Edexcel FP3 (Further Pure Mathematics 3) 2013 June

1. The hyperbola \(H\) has foci at \(( 5,0 )\) and \(( - 5,0 )\) and directrices with equations \(x = \frac { 9 } { 5 }\) and \(x = - \frac { 9 } { 5 }\).

Find a cartesian equation for \(H\).

2. Two skew lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } )
& l _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k } ) + \mu ( - 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) \end{aligned}$$ respectively, where \(\lambda\) and \(\mu\) are real parameters.

1. Find a vector in the direction of the common perpendicular to \(l _ { 1 }\) and \(l _ { 2 }\)
2. Find the shortest distance between these two lines.

1. The point \(P\) lies on the ellipse \(E\) with equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 9 } = 1$$ \(N\) is the foot of the perpendicular from point \(P\) to the line \(x = 8\)
\(M\) is the midpoint of \(P N\).

1. Sketch the graph of the ellipse \(E\), showing also the line \(x = 8\) and a possible position for the line \(P N\).
2. Find an equation of the locus of \(M\) as \(P\) moves around the ellipse.
3. Show that this locus is a circle and state its centre and radius.

1. The plane \(\Pi _ { 1 }\) has vector equation

$$\mathbf { r } = \left( \begin{array} { r } 1
- 1
2 \end{array} \right) + s \left( \begin{array} { l } 1
1
0 \end{array} \right) + t \left( \begin{array} { r } 1
2
- 2 \end{array} \right) ,$$ where \(s\) and \(t\) are real parameters. The plane \(\Pi _ { 1 }\) is transformed to the plane \(\Pi _ { 2 }\) by the transformation represented by the matrix \(\mathbf { T }\), where $$\mathbf { T } = \left( \begin{array} { r r r } 2 & 0 & 3
0 & 2 & - 1
0 & 1 & 2 \end{array} \right)$$ Find an equation of the plane \(\Pi _ { 2 }\) in the form r.n=p

5. $$I _ { n } = \int _ { 1 } ^ { 5 } x ^ { n } ( 2 x - 1 ) ^ { - \frac { 1 } { 2 } } \mathrm {~d} x , \quad n \geqslant 0$$

1. Prove that, for \(n \geqslant 1\), $$( 2 n + 1 ) I _ { n } = n I _ { n - 1 } + 3 \times 5 ^ { n } - 1$$
2. Using the reduction formula given in part (a), find the exact value of \(I _ { 2 }\)

6. It is given that \(\left( \begin{array} { l } 1
2
0 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { l l l } 4 & 2 & 3
2 & b & 0
a & 1 & 8 \end{array} \right)$$ and \(a\) and \(b\) are constants.

1. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 1
   2
   0 \end{array} \right)\).
2. Find the values of \(a\) and \(b\).
3. Find the other eigenvalues of \(\mathbf { A }\).

7. \begin{figure}[h] \end{figure} The curves shown in Figure 1 have equations $$y = 6 \cosh x \text { and } y = 9 - 2 \sinh x$$

1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), find exact values for the \(x\)-coordinates of the two points where the curves intersect. The finite region between the two curves is shown shaded in Figure 1.
2. Using calculus, find the area of the shaded region, giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are integers.

8. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 2, has equation $$y = 2 x ^ { \frac { 1 } { 2 } } , \quad 1 \leqslant x \leqslant 8$$

1. Show that the length \(s\) of curve \(C\) is given by the equation $$s = \int _ { 1 } ^ { 8 } \sqrt { } \left( 1 + \frac { 1 } { x } \right) \mathrm { d } x$$
2. Using the substitution \(x = \sinh ^ { 2 } u\), or otherwise, find an exact value for \(s\). Give your answer in the form \(a \sqrt { } 2 + \ln ( b + c \sqrt { } 2 )\) where \(a , b\) and \(c\) are integers.