AQA FP2 (Further Pure Mathematics 2) 2006 June

1

1. Given that $$\frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) } = A + B \left( \frac { 1 } { r } - \frac { 1 } { r + 1 } \right)$$ find the values of \(A\) and \(B\).
2. Hence find the value of $$\sum _ { r = 1 } ^ { 99 } \frac { r ^ { 2 } + r - 1 } { r ( r + 1 ) }$$

2 A curve has parametric equations $$x = t - \frac { 1 } { 3 } t ^ { 3 } , \quad y = t ^ { 2 }$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + t ^ { 2 } \right) ^ { 2 }$$
2. The arc of the curve between \(t = 1\) and \(t = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that \(S\), the surface area generated, is given by \(S = k \pi\), where \(k\) is a rational number to be found.

3 The curve \(C\) has equation $$y = \cosh x - 3 \sinh x$$

1. 1. The line \(y = - 1\) meets \(C\) at the point \(( k , - 1 )\). Show that $$\mathrm { e } ^ { 2 k } - \mathrm { e } ^ { k } - 2 = 0$$
   2. Hence find \(k\), giving your answer in the form \(\ln a\).
2. 1. Find the \(x\)-coordinate of the point where the curve \(C\) intersects the \(x\)-axis, giving your answer in the form \(p \ln a\).
   2. Show that \(C\) has no stationary points.
   3. Show that there is exactly one point on \(C\) for which \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 0\).

4

1. On one Argand diagram, sketch the locus of points satisfying:
   1. \(| z - 3 + 2 \mathrm { i } | = 4\);
   2. \(\quad \arg ( z - 1 ) = - \frac { 1 } { 4 } \pi\).
2. Indicate on your sketch the set of points satisfying both $$| z - 3 + 2 i | \leqslant 4$$ and $$\arg ( z - 1 ) = - \frac { 1 } { 4 } \pi$$ (1 mark)

5 The cubic equation $$z ^ { 3 } - 4 \mathrm { i } z ^ { 2 } + q z - ( 4 - 2 \mathrm { i } ) = 0$$ 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 \(\alpha = \beta + \gamma\), show that:
   1. \(\alpha = 2 \mathrm { i }\);
   2. \(\quad \beta \gamma = - ( 1 + 2 \mathrm { i } )\);
   3. \(\quad q = - ( 5 + 2 \mathrm { i } )\).
3. Show that \(\beta\) and \(\gamma\) are the roots of the equation $$z ^ { 2 } - 2 \mathrm { i } z - ( 1 + 2 \mathrm { i } ) = 0$$
4. Given that \(\beta\) is real, find \(\beta\) and \(\gamma\).

6

1. The function f is given by $$\mathrm { f } ( n ) = 15 ^ { n } - 8 ^ { n - 2 }$$ Express $$\mathrm { f } ( n + 1 ) - 8 \mathrm { f } ( n )$$ in the form \(k \times 15 ^ { n }\).
2. Prove by induction that \(15 ^ { n } - 8 ^ { n - 2 }\) is a multiple of 7 for all integers \(n \geqslant 2\).

7

1. Find the six roots of the equation \(z ^ { 6 } = 1\), giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \phi }\), where \(- \pi < \phi \leqslant \pi\).
2. It is given that \(w = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta \neq n \pi\).
   1. Show that \(\frac { w ^ { 2 } - 1 } { w } = 2 \mathrm { i } \sin \theta\).
   2. Show that \(\frac { w } { w ^ { 2 } - 1 } = - \frac { \mathrm { i } } { 2 \sin \theta }\).
   3. Show that \(\frac { 2 \mathrm { i } } { w ^ { 2 } - 1 } = \cot \theta - \mathrm { i }\).
   4. Given that \(z = \cot \theta - \mathrm { i }\), show that \(z + 2 \mathrm { i } = z w ^ { 2 }\).
3. 1. Explain why the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ has five roots.
   2. Find the five roots of the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ giving your answers in the form \(a + \mathrm { i } b\).