Edexcel FP3 (Further Pure Mathematics 3) 2009 June

1. Solve the equation

$$7 \operatorname { sech } x - \tanh x = 5$$ Give your answers in the form \(\ln a\) where \(a\) is a rational number.

2. \begin{figure}[h] \end{figure} The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to a fixed origin \(O\), as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } .$$ Calculate

1. \(\mathbf { b } \times \mathbf { c }\),
2. a.(b \(\times \mathbf { c ) }\),
3. the area of triangle \(O B C\),
4. the volume of the tetrahedron \(O A B C\).

3. $$\mathbf { M } = \left( \begin{array} { r r r } 6 & 1 & - 1
0 & 7 & 0
3 & - 1 & 2 \end{array} \right)$$

1. Show that 7 is an eigenvalue of the matrix \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
2. Find an eigenvector corresponding to the eigenvalue 7.

1. Given that \(y = \operatorname { arsinh } ( \sqrt { } x ) , x > 0\),
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer as a simplified fraction.
   2. Hence, or otherwise, find
   $$\int _ { \frac { 1 } { 4 } } ^ { 4 } \frac { 1 } { \sqrt { [ x ( x + 1 ) ] } } \mathrm { d } x$$ giving your answer in the form \(\ln \left( \frac { a + b \sqrt { } 5 } { 2 } \right)\), where \(a\) and \(b\) are integers.

5. $$I _ { n } = \int _ { 0 } ^ { 5 } \frac { x ^ { n } } { \sqrt { } \left( 25 - x ^ { 2 } \right) } d x , \quad n \geqslant 0$$

1. Find an expression for \(\int \frac { x } { \sqrt { } \left( 25 - x ^ { 2 } \right) } \mathrm { d } x , \quad 0 \leqslant x \leqslant 5\).
2. Using your answer to part (a), or otherwise, show that $$I _ { n } = \frac { 25 ( n - 1 ) } { n } I _ { n - 2 } \quad n \geqslant 2$$
3. Find \(I _ { 4 }\) in the form \(k \pi\), where \(k\) is a fraction.

1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\), where \(a\) and \(b\) are constants.

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.

1. Given that \(L\) and \(H\) meet, show that the \(x\)-coordinates of the points of intersection are the roots of the equation $$\left( a ^ { 2 } m ^ { 2 } - b ^ { 2 } \right) x ^ { 2 } + 2 a ^ { 2 } m c x + a ^ { 2 } \left( c ^ { 2 } + b ^ { 2 } \right) = 0$$ Hence, given that \(L\) is a tangent to \(H\),
2. show that \(a ^ { 2 } m ^ { 2 } = b ^ { 2 } + c ^ { 2 }\). The hyperbola \(H ^ { \prime }\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 16 } = 1\).
3. Find the equations of the tangents to \(H ^ { \prime }\) which pass through the point \(( 1,4 )\).

7. The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\mathbf { r } = \left( \begin{array} { r } 1
- 1
2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
3
4 \end{array} \right) \text { and } \quad \mathbf { r } = \left( \begin{array} { r } \alpha
- 4
0 \end{array} \right) + \mu \left( \begin{array} { l } 0
3
2 \end{array} \right) .$$ If the lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect, find

1. the value of \(\alpha\),
2. an equation for the plane containing the lines \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in the form \(a x + b y + c z + d = 0\), where \(a , b , c\) and \(d\) are constants. For other values of \(\alpha\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect and are skew lines.
   Given that \(\alpha = 2\),
3. find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).

1. A curve, which is part of an ellipse, has parametric equations

$$x = 3 \cos \theta , \quad y = 5 \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 } .$$ The curve is rotated through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the area of the surface generated is given by the integral $$k \pi \int _ { 0 } ^ { \alpha } \sqrt { } \left( 16 c ^ { 2 } + 9 \right) \mathrm { d } c , \quad \text { where } c = \cos \theta$$ and where \(k\) and \(\alpha\) are constants to be found.
2. Using the substitution \(c = \frac { 3 } { 4 } \sinh u\), or otherwise, evaluate the integral, showing all of your working and giving the final answer to 3 significant figures.