Edexcel F3 (Further Pure Mathematics 3) 2014 June

1. Given that \(y = \arctan \left( \frac { 2 x } { 3 } \right)\),
   1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in its simplest form.
   2. Use integration by parts to find
   $$\int \arctan \left( \frac { 2 x } { 3 } \right) \mathrm { d } x$$

2. The line with equation \(x = 9\) is a directrix of an ellipse with equation $$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 8 } = 1$$ where \(a\) is a positive constant. Find the two possible exact values of the constant \(a\).

3. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials,

1. prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x \equiv 1$$
2. find algebraically the exact solutions of the equation $$2 \sinh x + 7 \cosh x = 9$$ giving your answers as natural logarithms.

4. A non-singular matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { l l l } 3 & k & 0
k & 2 & 0
k & 0 & 1 \end{array} \right) \text {, where } k \text { is a constant. }$$

1. Find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\). The point \(A\) is mapped onto the point ( \(- 5,10,7\) ) by the transformation represented by the matrix $$\left( \begin{array} { l l l } 3 & 1 & 0
   1 & 2 & 0
   1 & 0 & 1 \end{array} \right)$$
2. Find the coordinates of the point \(A\).

1. Given that

$$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } \cos ^ { n } \theta \mathrm {~d} \theta , \quad n \geqslant 0$$

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

6. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$$ The line \(l\) is a tangent to \(H\) at the point \(P ( 4 \cosh \alpha , 2 \sinh \alpha )\), where \(\alpha\) is a constant, \(\alpha \neq 0\)

1. Using calculus, show that an equation for \(l\) is $$2 y \sinh \alpha - x \cosh \alpha + 4 = 0$$ The line \(l\) cuts the \(y\)-axis at the point \(A\).
2. Find the coordinates of \(A\) in terms of \(\alpha\). The point \(B\) has coordinates ( \(0,10 \sinh \alpha\) ) and the point \(S\) is the focus of \(H\) for which \(x > 0\)
3. Show that the line segment \(A S\) is perpendicular to the line segment \(B S\).

7. The curve \(C\) has parametric equations $$x = 3 t ^ { 2 } , \quad y = 12 t , \quad 0 \leqslant t \leqslant 4$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

1. Show that the area of the surface generated is $$\pi ( a \sqrt { 5 } + b )$$ where \(a\) and \(b\) are constants to be found.
2. Show that the length of the curve \(C\) is given by $$k \int _ { 0 } ^ { 4 } \sqrt { \left( t ^ { 2 } + 4 \right) } \mathrm { d } t$$ where \(k\) is a constant to be found.
3. Use the substitution \(t = 2 \sinh \theta\) to show that the exact value of the length of the curve \(C\) is $$24 \sqrt { 5 } + 12 \ln ( 2 + \sqrt { 5 } )$$

8. The line \(l\) has equation $$\mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) , \text { where } \lambda \text { is a scalar parameter, }$$ and the plane \(\Pi\) has equation $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 19$$

1. Find the coordinates of the point of intersection of \(l\) and \(\Pi\). The perpendicular to \(\Pi\) from the point \(A ( 2,1 , - 2 )\) meets \(\Pi\) at the point \(B\).
2. Verify that the coordinates of \(B\) are \(( 4,3 , - 6 )\). The point \(A ( 2,1 , - 2 )\) is reflected in the plane \(\Pi\) to give the image point \(A ^ { \prime }\).
3. Find the coordinates of the point \(A ^ { \prime }\).
4. Find an equation for the line obtained by reflecting the line \(l\) in the plane \(\Pi\), giving your answer in the form $$\mathbf { r } \times \mathbf { a } = \mathbf { b } ,$$ where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.