AQA FP2 (Further Pure Mathematics 2) 2013 January

1

1. Show that $$12 \cosh x - 4 \sinh x = 4 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x }$$
2. Solve the equation $$12 \cosh x - 4 \sinh x = 33$$ giving your answers in the form \(k \ln 2\).

2 Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), in an Argand diagram are given by $$\begin{aligned} & L _ { 1 } : | z + 6 - 5 \mathrm { i } | = 4 \sqrt { 2 }
& L _ { 2 } : \quad \arg ( z + \mathrm { i } ) = \frac { 3 \pi } { 4 } \end{aligned}$$ The point \(P\) represents the complex number \(- 2 + \mathrm { i }\).

1. Verify that the point \(P\) is a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
2. Sketch \(L _ { 1 }\) and \(L _ { 2 }\) on one Argand diagram.
3. The point \(Q\) is also a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\). Find the complex number that is represented by \(Q\).

3

1. Show that \(\frac { 1 } { 5 r - 2 } - \frac { 1 } { 5 r + 3 } = \frac { A } { ( 5 r - 2 ) ( 5 r + 3 ) }\), stating the value of the constant \(A\).
   (2 marks)
2. Hence use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) } = \frac { n } { 3 ( 5 n + 3 ) }$$
3. Find the value of $$\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) }$$ (1 mark)

4 The roots of the equation $$z ^ { 3 } - 5 z ^ { 2 } + k z - 4 = 0$$ are \(\alpha , \beta\) and \(\gamma\).

1. 1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha \beta \gamma\).
   2. Hence find the value of \(\alpha ^ { 2 } \beta \gamma + \alpha \beta ^ { 2 } \gamma + \alpha \beta \gamma ^ { 2 }\).
2. The value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\) is - 4 .
   1. Explain why \(\alpha , \beta\) and \(\gamma\) cannot all be real.
   2. By considering \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 }\), find the possible values of \(k\).

5

1. Using the definition \(\tanh y = \frac { \mathrm { e } ^ { y } - \mathrm { e } ^ { - y } } { \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } }\), show that, for \(| x | < 1\), $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$$
2. Hence, or otherwise, show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \tanh ^ { - 1 } x \right) = \frac { 1 } { 1 - x ^ { 2 } }\).
3. Use integration by parts to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } 4 \tanh ^ { - 1 } x \mathrm {~d} x = \ln \left( \frac { 3 ^ { m } } { 2 ^ { n } } \right)$$ where \(m\) and \(n\) are positive integers.

6 A curve is defined parametrically by $$x = t ^ { 3 } + 5 , \quad y = 6 t ^ { 2 } - 1$$ The arc length between the points where \(t = 0\) and \(t = 3\) on the curve is \(s\).

1. Show that \(s = \int _ { 0 } ^ { 3 } 3 t \sqrt { t ^ { 2 } + A } \mathrm {~d} t\), stating the value of the constant \(A\).
2. Hence show that \(s = 61\).
   \(7 \quad\) The polynomial \(\mathrm { p } ( n )\) is given by \(\mathrm { p } ( n ) = ( n - 1 ) ^ { 3 } + n ^ { 3 } + ( n + 1 ) ^ { 3 }\).
3. 1. Show that \(\mathrm { p } ( k + 1 ) - \mathrm { p } ( k )\), where \(k\) is a positive integer, is a multiple of 9 .
   2. Prove by induction that \(\mathrm { p } ( n )\) is a multiple of 9 for all integers \(n \geqslant 1\).
4. Using the result from part (a)(ii), show that \(n \left( n ^ { 2 } + 2 \right)\) is a multiple of 3 for any positive integer \(n\).

8

1. Express \(- 4 + 4 \sqrt { 3 } \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
2. 1. Solve the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   2. The roots of the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\) are represented by the points \(P , Q\) and \(R\) on an Argand diagram. Find the area of the triangle \(P Q R\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
3. By considering the roots of the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\), show that $$\cos \frac { 2 \pi } { 9 } + \cos \frac { 4 \pi } { 9 } + \cos \frac { 8 \pi } { 9 } = 0$$