S\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find, by integration, the exact value of the volume of the solid generated.

7. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( 9 \mathbf { i } + 13 \mathbf { j } - 3 \mathbf { k } ) + \lambda ( \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } )
& l _ { 2 } : \mathbf { r } = ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \end{aligned}$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet, find the position vector of their point of intersection.
2. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to 1 decimal place. Given that the point \(A\) has position vector \(4 \mathbf { i } + 16 \mathbf { j } - 3 \mathbf { k }\) and that the point \(P\) lies on \(l _ { 1 }\) such that \(A P\) is perpendicular to \(l _ { 1 }\),
3. find the exact coordinates of \(P\).

8. A bottle of water is put into a refrigerator. The temperature inside the refrigerator remains constant at \(3 ^ { \circ } \mathrm { C }\) and \(t\) minutes after the bottle is placed in the refrigerator the temperature of the water in the bottle is \(\theta ^ { \circ } \mathrm { C }\). The rate of change of the temperature of the water in the bottle is modelled by the differential equation, $$\frac { \mathrm { d } \theta } { \mathrm {~d} t } = \frac { ( 3 - \theta ) } { 125 }$$

1. By solving the differential equation, show that, $$\theta = A \mathrm { e } ^ { - 0.008 t } + 3$$ where \(A\) is a constant. Given that the temperature of the water in the bottle when it was put in the refrigerator was \(16 ^ { \circ } \mathrm { C }\),
2. find the time taken for the temperature of the water in the bottle to fall to \(10 ^ { \circ } \mathrm { C }\), giving your answer to the nearest minute.