AQA FP2 (Further Pure Mathematics 2) 2016 June

1

1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { 4 r - 1 }\), show that $$\mathrm { f } ( r ) - \mathrm { f } ( r + 1 ) = \frac { A } { ( 4 r - 1 ) ( 4 r + 3 ) }$$ where \(A\) is an integer.
2. Use the method of differences to find the value of \(\sum _ { r = 1 } ^ { 50 } \frac { 1 } { ( 4 r - 1 ) ( 4 r + 3 ) }\), giving your answer as a fraction in its simplest form.
   [0pt] [4 marks]

2 The cubic equation \(3 z ^ { 3 } + p z ^ { 2 } + 17 z + q = 0\), where \(p\) and \(q\) are real, has a root \(\alpha = 1 + 2 \mathrm { i }\).

1. 1. Write down the value of another non-real root, \(\beta\), of this equation.
   2. Hence find the value of \(\alpha \beta\).
2. Find the value of the third root, \(\gamma\), of this equation.
3. Find the values of \(p\) and \(q\).

3 The arc of the curve with equation \(y = 4 - \ln \left( 1 - x ^ { 2 } \right)\) from \(x = 0\) to \(x = \frac { 3 } { 4 }\) has length \(s\).

1. Show that \(s = \int _ { 0 } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x\).
2. Find the value of \(s\), giving your answer in the form \(p + \ln N\), where \(p\) is a rational number and \(N\) is an integer.
   [0pt] [6 marks]
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4

1. Given that \(y = \tan ^ { - 1 } \sqrt { ( 3 x ) }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in terms of \(x\).
2. Hence, or otherwise, show that \(\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 1 } { ( 1 + 3 x ) \sqrt { x } } \mathrm {~d} x = \frac { \sqrt { 3 } \pi } { n }\), where \(n\) is an integer.
   [0pt] [4 marks]

5

1. Find the modulus of the complex number \(- 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answer as an integer.
2. The locus of points, \(L\), satisfies the equation \(| z + 4 \sqrt { 3 } - 4 \mathrm { i } | = 4\).
   1. Sketch the locus \(L\) on the Argand diagram below.
   2. The complex number \(w\) lies on \(L\) so that \(- \pi < \arg w \leqslant \pi\). Find the least possible value of arg \(w\), giving your answer in terms of \(\pi\).
3. Solve the equation \(z ^ { 3 } = - 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   [0pt] [5 marks]

6

1. Given that \(y = \sinh x\), use the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(x = \ln \left( y + \sqrt { y ^ { 2 } + 1 } \right)\).
2. A curve has equation \(y = 6 \cosh ^ { 2 } x + 5 \sinh x\).
   1. Show that the curve has a single stationary point and find its \(x\)-coordinate, giving your answer in the form \(\ln p\), where \(p\) is a rational number.
   2. The curve lies entirely above the \(x\)-axis. The region bounded by the curve, the coordinate axes and the line \(x = \cosh ^ { - 1 } 2\) has area \(A\). Show that $$A = a \cosh ^ { - 1 } 2 + b \sqrt { 3 } + c$$ where \(a\), \(b\) and \(c\) are integers.
      [0pt] [5 marks] \(7 \quad\) Given that \(p \geqslant - 1\), prove by induction that, for all integers \(n \geqslant 1\), $$( 1 + p ) ^ { n } \geqslant 1 + n p$$ [6 marks]

8

1. By applying de Moivre's theorem to \(( \cos \theta + i \sin \theta ) ^ { 4 }\), where \(\cos \theta \neq 0\), show that $$( 1 + i \tan \theta ) ^ { 4 } + ( 1 - i \tan \theta ) ^ { 4 } = \frac { 2 \cos 4 \theta } { \cos ^ { 4 } \theta }$$
2. Hence show that \(z = \mathrm { i } \tan \frac { \pi } { 8 }\) satisfies the equation \(( 1 + z ) ^ { 4 } + ( 1 - z ) ^ { 4 } = 0\), and express the three other roots of this equation in the form \(\mathrm { i } \tan \phi\), where \(0 < \phi < \pi\).
3. Use the results from part (b) to find the values of:
   1. \(\tan ^ { 2 } \frac { \pi } { 8 } \tan ^ { 2 } \frac { 3 \pi } { 8 }\);
   2. \(\tan ^ { 2 } \frac { \pi } { 8 } + \tan ^ { 2 } \frac { 3 \pi } { 8 }\).
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