AQA FP2 (Further Pure Mathematics 2) 2008 June

1

1. Express $$5 \sinh x + \cosh x$$ in the form \(A \mathrm { e } ^ { x } + B \mathrm { e } ^ { - x }\), where \(A\) and \(B\) are integers.
2. Solve the equation $$5 \sinh x + \cosh x + 5 = 0$$ giving your answer in the form \(\ln a\), where \(a\) is a rational number.

2

1. Given that $$\frac { 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { A } { r ( r + 1 ) } + \frac { B } { ( r + 1 ) ( r + 2 ) }$$ show that \(A = \frac { 1 } { 2 }\) and find the value of \(B\).
2. Use the method of differences to find $$\sum _ { r = 10 } ^ { 98 } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$ giving your answer as a rational number.

3 The cubic equation $$z ^ { 3 } + q z + ( 18 - 12 i ) = 0$$ where \(q\) is a complex number, has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the value of:
   1. \(\alpha \beta \gamma\);
   2. \(\alpha + \beta + \gamma\).
2. Given that \(\beta + \gamma = 2\), find the value of:
   1. \(\alpha\);
   2. \(\quad \beta \gamma\);
   3. \(q\).
3. Given that \(\beta\) is of the form \(k \mathrm { i }\), where \(k\) is real, find \(\beta\) and \(\gamma\).

4

1. A circle \(C\) in the Argand diagram has equation $$| z + 5 - \mathrm { i } | = \sqrt { 2 }$$ Write down its radius and the complex number representing its centre.
2. A half-line \(L\) in the Argand diagram has equation $$\arg ( z + 2 \mathrm { i } ) = \frac { 3 \pi } { 4 }$$ Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) lies on \(L\).
3. 1. Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) also lies on \(C\).
   2. Hence show that \(L\) touches \(C\).
   3. Sketch \(L\) and \(C\) on one Argand diagram.
4. The complex number \(z _ { 2 }\) lies on \(C\) and is such that \(\arg \left( z _ { 2 } + 2 \mathrm { i } \right)\) has as great a value as possible. Indicate the position of \(z _ { 2 }\) on your sketch.

5

1. Use the definition \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) to show that \(\cosh 2 x = 2 \cosh ^ { 2 } x - 1\).
   (2 marks)
2. 1. The arc of the curve \(y = \cosh x\) between \(x = 0\) and \(x = \ln a\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that \(S\), the surface area generated, is given by $$S = 2 \pi \int _ { 0 } ^ { \ln a } \cosh ^ { 2 } x \mathrm {~d} x$$
   2. Hence show that $$S = \pi \left( \ln a + \frac { a ^ { 4 } - 1 } { 4 a ^ { 2 } } \right)$$

6 By using the substitution \(u = x - 2\), or otherwise, find the exact value of $$\int _ { - 1 } ^ { 5 } \frac { \mathrm {~d} x } { \sqrt { 32 + 4 x - x ^ { 2 } } }$$

7

1. Explain why \(n ( n + 1 )\) is a multiple of 2 when \(n\) is an integer.
2. 1. Given that $$\mathrm { f } ( n ) = n \left( n ^ { 2 } + 5 \right)$$ show that \(\mathrm { f } ( k + 1 ) - \mathrm { f } ( k )\), where \(k\) is a positive integer, is a multiple of 6 .
   2. Prove by induction that \(\mathrm { f } ( n )\) is a multiple of 6 for all integers \(n \geqslant 1\).

8

1. 1. Expand $$\left( z + \frac { 1 } { z } \right) \left( z - \frac { 1 } { z } \right)$$
   2. Hence, or otherwise, expand $$\left( z + \frac { 1 } { z } \right) ^ { 4 } \left( z - \frac { 1 } { z } \right) ^ { 2 }$$
2. 1. Use De Moivre's theorem to show that if \(z = \cos \theta + \mathrm { i } \sin \theta\) then $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
   2. Write down a corresponding result for \(z ^ { n } - \frac { 1 } { z ^ { n } }\).
3. Hence express \(\cos ^ { 4 } \theta \sin ^ { 2 } \theta\) in the form $$A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$ where \(A , B , C\) and \(D\) are rational numbers.
4. Find \(\int \cos ^ { 4 } \theta \sin ^ { 2 } \theta d \theta\).