Edexcel F3 (Further Pure Mathematics 3) 2017 June

1. Solve the equation

$$18 \cosh x + 14 \sinh x = 11 + \mathrm { e } ^ { x }$$ Give your answers in the form \(\ln a\), where \(a\) is rational.

2. $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 3 & a
2 & 0 & 1
1 & - 2 & 1 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 4
3 & - 2 & 3
1 & 2 & b \end{array} \right)$$ where \(a\) and \(b\) are constants.

1. Write down \(\mathbf { A } ^ { \mathrm { T } }\) in terms of \(a\).
2. Calculate \(\mathbf { A B }\), giving your answer in terms of \(a\) and \(b\).
3. Hence show that $$( \mathbf { A B } ) ^ { \mathrm { T } } = \mathbf { B } ^ { \mathrm { T } } \mathbf { A } ^ { \mathrm { T } }$$

3. Given that $$y = x - \operatorname { artanh } \left( \frac { 2 x } { 1 + x ^ { 2 } } \right)$$

1. show that $$1 - \frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { k } { 1 - x ^ { 2 } }$$ where \(k\) is a constant to be found.
2. Hence, or otherwise, show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + x \left( 1 - \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = 0$$

4. $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 1 & 3
1 & 5 & 1
3 & 1 & 1 \end{array} \right)$$

1. Show that 6 is an eigenvalue of the matrix \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
2. Find a normalised eigenvector corresponding to the eigenvalue 6

5. $$I _ { n } = \int \operatorname { cosec } ^ { n } x \mathrm {~d} x , \quad 0 < x < \frac { \pi } { 2 } , \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 2\) $$I _ { n } = \frac { n - 2 } { n - 1 } I _ { n - 2 } - \frac { 1 } { n - 1 } \cot x \operatorname { cosec } ^ { n - 2 } x$$
2. Hence, or otherwise, find $$\int \operatorname { cosec } ^ { 4 } x \mathrm {~d} x$$ giving your answer in terms of \(\cot x\).

1. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)
   and the ellipse \(E\) has equation \(\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1\)
   where \(a > b > 0\)
   The line \(l\) is a tangent to hyperbola \(H\) at the point \(P ( a \sec \theta , b \tan \theta )\), where \(0 < \theta < \frac { \pi } { 2 }\)
   1. Using calculus, show that an equation for \(l\) is
   $$b x \sec \theta - a y \tan \theta = a b$$ Given that the point \(F\) is the focus of ellipse \(E\) for which \(x > 0\) and that the line \(l\) passes through \(F\),
2. show that \(l\) is parallel to the line \(y = x\)

1. (a) Find

$$\int \frac { 5 + x } { \sqrt { 4 - 3 x ^ { 2 } } } \mathrm {~d} x$$ (b) Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { 5 + x } { \sqrt { 4 - 3 x ^ { 2 } } } d x$$ giving your answer in the form \(p \pi \sqrt { 3 } + q\), where \(p\) and \(q\) are rational numbers to be found.

8. The curve \(C\) has parametric equations $$x = \theta - \sin \theta , \quad y = 1 - \cos \theta , \quad 0 \leqslant \theta \leqslant 2 \pi$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).

1. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { 2 \pi } ( 1 - \cos \theta ) ^ { \frac { 3 } { 2 } } \mathrm {~d} \theta$$
2. Hence find the exact value of \(S\).

9 With respect to a fixed origin \(O\), the points \(A ( - 1,5,1 ) , B ( 1,0,3 ) , C ( 2 , - 1,2 )\) and \(D ( 3,6 , - 1 )\) are the vertices of a tetrahedron.

1. Find the volume of the tetrahedron \(A B C D\). The plane \(\Pi\) contains the points \(A , B\) and \(C\).
2. Find a cartesian equation of \(\Pi\). The point \(T\) lies on the plane \(\Pi\). The line \(D T\) is perpendicular to \(\Pi\).
3. Find the exact coordinates of the point \(T\).