Edexcel FP3 (Further Pure Mathematics 3) 2013 June

1. A hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 25 } = 1 , \quad \text { where } a \text { is a positive constant. }$$ The foci of \(H\) are at the points with coordinates \(( 13,0 )\) and \(( - 13,0 )\).
Find

1. the value of the constant \(a\),
2. the equations of the directrices of \(H\).

2. (a) Find $$\int \frac { 1 } { \sqrt { } \left( 4 x ^ { 2 } + 9 \right) } d x$$ (b) Use your answer to part (a) to find the exact value of $$\int _ { - 3 } ^ { 3 } \frac { 1 } { \sqrt { \left( 4 x ^ { 2 } + 9 \right) } } d x$$ giving your answer in the form \(k \ln ( a + b \sqrt { } 5 )\), where \(a\) and \(b\) are integers and \(k\) is a constant.

3. The curve with parametric equations $$x = \cosh 2 \theta , \quad y = 4 \sinh \theta , \quad 0 \leqslant \theta \leqslant 1$$ is rotated through \(2 \pi\) radians about the \(x\)-axis.
Show that the area of the surface generated is \(\lambda \left( \cosh ^ { 3 } \alpha - 1 \right)\), where \(\alpha = 1\) and \(\lambda\) is a constant to be found.

4. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation $$y = 40 \operatorname { arcosh } x - 9 x , \quad x \geqslant 1$$ Use calculus to find the exact coordinates of the turning point of the curve, giving your answer in the form \(\left( \frac { p } { q } , r \ln 3 + s \right)\), where \(p , q , r\) and \(s\) are integers.

1. The matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } 1 & 1 & a
2 & b & c
- 1 & 0 & 1 \end{array} \right) , \text { where } a , b \text { and } c \text { are constants. }$$

1. Given that \(\mathbf { j } + \mathbf { k }\) and \(\mathbf { i } - \mathbf { k }\) are two of the eigenvectors of \(\mathbf { M }\), find
   1. the values of \(a , b\) and \(c\),
   2. the eigenvalues which correspond to the two given eigenvectors.
2. The matrix \(\mathbf { P }\) is given by $$\mathbf { P } = \left( \begin{array} { r r r } 1 & 1 & 0
   2 & 1 & d
   - 1 & 0 & 1 \end{array} \right) \text {, where } d \text { is constant, } d \neq - 1$$ Find
   1. the determinant of \(\mathbf { P }\) in terms of \(d\),
   2. the matrix \(\mathbf { P } ^ { - 1 }\) in terms of \(d\).

1. Given that

$$I _ { n } = \int _ { 0 } ^ { 4 } x ^ { n } \sqrt { } \left( 16 - x ^ { 2 } \right) \mathrm { d } x , \quad n \geqslant 0$$

1. prove that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = 16 ( n - 1 ) I _ { n - 2 }$$
2. Hence, showing each step of your working, find the exact value of \(I _ { 5 }\)

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , \quad a > b > 0$$ The line \(l\) is a normal to \(E\) at a point \(P ( a \cos \theta , b \sin \theta ) , \quad 0 < \theta < \frac { \pi } { 2 }\)

1. Using calculus, show that an equation for \(l\) is $$a x \sin \theta - b y \cos \theta = \left( a ^ { 2 } - b ^ { 2 } \right) \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at \(A\) and the \(y\)-axis at \(B\).
2. Show that the area of the triangle \(O A B\), where \(O\) is the origin, may be written as \(k \sin 2 \theta\), giving the value of the constant \(k\) in terms of \(a\) and \(b\).
3. Find, in terms of \(a\) and \(b\), the exact coordinates of the point \(P\), for which the area of the triangle \(O A B\) is a maximum.

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

$$\mathbf { r } . ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$

1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) , \text { where } \lambda \text { and } \mu \text { are scalar parameters. }$$
2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer to the nearest degree.
3. Find an equation of the line of intersection of the two planes in the form \(\mathbf { r } \times \mathbf { a } = \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors.