AQA FP2 (Further Pure Mathematics 2) 2012 June

1

1. Sketch the curve \(y = \cosh x\).
2. Solve the equation $$6 \cosh ^ { 2 } x - 7 \cosh x - 5 = 0$$ giving your answers in logarithmic form.

2

1. Draw on the Argand diagram below:
   1. the locus of points for which $$| z - 2 - 3 \mathrm { i } | = 2$$
   2. the locus of points for which $$| z + 2 - \mathrm { i } | = | z - 2 |$$
2. Indicate on your diagram the points satisfying both $$| z - 2 - 3 \mathrm { i } | = 2$$ and $$| z + 2 - \mathrm { i } | \leqslant | z - 2 |$$ (l mark)
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3

1. Show that $$\frac { 2 ^ { r + 1 } } { r + 2 } - \frac { 2 ^ { r } } { r + 1 } = \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$
2. Hence find $$\sum _ { r = 1 } ^ { 30 } \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$ giving your answer in the form \(2 ^ { n } - 1\), where \(n\) is an integer.

4 The cubic equation $$z ^ { 3 } + p z + q = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. 1. Write down the value of \(\alpha + \beta + \gamma\).
   2. Express \(\alpha \beta \gamma\) in terms of \(q\).
2. Show that $$\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 \alpha \beta \gamma$$
3. Given that \(\alpha = 4 + 7 \mathrm { i }\) and that \(p\) and \(q\) are real, find the values of:
   1. \(\beta\) and \(\gamma\);
   2. \(p\) and \(q\).
4. Find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).

5 The function f , where \(\mathrm { f } ( x ) = \sec x\), has domain \(0 \leqslant x < \frac { \pi } { 2 }\) and has inverse function \(\mathrm { f } ^ { - 1 }\), where \(\mathrm { f } ^ { - 1 } ( x ) = \sec ^ { - 1 } x\).

1. Show that $$\sec ^ { - 1 } x = \cos ^ { - 1 } \frac { 1 } { x }$$
2. Hence show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \sec ^ { - 1 } x \right) = \frac { 1 } { \sqrt { x ^ { 4 } - x ^ { 2 } } }$$

6

1. Show that $$\frac { 1 } { 4 } ( \cosh 4 x + 2 \cosh 2 x + 1 ) = \cosh ^ { 2 } x \cosh 2 x$$
2. Show that, if \(y = \cosh ^ { 2 } x\), then $$1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \cosh ^ { 2 } 2 x$$
3. The arc of the curve \(y = \cosh ^ { 2 } x\) between the points where \(x = 0\) and \(x = \ln 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the area \(S\) of the curved surface formed is given by $$S = \frac { \pi } { 256 } ( a \ln 2 + b )$$ where \(a\) and \(b\) are integers.

7

1. Prove by induction that, for all integers \(n \geqslant 1\), $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots + \frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$
2. Find the smallest integer \(n\) for which the sum of the series differs from 1 by less than \(10 ^ { - 5 }\).

8

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. 1. Expand \(\left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 4 }\).
   2. Show that $$\cos ^ { 4 } 2 \theta = A \cos 8 \theta + B \cos 4 \theta + C$$ where \(A , B\) and \(C\) are rational numbers.
3. Hence solve the equation $$8 \cos ^ { 4 } 2 \theta = \cos 8 \theta + 5$$ for \(0 \leqslant \theta \leqslant \pi\), giving each solution in the form \(k \pi\).
4. Show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 4 } 2 \theta d \theta = \frac { 3 \pi } { 16 }$$