AQA FP2 (Further Pure Mathematics 2) 2010 June

1

1. Show that $$9 \sinh x - \cosh x = 4 \mathrm { e } ^ { x } - 5 \mathrm { e } ^ { - x }$$
2. Given that $$9 \sinh x - \cosh x = 8$$ find the exact value of \(\tanh x\).

2

1. Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions.
2. Use the method of differences to find $$\sum _ { r = 1 } ^ { 48 } \frac { 1 } { r ( r + 2 ) }$$ giving your answer as a rational number.

3 Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), in an Argand diagram are given by $$\begin{aligned} & L _ { 1 } : | z + 1 + 3 \mathrm { i } | = | z - 5 - 7 \mathrm { i } |
& L _ { 2 } : \arg z = \frac { \pi } { 4 } \end{aligned}$$

1. Verify that the point represented by the complex number \(2 + 2 \mathrm { i }\) is a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
2. Sketch \(L _ { 1 }\) and \(L _ { 2 }\) on one Argand diagram.
3. Shade on your Argand diagram the region satisfying
   both $$| z + 1 + 3 i | \leqslant | z - 5 - 7 i |$$ and $$\frac { \pi } { 4 } \leqslant \arg z \leqslant \frac { \pi } { 2 }$$

4 The roots of the cubic equation $$z ^ { 3 } - 2 z ^ { 2 } + p z + 10 = 0$$ are \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 4\).

1. Write down the value of \(\alpha + \beta + \gamma\).
2. 1. Explain why \(\alpha ^ { 3 } - 2 \alpha ^ { 2 } + p \alpha + 10 = 0\).
   2. Hence show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = p + 13$$
   3. Deduce that \(p = - 3\).
3. 1. Find the real root \(\alpha\) of the cubic equation \(z ^ { 3 } - 2 z ^ { 2 } - 3 z + 10 = 0\).
   2. Find the values of \(\beta\) and \(\gamma\).

5

1. Using the identities $$\cosh ^ { 2 } t - \sinh ^ { 2 } t = 1 , \quad \tanh t = \frac { \sinh t } { \cosh t } \quad \text { and } \quad \operatorname { sech } t = \frac { 1 } { \cosh t }$$ show that:
   1. \(\tanh ^ { 2 } t + \operatorname { sech } ^ { 2 } t = 1\);
   2. \(\frac { \mathrm { d } } { \mathrm { d } t } ( \tanh t ) = \operatorname { sech } ^ { 2 } t\);
   3. \(\frac { \mathrm { d } } { \mathrm { d } t } ( \operatorname { sech } t ) = - \operatorname { sech } t \tanh t\).
2. A curve \(C\) is given parametrically by $$x = \operatorname { sech } t , y = 4 - \tanh t$$
   1. Show that the arc length, \(s\), of \(C\) between the points where \(t = 0\) and \(t = \frac { 1 } { 2 } \ln 3\) is given by $$s = \int _ { 0 } ^ { \frac { 1 } { 2 } \ln 3 } \operatorname { sech } t \mathrm {~d} t$$
   2. Using the substitution \(u = \mathrm { e } ^ { t }\), find the exact value of \(s\).

6

1. Show that \(\frac { 1 } { ( k + 2 ) ! } - \frac { k + 1 } { ( k + 3 ) ! } = \frac { 2 } { ( k + 3 ) ! }\).
2. Prove by induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \frac { r \times 2 ^ { r } } { ( r + 2 ) ! } = 1 - \frac { 2 ^ { n + 1 } } { ( n + 2 ) ! }$$ (6 marks)
   \includegraphics[max width=\textwidth, alt={}]{d417bc62-f92a-4c90-a4c2-435f38e46edc-7_2010_1711_693_152}

7

1. 1. Express each of the numbers \(1 + \sqrt { 3 } \mathrm { i }\) and \(1 - \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\).
   2. Hence express $$( 1 + \sqrt { 3 } i ) ^ { 8 } ( 1 - i ) ^ { 5 }$$ in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\).
2. Solve the equation $$z ^ { 3 } = ( 1 + \sqrt { 3 } \mathrm { i } ) ^ { 8 } ( 1 - \mathrm { i } ) ^ { 5 }$$ giving your answers in the form \(a \sqrt { 2 } \mathrm { e } ^ { \mathrm { i } \theta }\), where \(a\) is a positive integer and \(- \pi < \theta \leqslant \pi\).