Edexcel FP3 (Further Pure Mathematics 3) 2015 June

1. Solve the equation

$$2 \cosh ^ { 2 } x - 3 \sinh x = 1$$ giving your answers in terms of natural logarithms.

2. A curve has equation $$y = \cosh x , \quad 1 \leqslant x \leqslant \ln 5$$ Find the exact length of this curve. Give your answer in terms of e .

3. $$\mathbf { A } = \left( \begin{array} { l l l } 2 & 1 & 0
1 & 2 & 1
0 & 1 & 2 \end{array} \right)$$

1. Find the eigenvalues of \(\mathbf { A }\).
2. Find a normalised eigenvector for each of the eigenvalues of \(\mathbf { A }\).
3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).

1. The curve \(C\) has equation

$$y = \frac { 1 } { \sqrt { x ^ { 2 } + 2 x - 3 } } , \quad x > 1$$

1. Find \(\int y \mathrm {~d} x\) The region \(R\) is bounded by the curve \(C\), the \(x\)-axis and the lines with equations \(x = 2\) and \(x = 3\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find the volume of the solid generated. Give your answer in the form \(p \pi \ln q\), where \(p\) and \(q\) are rational numbers to be found.

5. The points \(A , B\) and \(C\) have position vectors \(\left( \begin{array} { l } 1
3
2 \end{array} \right) , \left( \begin{array} { r } - 1
0
1 \end{array} \right)\) and \(\left( \begin{array} { l } 2
1
0 \end{array} \right)\) respectively.

1. Find a vector equation of the straight line \(A B\).
2. Find a cartesian form of the equation of the straight line \(A B\). The plane \(\Pi\) contains the points \(A , B\) and \(C\).
3. Find a vector equation of \(\Pi\) in the form r.n \(= p\).
4. Find the perpendicular distance from the origin to \(\Pi\).

1. The hyperbola \(H\) is given by the equation \(x ^ { 2 } - y ^ { 2 } = 1\)
   1. Write down the equations of the two asymptotes of \(H\).
   2. Show that an equation of the tangent to \(H\) at the point \(P ( \cosh t , \sinh t )\) is
   $$y \sinh t = x \cosh t - 1$$ The tangent at \(P\) meets the asymptotes of \(H\) at the points \(Q\) and \(R\).
2. Show that \(P\) is the midpoint of \(Q R\).
3. Show that the area of the triangle \(O Q R\), where \(O\) is the origin, is independent of \(t\).

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

1. Prove that for \(n \geqslant 2\) $$I _ { n } = \frac { 1 } { n } \left( - \sin ^ { n - 1 } x \cos x + ( n - 1 ) I _ { n - 2 } \right)$$ Given that \(n\) is an odd number, \(n \geqslant 3\)
2. show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { n } x \mathrm {~d} x = \frac { ( n - 1 ) ( n - 3 ) \ldots 6.4 .2 } { n ( n - 2 ) ( n - 4 ) \ldots 7.5 .3 }$$
3. Hence find \(\int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { 5 } x \cos ^ { 2 } x d x\)

1. The ellipse \(E\) has equation \(x ^ { 2 } + 4 y ^ { 2 } = 4\)
   1. (i) Find the coordinates of the foci, \(F _ { 1 }\) and \(F _ { 2 }\), of \(E\).
      (ii) Write down the equations of the directrices of \(E\).
   2. Given that the point \(P\) lies on the ellipse, show that
   $$\left| P F _ { 1 } \right| + \left| P F _ { 2 } \right| = 4$$ A chord of an ellipse is a line segment joining two points on the ellipse.
   The set of midpoints of the parallel chords of \(E\) with gradient \(m\), where \(m\) is a constant, lie on a straight line \(l\).
2. Find an equation of \(l\).