AQA FP2 (Further Pure Mathematics 2) 2008 January

1

1. Express \(4 + 4 \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
2. Solve the equation $$z ^ { 5 } = 4 + 4 i$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).

2

1. Show that $$( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 } = 24 r ^ { 2 } + 2$$
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$

3 A circle \(C\) and a half-line \(L\) have equations $$| z - 2 \sqrt { 3 } - \mathrm { i } | = 4$$ and $$\arg ( z + i ) = \frac { \pi } { 6 }$$ respectively.

1. Show that:
   1. the circle \(C\) passes through the point where \(z = - \mathrm { i }\);
   2. the half-line \(L\) passes through the centre of \(C\).
2. On one Argand diagram, sketch \(C\) and \(L\).
3. Shade on your sketch the set of points satisfying both $$| z - 2 \sqrt { 3 } - \mathrm { i } | \leqslant 4$$ and $$0 \leqslant \arg ( z + i ) \leqslant \frac { \pi } { 6 }$$

4 The cubic equation $$z ^ { 3 } + \mathrm { i } z ^ { 2 } + 3 z - ( 1 + \mathrm { i } ) = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of:
   1. \(\alpha + \beta + \gamma\);
   2. \(\alpha \beta + \beta \gamma + \gamma \alpha\);
   3. \(\alpha \beta \gamma\).
2. Find the value of:
   1. \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\);
   2. \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\);
   3. \(\alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }\).
3. Hence write down a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).

5 Prove by induction that for all integers \(n \geqslant 1\) $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + 1 \right) ( r ! ) = n ( n + 1 ) !$$

6

1. 1. By applying De Moivre's theorem to \(( \cos \theta + \mathrm { i } \sin \theta ) ^ { 3 }\), show that $$\cos 3 \theta = \cos ^ { 3 } \theta - 3 \cos \theta \sin ^ { 2 } \theta$$
   2. Find a similar expression for \(\sin 3 \theta\).
   3. Deduce that $$\tan 3 \theta = \frac { \tan ^ { 3 } \theta - 3 \tan \theta } { 3 \tan ^ { 2 } \theta - 1 }$$
2. 1. Hence show that \(\tan \frac { \pi } { 12 }\) is a root of the cubic equation $$x ^ { 3 } - 3 x ^ { 2 } - 3 x + 1 = 0$$
   2. Find two other values of \(\theta\), where \(0 < \theta < \pi\), for which \(\tan \theta\) is a root of this cubic equation.
3. Hence show that $$\tan \frac { \pi } { 12 } + \tan \frac { 5 \pi } { 12 } = 4$$

7

1. Given that \(y = \ln \tanh \frac { x } { 2 }\), where \(x > 0\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \operatorname { cosech } x$$
2. A curve has equation \(y = \ln \tanh \frac { x } { 2 }\), where \(x > 0\). The length of the arc of the curve between the points where \(x = 1\) and \(x = 2\) is denoted by \(s\).
   1. Show that $$s = \int _ { 1 } ^ { 2 } \operatorname { coth } x \mathrm {~d} x$$
   2. Hence show that \(s = \ln ( 2 \cosh 1 )\).