Edexcel C4 (Core Mathematics 4) 2008 June

1. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { e } ^ { 0.5 x ^ { 2 } }\). The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 2\).

1. Complete the table with the values of \(y\) corresponding to \(x = 0.8\) and \(x = 1.6\).

   \(x\)	0	0.4	0.8	1.2	1.6	2
   \(y\)	\(\mathrm { e } ^ { 0 }\)	\(\mathrm { e } ^ { 0.08 }\)		\(\mathrm { e } ^ { 0.72 }\)		\(\mathrm { e } ^ { 2 }\)

2. Use the trapezium rule with all the values in the table to find an approximate value for the area of \(R\), giving your answer to 4 significant figures.

2. (a) Use integration by parts to find \(\int x \mathrm { e } ^ { x } \mathrm {~d} x\).
(b) Hence find \(\int x ^ { 2 } \mathrm { e } ^ { x } \mathrm {~d} x\).

3. \begin{figure}[h] \end{figure} Figure 2 shows a right circular cylindrical metal rod which is expanding as it is heated. After \(t\) seconds the radius of the rod is \(x \mathrm {~cm}\) and the length of the rod is \(5 x \mathrm {~cm}\). The cross-sectional area of the rod is increasing at the constant rate of \(0.032 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\).

1. Find \(\frac { \mathrm { d } x } { \mathrm {~d} t }\) when the radius of the rod is 2 cm , giving your answer to 3 significant figures.
2. Find the rate of increase of the volume of the rod when \(x = 2\).
   \section*{LU}

4. A curve has equation \(3 x ^ { 2 } - y ^ { 2 } + x y = 4\). The points \(P\) and \(Q\) lie on the curve. The gradient of the tangent to the curve is \(\frac { 8 } { 3 }\) at \(P\) and at \(Q\).

1. Use implicit differentiation to show that \(y - 2 x = 0\) at \(P\) and at \(Q\).
2. Find the coordinates of \(P\) and \(Q\).

5. (a) Expand \(\frac { 1 } { \sqrt { } ( 4 - 3 x ) }\), where \(| x | < \frac { 4 } { 3 }\), in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\). Simplify each term.
(b) Hence, or otherwise, find the first 3 terms in the expansion of \(\frac { x + 8 } { \sqrt { } ( 4 - 3 x ) }\) as a series in ascending powers of \(x\).

6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$\begin{array} { l l } l _ { 1 } : & \mathbf { r } = ( - 9 \mathbf { i } + 10 \mathbf { k } ) + \lambda ( 2 \mathbf { i } + \mathbf { j } - \mathbf { k } )
l _ { 2 } : & \mathbf { r } = ( 3 \mathbf { i } + \mathbf { j } + 17 \mathbf { k } ) + \mu ( 3 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) \end{array}$$ 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 \(A\) has position vector \(5 \mathbf { i } + 7 \mathbf { j } + 3 \mathbf { k }\).
3. Show that \(A\) lies on \(l _ { 1 }\). The point \(B\) is the image of \(A\) after reflection in the line \(l _ { 2 }\).
4. Find the position vector of \(B\).

7. (a) Express \(\frac { 2 } { 4 - y ^ { 2 } }\) in partial fractions.
(b) Hence obtain the solution of $$2 \cot x \frac { \mathrm {~d} y } { \mathrm {~d} x } = \left( 4 - y ^ { 2 } \right)$$ for which \(y = 0\) at \(x = \frac { \pi } { 3 }\), giving your answer in the form \(\sec ^ { 2 } x = \mathrm { g } ( y )\).

8. \begin{figure}[h] \end{figure} Figure 3 shows the curve \(C\) with parametric equations $$x = 8 \cos t , \quad y = 4 \sin 2 t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 } .$$ The point \(P\) lies on \(C\) and has coordinates \(( 4,2 \sqrt { } 3 )\).

1. Find the value of \(t\) at the point \(P\). The line \(l\) is a normal to \(C\) at \(P\).
2. Show that an equation for \(l\) is \(y = - x \sqrt { 3 } + 6 \sqrt { 3 }\). The finite region \(R\) is enclosed by the curve \(C\), the \(x\)-axis and the line \(x = 4\), as shown shaded in Figure 3.
3. Show that the area of \(R\) is given by the integral \(\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } 64 \sin ^ { 2 } t \cos t \mathrm {~d} t\).
4. Use this integral to find the area of \(R\), giving your answer in the form \(a + b \sqrt { } 3\), where \(a\) and \(b\) are constants to be determined.