Edexcel C4 (Core Mathematics 4) 2009 June

1. $$f ( x ) = \frac { 1 } { \sqrt { ( 4 + x ) } } , \quad | x | < 4$$ 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.
(6)

2. \begin{figure}[h] \end{figure} Figure 1 shows the finite region \(R\) bounded by the \(x\)-axis, the \(y\)-axis and the curve with equation \(y = 3 \cos \left( \frac { x } { 3 } \right) , 0 \leqslant x \leqslant \frac { 3 \pi } { 2 }\).
The table shows corresponding values of \(x\) and \(y\) for \(y = 3 \cos \left( \frac { x } { 3 } \right)\).

1. Complete the table above giving the missing value of \(y\) to 5 decimal places.
2. Using the trapezium rule, with all the values of \(y\) from the completed table, find an approximation for the area of \(R\), giving your answer to 3 decimal places.
3. Use integration to find the exact area of \(R\).

3. $$\mathrm { f } ( x ) = \frac { 4 - 2 x } { ( 2 x + 1 ) ( x + 1 ) ( x + 3 ) } = \frac { A } { 2 x + 1 } + \frac { B } { x + 1 } + \frac { C } { x + 3 }$$

1. Find the values of the constants \(A , B\) and \(C\).
2. 1. Hence find \(\int f ( x ) \mathrm { d } x\).
   2. Find \(\int _ { 0 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x\) in the form \(\ln k\), where \(k\) is a constant.

4. The curve \(C\) has the equation \(y \mathrm { e } ^ { - 2 x } = 2 x + y ^ { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) on \(C\) has coordinates \(( 0,1 )\).
2. Find the equation of the normal to \(C\) at \(P\), giving your answer 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 the curve with parametric equations $$x = 2 \cos 2 t , \quad y = 6 \sin t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$

1. Find the gradient of the curve at the point where \(t = \frac { \pi } { 3 }\).
2. Find a cartesian equation of the curve in the form $$y = \mathrm { f } ( x ) , \quad - k \leqslant x \leqslant k$$ stating the value of the constant \(k\).
3. Write down the range of \(\mathrm { f } ( x )\).

6. (a) Find \(\int \sqrt { } ( 5 - x ) \mathrm { d } x\).
(2) \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve with equation $$y = ( x - 1 ) \sqrt { } ( 5 - x ) , \quad 1 \leqslant x \leqslant 5$$ (b) (i) Using integration by parts, or otherwise, find $$\int ( x - 1 ) \sqrt { } ( 5 - x ) \mathrm { d } x$$ (ii) Hence find \(\int _ { 1 } ^ { 5 } ( x - 1 ) \sqrt { } ( 5 - x ) \mathrm { d } x\).

7. Relative to a fixed origin \(O\), the point \(A\) has position vector \(( 8 \mathbf { i } + 13 \mathbf { j } - 2 \mathbf { k } )\), the point \(B\) has position vector ( \(10 \mathbf { i } + 14 \mathbf { j } - 4 \mathbf { k }\) ), and the point \(C\) has position vector \(( 9 \mathbf { i } + 9 \mathbf { j } + 6 \mathbf { k } )\). The line \(l\) passes through the points \(A\) and \(B\).

1. Find a vector equation for the line \(l\).
2. Find \(| \overrightarrow { C B } |\).
3. Find the size of the acute angle between the line segment \(C B\) and the line \(l\), giving your answer in degrees to 1 decimal place.
4. Find the shortest distance from the point \(C\) to the line \(l\). The point \(X\) lies on \(l\). Given that the vector \(\overrightarrow { C X }\) is perpendicular to \(l\),
5. find the area of the triangle \(C X B\), giving your answer to 3 significant figures.

8. (a) Using the identity \(\cos 2 \theta = 1 - 2 \sin ^ { 2 } \theta\), find \(\int \sin ^ { 2 } \theta \mathrm {~d} \theta\). \begin{figure}[h] \end{figure} Figure 4 shows part of the curve \(C\) with parametric equations $$x = \tan \theta , \quad y = 2 \sin 2 \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$ The finite shaded region \(S\) shown in Figure 4 is bounded by \(C\), the line \(x = \frac { 1 } { \sqrt { 3 } }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
(b) Show that the volume of the solid of revolution formed is given by the integral $$k \int _ { 0 } ^ { \frac { \pi } { 6 } } \sin ^ { 2 } \theta \mathrm {~d} \theta$$ where \(k\) is a constant.
(c) Hence find the exact value for this volume, giving your answer in the form \(p \pi ^ { 2 } + q \pi \sqrt { } 3\), where \(p\) and \(q\) are constants.