AQA FP2 (Further Pure Mathematics 2) 2011 June

Question 1
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1
  1. Draw on the same Argand diagram:
    1. the locus of points for which $$| z - 2 - 5 \mathrm { i } | = 5$$
    2. the locus of points for which $$\arg ( z + 2 i ) = \frac { \pi } { 4 }$$
  2. Indicate on your diagram the set of points satisfying both $$| z - 2 - 5 i | \leqslant 5$$ and $$\arg ( z + 2 \mathrm { i } ) = \frac { \pi } { 4 }$$ (2 marks)
Question 2
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2
  1. Use the definitions of \(\cosh \theta\) and \(\sinh \theta\) in terms of \(\mathrm { e } ^ { \theta }\) to show that $$\cosh x \cosh y - \sinh x \sinh y = \cosh ( x - y )$$
  2. It is given that \(x\) satisfies the equation $$\cosh ( x - \ln 2 ) = \sinh x$$
    1. Show that \(\tanh x = \frac { 5 } { 7 }\).
    2. Express \(x\) in the form \(\frac { 1 } { 2 } \ln a\).
Question 3
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3
  1. Show that $$( r + 1 ) ! - ( r - 1 ) ! = \left( r ^ { 2 } + r - 1 \right) ( r - 1 ) !$$
  2. Hence show that $$\sum _ { r = 1 } ^ { n } \left( r ^ { 2 } + r - 1 \right) ( r - 1 ) ! = ( n + 2 ) n ! - 2$$ (4 marks) The cubic equation $$z ^ { 3 } - 2 z ^ { 2 } + k = 0 \quad ( k \neq 0 )$$ has roots \(\alpha , \beta\) and \(\gamma\).
    1. Write down the values of \(\alpha + \beta + \gamma\) and \(\alpha \beta + \beta \gamma + \gamma \alpha\).
    2. Show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 4\).
    3. Explain why \(\alpha ^ { 3 } - 2 \alpha ^ { 2 } + k = 0\).
    4. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 8 - 3 k\).
  4. Given that \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } = 0\) :
    1. show that \(k = 2\);
    2. find the value of \(\alpha ^ { 5 } + \beta ^ { 5 } + \gamma ^ { 5 }\).
Question 5
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5
  1. The arc of the curve \(y ^ { 2 } = x ^ { 2 } + 8\) between the points where \(x = 0\) and \(x = 6\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the area \(S\) of the curved surface formed is given by $$S = 2 \sqrt { 2 } \pi \int _ { 0 } ^ { 6 } \sqrt { x ^ { 2 } + 4 } \mathrm {~d} x$$
  2. By means of the substitution \(x = 2 \sinh \theta\), show that $$S = \pi \left( 24 \sqrt { 5 } + 4 \sqrt { 2 } \sinh ^ { - 1 } 3 \right)$$
Question 6
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6
  1. Show that $$( k + 1 ) \left( 4 ( k + 1 ) ^ { 2 } - 1 \right) = 4 k ^ { 3 } + 12 k ^ { 2 } + 11 k + 3$$
  2. Prove by induction that, for all integers \(n \geqslant 1\), $$1 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } + \ldots + ( 2 n - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
Question 7
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7
    1. Use de Moivre's Theorem to show that $$\cos 5 \theta = \cos ^ { 5 } \theta - 10 \cos ^ { 3 } \theta \sin ^ { 2 } \theta + 5 \cos \theta \sin ^ { 4 } \theta$$ and find a similar expression for \(\sin 5 \theta\).
    2. Deduce that $$\tan 5 \theta = \frac { \tan \theta \left( 5 - 10 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta \right) } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$
  2. Explain why \(t = \tan \frac { \pi } { 5 }\) is a root of the equation $$t ^ { 4 } - 10 t ^ { 2 } + 5 = 0$$ and write down the three other roots of this equation in trigonometrical form.
    (3 marks)
  3. Deduce that $$\tan \frac { \pi } { 5 } \tan \frac { 2 \pi } { 5 } = \sqrt { 5 }$$