AQA FP2 (Further Pure Mathematics 2) 2007 January

1

1. Given that $$4 \cosh ^ { 2 } x = 7 \sinh x + 1$$ find the two possible values of \(\sinh x\).
2. Hence obtain the two possible values of \(x\), giving your answers in the form \(\ln p\).

2

1. Sketch on one diagram:
   1. the locus of points satisfying \(| z - 4 + 2 \mathrm { i } | = 2\);
   2. the locus of points satisfying \(| z | = | z - 3 - 2 \mathrm { i } |\).
2. Shade on your sketch the region in which
   both $$| z - 4 + 2 i | \leqslant 2$$ and $$| z | \leqslant | z - 3 - 2 \mathrm { i } |$$

3 The cubic equation $$z ^ { 3 } + 2 ( 1 - \mathrm { i } ) z ^ { 2 } + 32 ( 1 + \mathrm { i } ) = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. It is given that \(\alpha\) is of the form \(k \mathrm { i }\), where \(k\) is real. By substituting \(z = k \mathrm { i }\) into the equation, show that \(k = 4\).
2. Given that \(\beta = - 4\), find the value of \(\gamma\).

4

1. Given that \(y = \operatorname { sech } t\), show that:
   1. \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - \operatorname { sech } t \tanh t\);
   2. \(\left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \operatorname { sech } ^ { 2 } t - \operatorname { sech } ^ { 4 } t\).
2. The diagram shows a sketch of part of the curve given parametrically by $$x = t - \tanh t \quad y = \operatorname { sech } t$$
   \includegraphics[max width=\textwidth, alt={}]{1891766e-7744-49ac-82b6-7e51cb63b381-3_424_625_863_703}
   The curve meets the \(y\)-axis at the point \(K\), and \(P ( x , y )\) is a general point on the curve. The arc length \(K P\) is denoted by \(s\). Show that:
   1. \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \tanh ^ { 2 } t\);
   2. \(s = \ln \cosh t\);
   3. \(y = \mathrm { e } ^ { - s }\).
3. The arc \(K P\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the surface area generated is $$2 \pi \left( 1 - \mathrm { e } ^ { - S } \right)$$ (4 marks)

5

1. Prove by induction that, if \(n\) is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
2. Find the value of \(\left( \cos \frac { \pi } { 6 } + \mathrm { i } \sin \frac { \pi } { 6 } \right) ^ { 6 }\).
3. Show that $$( \cos \theta + \mathrm { i } \sin \theta ) ( 1 + \cos \theta - \mathrm { i } \sin \theta ) = 1 + \cos \theta + \mathrm { i } \sin \theta$$
4. Hence show that $$\left( 1 + \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right) ^ { 6 } + \left( 1 + \cos \frac { \pi } { 6 } - i \sin \frac { \pi } { 6 } \right) ^ { 6 } = 0$$

6

1. Find the three roots of \(z ^ { 3 } = 1\), giving the non-real roots in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leqslant \pi\).
2. Given that \(\omega\) is one of the non-real roots of \(z ^ { 3 } = 1\), show that $$1 + \omega + \omega ^ { 2 } = 0$$
3. By using the result in part (b), or otherwise, show that:
   1. \(\frac { \omega } { \omega + 1 } = - \frac { 1 } { \omega }\);
   2. \(\frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } = - \omega\);
   3. \(\left( \frac { \omega } { \omega + 1 } \right) ^ { k } + \left( \frac { \omega ^ { 2 } } { \omega ^ { 2 } + 1 } \right) ^ { k } = ( - 1 ) ^ { k } 2 \cos \frac { 2 } { 3 } k \pi\), where \(k\) is an integer.

7

1. Use the identity \(\tan ( A - B ) = \frac { \tan A - \tan B } { 1 + \tan A \tan B }\) with \(A = ( r + 1 ) x\) and \(B = r x\) to show that $$\tan r x \tan ( r + 1 ) x = \frac { \tan ( r + 1 ) x } { \tan x } - \frac { \tan r x } { \tan x } - 1$$ (4 marks)
2. Use the method of differences to show that $$\tan \frac { \pi } { 50 } \tan \frac { 2 \pi } { 50 } + \tan \frac { 2 \pi } { 50 } \tan \frac { 3 \pi } { 50 } + \ldots + \tan \frac { 19 \pi } { 50 } \tan \frac { 20 \pi } { 50 } = \frac { \tan \frac { 2 \pi } { 5 } } { \tan \frac { \pi } { 50 } } - 20$$