Edexcel F3 (Further Pure Mathematics 3) 2015 June

1. Find the exact values of \(x\) for which

$$\cosh 2 x - 7 \sinh x = 5$$ giving your answers as natural logarithms.

2. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive constants.
The hyperbola \(H\) has eccentricity \(\frac { \sqrt { 21 } } { 4 }\) and passes through the point (12, 5).
Find

1. the value of \(a\) and the value of \(b\),
2. the coordinates of the foci of \(H\).

1. \(\mathbf { M } = \left( \begin{array} { r r r } 0 & 1 & 9
   1 & 4 & k
   1 & 0 & - 3 \end{array} \right)\), where \(k\) is a constant.

Given that \(\left( \begin{array} { r } 7
19
1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),

1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { r } 7
   19
   1 \end{array} \right)\),
2. show that \(k = - 7\)
3. find the other two eigenvalues of the matrix \(\mathbf { M }\). The image of the vector \(\left( \begin{array} { c } p
   q
   r \end{array} \right)\) under the transformation represented by \(\mathbf { M }\) is \(\left( \begin{array} { r } - 6
   21
   5 \end{array} \right)\).
4. Find the values of the constants \(p , q\) and \(r\).

4. $$I _ { n } = \int \cosh ^ { n } x \mathrm {~d} x , \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 2\) $$n I _ { n } = \sinh x \cosh ^ { n - 1 } x + ( n - 1 ) I _ { n - 2 }$$
2. Hence find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 5 } x \mathrm {~d} x$$

1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1\)

The line \(L\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants.
Given that \(L\) is a tangent to \(E\),

1. show that $$c ^ { 2 } - 25 m ^ { 2 } = 9$$
2. find the equations of the tangents to \(E\) which pass through the point \(( 3,4 )\).

6. \begin{figure}[h] \end{figure} Figure 1 shows the curve \(C\) with parametric equations $$x = 2 \cos \theta - \cos 2 \theta , y = 2 \sin \theta - \sin 2 \theta , \quad 0 \leqslant \theta \leqslant \pi$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} \theta } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} \theta } \right) ^ { 2 } = 8 ( 1 - \cos \theta )$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find the area of the surface generated, giving your answer in the form \(k \pi\), where \(k\) is a rational number.

1. The plane \(\Pi _ { 1 }\) contains the point \(( 3,3 , - 2 )\) and the line \(\frac { x - 1 } { 2 } = \frac { y - 2 } { - 1 } = \frac { z + 1 } { 4 }\)
   1. Show that a cartesian equation of the plane \(\Pi _ { 1 }\) is
   $$3 x - 10 y - 4 z = - 13$$ The plane \(\Pi _ { 2 }\) is parallel to the plane \(\Pi _ { 1 }\)
   The point ( \(\alpha , 1,1\) ), where \(\alpha\) is a constant, lies in \(\Pi _ { 2 }\)
   Given that the shortest distance between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) is \(\frac { 1 } { \sqrt { 5 } }\)
2. find the possible values of \(\alpha\).

1. (a) Show that, under the substitution \(x = \frac { 3 } { 4 } \sinh u\),

$$\int \frac { x ^ { 2 } } { \sqrt { 16 x ^ { 2 } + 9 } } \mathrm {~d} x = k \int ( \cosh 2 u - 1 ) \mathrm { d } u$$ where \(k\) is a constant to be determined.
(b) Hence show that $$\int _ { 0 } ^ { 1 } \frac { 64 x ^ { 2 } } { \sqrt { 16 x ^ { 2 } + 9 } } \mathrm {~d} x = p + q \ln 3$$ where \(p\) and \(q\) are rational numbers to be found.