AQA FP2 (Further Pure Mathematics 2) 2009 January

1

1. Use the definitions \(\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right)\) and \(\cosh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } + \mathrm { e } ^ { - \theta } \right)\) to show that $$1 + 2 \sinh ^ { 2 } \theta = \cosh 2 \theta$$
2. Solve the equation $$3 \cosh 2 \theta = 2 \sinh \theta + 11$$ giving each of your answers in the form \(\ln p\).

2

1. Indicate on an Argand diagram the region for which \(| z - 4 \mathrm { i } | \leqslant 2\).
2. The complex number \(z\) satisfies \(| z - 4 \mathrm { i } | \leqslant 2\). Find the range of possible values of \(\arg z\).

3

1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { 4 } r ^ { 2 } ( r + 1 ) ^ { 2 }\), show that $$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = r ^ { 3 }$$
2. Use the method of differences to show that $$\sum _ { r = n } ^ { 2 n } r ^ { 3 } = \frac { 3 } { 4 } n ^ { 2 } ( n + 1 ) ( 5 n + 1 )$$

4 It is given that \(\alpha , \beta\) and \(\gamma\) satisfy the equations $$\begin{aligned} & \alpha + \beta + \gamma = 1
& \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 5
& \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 23 \end{aligned}$$

1. Show that \(\alpha \beta + \beta \gamma + \gamma \alpha = 3\).
2. Use the identity $$( \alpha + \beta + \gamma ) \left( \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } - \alpha \beta - \beta \gamma - \gamma \alpha \right) = \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } - 3 \alpha \beta \gamma$$ to find the value of \(\alpha \beta \gamma\).
3. Write down a cubic equation, with integer coefficients, whose roots are \(\alpha , \beta\) and \(\gamma\).
4. Explain why this cubic equation has two non-real roots.
5. Given that \(\alpha\) is real, find the values of \(\alpha , \beta\) and \(\gamma\).

5

1. Given that \(u = \cosh ^ { 2 } x\), show that \(\frac { \mathrm { d } u } { \mathrm {~d} x } = \sinh 2 x\).
2. Hence show that $$\int _ { 0 } ^ { 1 } \frac { \sinh 2 x } { 1 + \cosh ^ { 4 } x } \mathrm {~d} x = \tan ^ { - 1 } \left( \cosh ^ { 2 } 1 \right) - \frac { \pi } { 4 }$$

6 Prove by induction that $$\frac { 2 \times 1 } { 2 \times 3 } + \frac { 2 ^ { 2 } \times 2 } { 3 \times 4 } + \frac { 2 ^ { 3 } \times 3 } { 4 \times 5 } + \ldots + \frac { 2 ^ { n } \times n } { ( n + 1 ) ( n + 2 ) } = \frac { 2 ^ { n + 1 } } { n + 2 } - 1$$ for all integers \(n \geqslant 1\).

7

1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \cosh ^ { - 1 } \frac { 1 } { x } \right) = \frac { - 1 } { x \sqrt { 1 - x ^ { 2 } } }$$ (3 marks)
2. A curve has equation $$y = \sqrt { 1 - x ^ { 2 } } - \cosh ^ { - 1 } \frac { 1 } { x } \quad ( 0 < x < 1 )$$ Show that:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \sqrt { 1 - x ^ { 2 } } } { x }\);
      (4 marks)
   2. the length of the arc of the curve from the point where \(x = \frac { 1 } { 4 }\) to the point where $$x = \frac { 3 } { 4 } \text { is } \ln 3 .$$ (5 marks)

8

1. Show that $$\left( z ^ { 4 } - \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z ^ { 4 } - \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 8 } - 2 z ^ { 4 } \cos \theta + 1$$ (2 marks)
2. Hence solve the equation $$z ^ { 8 } - z ^ { 4 } + 1 = 0$$ giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \phi }\), where \(- \pi < \phi \leqslant \pi\).
3. Indicate the roots on an Argand diagram.