AQA FP2 (Further Pure Mathematics 2) 2010 January

1

1. Use the definitions \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) and \(\sinh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } - \mathrm { e } ^ { - x } \right)\) to show that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1$$
2. 1. Express $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x$$ in terms of \(\cosh x\).
   2. Sketch the curve \(y = \cosh x\).
   3. Hence solve the equation $$5 \cosh ^ { 2 } x + 3 \sinh ^ { 2 } x = 9.5$$ giving your answers in logarithmic form.

2

1. On the same Argand diagram, draw:
   1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 4\);
   2. the locus of points satisfying \(| z | = | z - 2 \mathrm { i } |\).
2. Indicate on your sketch the set of points satisfying both $$| z - 4 + 2 i | \leqslant 4$$ and $$| z | \geqslant | z - 2 \mathrm { i } |$$

3 The cubic equation $$2 z ^ { 3 } + p z ^ { 2 } + q z + 16 = 0$$ where \(p\) and \(q\) are real, has roots \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha = 2 + 2 \sqrt { 3 } \mathrm { i }\).

1. 1. Write down another root, \(\beta\), of the equation.
   2. Find the third root, \(\gamma\).
   3. Find the values of \(p\) and \(q\).
2. 1. Express \(\alpha\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. Show that $$( 2 + 2 \sqrt { 3 } \mathrm { i } ) ^ { n } = 4 ^ { n } \left( \cos \frac { n \pi } { 3 } + \mathrm { i } \sin \frac { n \pi } { 3 } \right)$$
   3. Show that $$\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } = 2 ^ { 2 n + 1 } \cos \frac { n \pi } { 3 } + \left( - \frac { 1 } { 2 } \right) ^ { n }$$ where \(n\) is an integer.

4 A curve \(C\) is given parametrically by the equations $$x = \frac { 1 } { 2 } \cosh 2 t , \quad y = 2 \sinh t$$

1. Express $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 }$$ in terms of \(\cosh t\).
2. The arc of \(C\) from \(t = 0\) to \(t = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
   1. Show that \(S\), the area of the curved surface generated, is given by $$S = 8 \pi \int _ { 0 } ^ { 1 } \sinh t \cosh ^ { 2 } t \mathrm {~d} t$$
   2. Find the exact value of \(S\).

5 The sum to \(r\) terms, \(S _ { r }\), of a series is given by $$S _ { r } = r ^ { 2 } ( r + 1 ) ( r + 2 )$$ Given that \(u _ { r }\) is the \(r\) th term of the series whose sum is \(S _ { r }\), show that:

1. 1. \(u _ { 1 } = 6\);
   2. \(u _ { 2 } = 42\);
   3. \(\quad u _ { n } = n ( n + 1 ) ( 4 n - 1 )\).
2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } = 3 n ^ { 2 } ( n + 1 ) ( 5 n + 2 )$$

6

1. Show that the substitution \(t = \tan \theta\) transforms the integral $$\int \frac { \mathrm { d } \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta }$$ into $$\int \frac { \mathrm { d } t } { 9 + t ^ { 2 } }$$
2. Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { d \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta } = \frac { \pi } { 18 }$$

7 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 , \quad u _ { k + 1 } = 2 u _ { k } + 1$$

1. Prove by induction that, for all \(n \geqslant 1\), $$u _ { n } = 3 \times 2 ^ { n - 1 } - 1$$
2. Show that $$\sum _ { r = 1 } ^ { n } u _ { r } = u _ { n + 1 } - ( n + 2 )$$

8

1. 1. Show that \(\omega = \mathrm { e } ^ { \frac { 2 \pi \mathrm { i } } { 7 } }\) is a root of the equation \(z ^ { 7 } = 1\).
   2. Write down the five other non-real roots in terms of \(\omega\).
2. Show that $$1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } + \omega ^ { 5 } + \omega ^ { 6 } = 0$$
3. Show that:
   1. \(\quad \omega ^ { 2 } + \omega ^ { 5 } = 2 \cos \frac { 4 \pi } { 7 }\);
   2. \(\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }\).