Edexcel C4 (Core Mathematics 4) 2007 June

1. $$f ( x ) = ( 3 + 2 x ) ^ { - 3 } , \quad | x | < \frac { 3 } { 2 }$$ Find the binomial expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), as far as the term in \(x ^ { 3 }\).
Give each coefficient as a simplified fraction.
(5)

2. Use the substitution \(u = 2 ^ { x }\) to find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 2 ^ { x } } { \left( 2 ^ { x } + 1 \right) ^ { 2 } } d x$$

3. (a) Find \(\int x \cos 2 x d x\).
(b) Hence, using the identity \(\cos 2 x = 2 \cos ^ { 2 } x - 1\), deduce \(\int x \cos ^ { 2 } x \mathrm {~d} x\).

4. $$\frac { 2 \left( 4 x ^ { 2 } + 1 \right) } { ( 2 x + 1 ) ( 2 x - 1 ) } \equiv A + \frac { B } { ( 2 x + 1 ) } + \frac { C } { ( 2 x - 1 ) } .$$

1. Find the values of the constants \(A , B\) and \(C\).
2. Hence show that the exact value of \(\int _ { 1 } ^ { 2 } \frac { 2 \left( 4 x ^ { 2 } + 1 \right) } { ( 2 x + 1 ) ( 2 x - 1 ) } \mathrm { d } x\) is \(2 + \ln k\), giving the value of the constant \(k\).

5. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1
0
- 1 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
0 \end{array} \right)\). The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 1
3
6 \end{array} \right) + \mu \left( \begin{array} { r } 2
1
- 1 \end{array} \right)\).

1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(A\) is on \(l _ { 1 }\) where \(\lambda = 1\), and the point \(B\) is on \(l _ { 2 }\) where \(\mu = 2\).
2. Find the cosine of the acute angle between \(A B\) and \(l _ { 1 }\).

6. A curve has parametric equations $$x = \tan ^ { 2 } t , \quad y = \sin t , \quad 0 < t < \frac { \pi } { 2 }$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). You need not simplify your answer.
2. Find an equation of the tangent to the curve at the point where \(t = \frac { \pi } { 4 }\). Give your answer in the form \(y = a x + b\), where \(a\) and \(b\) are constants to be determined.
3. Find a cartesian equation of the curve in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
   \section*{LU}

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

1. Given that \(y = \sqrt { } ( \tan x )\), complete the table with the values of \(y\) corresponding to \(x = \frac { \pi } { 16 } , \frac { \pi } { 8 }\) and \(\frac { 3 \pi } { 16 }\), giving your answers to 5 decimal places.

   \(x\)	0	\(\frac { \pi } { 16 }\)	\(\frac { \pi } { 8 }\)	\(\frac { 3 \pi } { 16 }\)	\(\frac { \pi } { 4 }\)
   \(y\)	0				1

2. Use the trapezium rule with all the values of \(y\) in the completed table to obtain an estimate for the area of the shaded region \(R\), giving your answer to 4 decimal places. The region \(R\) is rotated through \(2 \pi\) radians around the \(x\)-axis to generate a solid of revolution.
3. Use integration to find an exact value for the volume of the solid generated. \section*{LO}

8. A population growth is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = k P ,$$ where \(P\) is the population, \(t\) is the time measured in days and \(k\) is a positive constant.
Given that the initial population is \(P _ { 0 }\),

1. solve the differential equation, giving \(P\) in terms of \(P _ { 0 } , k\) and \(t\). Given also that \(k = 2.5\),
2. find the time taken, to the nearest minute, for the population to reach \(2 P _ { 0 }\). In an improved model the differential equation is given as $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \lambda P \cos \lambda t$$ where \(P\) is the population, \(t\) is the time measured in days and \(\lambda\) is a positive constant.
   Given, again, that the initial population is \(P _ { 0 }\) and that time is measured in days,
3. solve the second differential equation, giving \(P\) in terms of \(P _ { 0 } , \lambda\) and \(t\). Given also that \(\lambda = 2.5\),
4. find the time taken, to the nearest minute, for the population to reach \(2 P _ { 0 }\) for the first time, using the improved model.