Edexcel FP3 (Further Pure Mathematics 3) 2012 June

1. The hyperbola \(H\) has equation

$$\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$$ Find

1. the coordinates of the foci of \(H\),
2. the equations of the directrices of \(H\).

2. \begin{figure}[h] \end{figure} The curve \(C\), shown in Figure 1, has equation $$y = \frac { 1 } { 3 } \cosh 3 x , \quad 0 \leqslant x \leqslant \ln a$$ where \(a\) is a constant and \(a > 1\) Using calculus, show that the length of curve \(C\) is $$k \left( a ^ { 3 } - \frac { 1 } { a ^ { 3 } } \right)$$ and state the value of the constant \(k\).

3. The position vectors of the points \(A , B\) and \(C\) relative to an origin \(O\) are \(\mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , 7 \mathbf { i } - 3 \mathbf { k }\) and \(4 \mathbf { i } + 4 \mathbf { j }\) respectively. Find

1. \(\overrightarrow { A C } \times \overrightarrow { B C }\),
2. the area of triangle \(A B C\),
3. an equation of the plane \(A B C\) in the form \(\mathbf { r } . \mathbf { n } = p\)

4. $$I _ { n } = \int _ { 0 } ^ { \frac { \pi } { 4 } } x ^ { n } \sin 2 x \mathrm {~d} x , \quad n \geqslant 0$$

1. Prove that, for \(n \geqslant 2\), $$I _ { n } = \frac { 1 } { 4 } n \left( \frac { \pi } { 4 } \right) ^ { n - 1 } - \frac { 1 } { 4 } n ( n - 1 ) I _ { n - 2 }$$
2. Find the exact value of \(I _ { 2 }\)
3. Show that \(I _ { 4 } = \frac { 1 } { 64 } \left( \pi ^ { 3 } - 24 \pi + 48 \right)\)

1. (a) Differentiate \(x \operatorname { arsinh } 2 x\) with respect to \(x\).
   (b) Hence, or otherwise, find the exact value of

$$\int _ { 0 } ^ { \sqrt { 2 } } \operatorname { arsinh } 2 x \mathrm {~d} x$$ giving your answer in the form \(A \ln B + C\), where \(A , B\) and \(C\) are real.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ The line \(l _ { 1 }\) is a tangent to \(E\) at the point \(P ( a \cos \theta , b \sin \theta )\).

1. Using calculus, show that an equation for \(l _ { 1 }\) is $$\frac { x \cos \theta } { a } + \frac { y \sin \theta } { b } = 1$$ The circle \(C\) has equation $$x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$$ The line \(l _ { 2 }\) is a tangent to \(C\) at the point \(Q ( a \cos \theta , a \sin \theta )\).
2. Find an equation for the line \(l _ { 2 }\). Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(R\),
3. find, in terms of \(a , b\) and \(\theta\), the coordinates of \(R\).
4. Find the locus of \(R\), as \(\theta\) varies.

7. $$\mathrm { f } ( x ) = 5 \cosh x - 4 \sinh x , \quad x \in \mathbb { R }$$

1. Show that \(\mathrm { f } ( x ) = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + 9 \mathrm { e } ^ { - x } \right)\) Hence
2. solve \(\mathrm { f } ( x ) = 5\)
3. show that \(\int _ { \frac { 1 } { 2 } \ln 3 } ^ { \ln 3 } \frac { 1 } { 5 \cosh x - 4 \sinh x } \mathrm {~d} x = \frac { \pi } { 18 }\)

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

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0
1 & 2 & 0
- 1 & 0 & 4 \end{array} \right)$$

1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
2. For the eigenvalue 4, find a corresponding eigenvector. The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { M }\). The equation of \(l _ { 1 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\).
3. Find a vector equation for the line \(l _ { 2 }\).