Edexcel C4 (Core Mathematics 4) 2007 January

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

2. \begin{figure}[h] \end{figure} The curve with equation \(y = \frac { 1 } { 3 ( 1 + 2 x ) } , x > - \frac { 1 } { 2 }\), is shown in Figure 1.
The region bounded by the lines \(x = - \frac { 1 } { 4 } , x = \frac { 1 } { 2 }\), the \(x\)-axis and the curve is shown shaded in Figure 1. This region is rotated through 360 degrees about the \(x\)-axis.

1. Use calculus to find the exact value of the volume of the solid generated. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{d366e541-15f6-4fb5-9afb-faf6120f1a1c-03_383_447_1411_753}
   \end{figure} Figure 2 shows a paperweight with axis of symmetry \(A B\) where \(A B = 3 \mathrm {~cm}\). \(A\) is a point on the top surface of the paperweight, and \(B\) is a point on the base of the paperweight. The paperweight is geometrically similar to the solid in part (a).
2. Find the volume of this paperweight.

1. A curve has parametric equations

$$x = 7 \cos t - \cos 7 t , y = 7 \sin t - \sin 7 t , \quad \frac { \pi } { 8 } < t < \frac { \pi } { 3 }$$

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 normal to the curve at the point where \(t = \frac { \pi } { 6 }\). Give your answer in its simplest exact form.

4. (a) Express \(\frac { 2 x - 1 } { ( x - 1 ) ( 2 x - 3 ) }\) in partial fractions.
(b) Given that \(x \geqslant 2\), find the general solution of the differential equation $$( 2 x - 3 ) ( x - 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = ( 2 x - 1 ) y$$ (c) Hence find the particular solution of this differential equation that satisfies \(y = 10\) at \(x = 2\), giving your answer in the form \(y = \mathrm { f } ( x )\).

5. A set of curves is given by the equation \(\sin x + \cos y = 0.5\).

1. Use implicit differentiation to find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\). For \(- \pi < x < \pi\) and \(- \pi < y < \pi\),
2. find the coordinates of the points where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).

6. (a) Given that \(y = 2 ^ { x }\), and using the result \(2 ^ { x } = \mathrm { e } ^ { x \ln 2 }\), or otherwise, show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 ^ { x } \ln 2\).
(b) Find the gradient of the curve with equation \(y = 2 ^ { \left( x ^ { 2 } \right) }\) at the point with coordinates \(( 2,16 )\).

7. The point \(A\) has position vector \(\mathbf { a } = 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k }\) and the point \(B\) has position vector \(\mathbf { b } = \mathbf { i } + \mathbf { j } - 4 \mathbf { k }\), relative to an origin \(O\).

1. Find the position vector of the point \(C\), with position vector \(\mathbf { c }\), given by $$\mathbf { c } = \mathbf { a } + \mathbf { b } .$$
2. Show that \(O A C B\) is a rectangle, and find its exact area. The diagonals of the rectangle, \(A B\) and \(O C\), meet at the point \(D\).
3. Write down the position vector of the point \(D\).
4. Find the size of the angle \(A D C\).

8. $$I = \int _ { 0 } ^ { 5 } \mathrm { e } ^ { \sqrt { } ( 3 x + 1 ) } \mathrm { d } x$$

1. Given that \(y = \mathrm { e } ^ { \sqrt { } ( 3 x + 1 ) }\), complete the table with the values of \(y\) corresponding to \(x = 2\), 3 and 4.

   \(x\)	0	1	2	3	4	5
   \(y\)	\(\mathrm { e } ^ { 1 }\)	\(\mathrm { e } ^ { 2 }\)				\(\mathrm { e } ^ { 4 }\)

2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the original integral \(I\), giving your answer to 4 significant figures.
3. Use the substitution \(t = \sqrt { } ( 3 x + 1 )\) to show that \(I\) may be expressed as \(\int _ { a } ^ { b } k t e ^ { t } \mathrm {~d} t\), giving the values of \(a , b\) and \(k\).
4. Use integration by parts to evaluate this integral, and hence find the value of \(I\) correct to 4 significant figures, showing all the steps in your working.