AQA FP2 (Further Pure Mathematics 2) 2014 June

1

1. Express - 9 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   [0pt] [2 marks]
2. Solve the equation \(z ^ { 4 } + 9 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
   [0pt] [5 marks]

2

1. Sketch, on the Argand diagram below, the locus \(L\) of points satisfying $$\arg ( z - 2 \mathrm { i } ) = \frac { 2 \pi } { 3 }$$
2. 1. A circle \(C\), of radius 3, has its centre lying on \(L\) and touches the line \(\operatorname { Im } ( z ) = 2\). Sketch \(C\) on the Argand diagram used in part (a).
   2. Find the centre of \(C\), giving your answer in the form \(a + b \mathrm { i }\).
      [0pt] [3 marks]

3

1. Express \(( k + 1 ) ^ { 2 } + 5 ( k + 1 ) + 8\) in the form \(k ^ { 2 } + a k + b\), where \(a\) and \(b\) are constants.
2. Prove by induction that, for all integers \(n \geqslant 1\), $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) \left( \frac { 1 } { 2 } \right) ^ { r - 1 } = 16 - \left( n ^ { 2 } + 5 n + 8 \right) \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$$

4 The roots of the equation $$z ^ { 3 } + 2 z ^ { 2 } + 3 z - 4 = 0$$ are \(\alpha , \beta\) and \(\gamma\).

1. 1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
   2. Hence show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 2\).
2. Find the value of:
   1. \(( \alpha + \beta ) ( \beta + \gamma ) + ( \beta + \gamma ) ( \gamma + \alpha ) + ( \gamma + \alpha ) ( \alpha + \beta )\);
   2. \(( \alpha + \beta ) ( \beta + \gamma ) ( \gamma + \alpha )\).
3. Find a cubic equation whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).

5

1. Using the definition \(\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right)\), prove the identity $$4 \sinh ^ { 3 } \theta + 3 \sinh \theta = \sinh 3 \theta$$
2. Given that \(x = \sinh \theta\) and \(16 x ^ { 3 } + 12 x - 3 = 0\), find the value of \(\theta\) in terms of a natural logarithm.
3. Hence find the real root of the equation \(16 x ^ { 3 } + 12 x - 3 = 0\), giving your answer in the form \(2 ^ { p } - 2 ^ { q }\), where \(p\) and \(q\) are rational numbers.
   [0pt] [2 marks]

6

1. 1. Use De Moivre's Theorem to show that if \(z = \cos \theta + \mathrm { i } \sin \theta\), then $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$
   2. Write down a similar expression for \(z ^ { n } + \frac { 1 } { z ^ { n } }\).
2. 1. Expand \(\left( z - \frac { 1 } { z } \right) ^ { 2 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\) in terms of \(z\).
   2. Hence show that $$8 \sin ^ { 2 } \theta \cos ^ { 2 } \theta = A + B \cos 4 \theta$$ where \(A\) and \(B\) are integers.
3. Hence, by means of the substitution \(x = 2 \sin \theta\), find the exact value of $$\int _ { 1 } ^ { 2 } x ^ { 2 } \sqrt { 4 - x ^ { 2 } } \mathrm {~d} x$$ \includegraphics[max width=\textwidth, alt={}, center]{5287255f-5ac4-401a-b850-758257412ff7-14_1180_1707_1525_153}

7

1. Given that \(y = \tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right)\) and \(x \neq 1\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
   [0pt] [4 marks]
2. Hence, given that \(x < 1\), show that \(\tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right) - \tan ^ { - 1 } x = \frac { \pi } { 4 }\).
   [0pt] [3 marks]

8 A curve has equation \(y = 2 \sqrt { x - 1 }\), where \(x > 1\). The length of the arc of the curve between the points on the curve where \(x = 2\) and \(x = 9\) is denoted by \(s\).

1. Show that \(s = \int _ { 2 } ^ { 9 } \sqrt { \frac { x } { x - 1 } } \mathrm {~d} x\).
2. 1. Show that \(\cosh ^ { - 1 } 3 = 2 \ln ( 1 + \sqrt { 2 } )\).
   2. Use the substitution \(x = \cosh ^ { 2 } \theta\) to show that $$s = m \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } )$$ where \(m\) is an integer.
      [0pt] [6 marks]
      \includegraphics[max width=\textwidth, alt={}]{5287255f-5ac4-401a-b850-758257412ff7-20_1638_1709_1069_153}
      \includegraphics[max width=\textwidth, alt={}, center]{5287255f-5ac4-401a-b850-758257412ff7-24_2489_1728_221_141}