AQA FP2 (Further Pure Mathematics 2) 2015 June

1

1. Express \(\frac { 1 } { ( r + 2 ) r ! }\) in the form \(\frac { A } { ( r + 1 ) ! } + \frac { B } { ( r + 2 ) ! }\), where \(A\) and \(B\) are integers.
   [0pt] [3 marks]
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 2 ) r ! }\).
   [0pt] [2 marks]

2

1. Sketch the graph of \(y = \tanh x\) and state the equations of its asymptotes.
2. Use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that $$\operatorname { sech } ^ { 2 } x + \tanh ^ { 2 } x = 1$$
3. Solve the equation \(6 \operatorname { sech } ^ { 2 } x = 4 + \tanh x\), giving your answers in terms of natural logarithms.
   [0pt] [5 marks] \section*{Answer space for question 2}
4. 
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3 A curve \(C\) is defined parametrically by $$x = \frac { t ^ { 2 } + 1 } { t } , \quad y = 2 \ln t$$

1. Show that \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + \frac { 1 } { t ^ { 2 } } \right) ^ { 2 }\).
2. The arc of \(C\) from \(t = 1\) to \(t = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the area of the surface generated, giving your answer in the form \(\pi ( m \ln 2 + n )\), where \(m\) and \(n\) are integers.
   [0pt] [5 marks]

4 The expression \(\mathrm { f } ( n )\) is given by \(\mathrm { f } ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }\).

1. Show that \(\mathrm { f } ( k + 1 ) - 16 \mathrm { f } ( k )\) can be expressed in the form \(A \times 3 ^ { 3 k }\), where \(A\) is an integer.
2. Prove by induction that \(\mathrm { f } ( n )\) is a multiple of 11 for all integers \(n \geqslant 1\).

5 The locus of points, \(L\), satisfies the equation $$| z - 2 + 4 \mathrm { i } | = | z |$$

1. Sketch \(L\) on the Argand diagram below.
2. The locus \(L\) cuts the real axis at \(A\) and the imaginary axis at \(B\).
   1. Show that the complex number represented by \(C\), the midpoint of \(A B\), is $$\frac { 5 } { 2 } - \frac { 5 } { 4 } \mathrm { i }$$
   2. The point \(O\) is the origin of the Argand diagram. Find the equation of the circle that passes through the points \(O , A\) and \(B\), giving your answer in the form \(| z - \alpha | = k\).
      [0pt] [2 marks] \section*{(a)}
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6

1. Given that \(y = ( x - 2 ) \sqrt { 5 + 4 x - x ^ { 2 } } + 9 \sin ^ { - 1 } \left( \frac { x - 2 } { 3 } \right)\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 5 + 4 x - x ^ { 2 } }$$ where \(k\) is an integer.
2. Hence show that $$\int _ { 2 } ^ { \frac { 7 } { 2 } } \sqrt { 5 + 4 x - x ^ { 2 } } \mathrm {~d} x = p \sqrt { 3 } + q \pi$$ where \(p\) and \(q\) are rational numbers.
   [0pt] [3 marks]

7 The cubic equation \(27 z ^ { 3 } + k z ^ { 2 } + 4 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\).
2. 1. In the case where \(\beta = \gamma\), find the roots of the equation.
   2. Find the value of \(k\) in this case.
3. 1. In the case where \(\alpha = 1 - \mathrm { i }\), find \(\alpha ^ { 2 }\) and \(\alpha ^ { 3 }\).
   2. Hence find the value of \(k\) in this case.
4. In the case where \(k = - 12\), find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } + 1 , \frac { 1 } { \beta } + 1\) and \(\frac { 1 } { \gamma } + 1\).
   [0pt] [5 marks]

8 The complex number \(\omega\) is given by \(\omega = \cos \frac { 2 \pi } { 5 } + \mathrm { i } \sin \frac { 2 \pi } { 5 }\).

1. 1. Verify that \(\omega\) is a root of the equation \(z ^ { 5 } = 1\).
   2. Write down the three other non-real roots of \(z ^ { 5 } = 1\), in terms of \(\omega\).
2. 1. Show that \(1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } = 0\).
   2. Hence show that \(\left( \omega + \frac { 1 } { \omega } \right) ^ { 2 } + \left( \omega + \frac { 1 } { \omega } \right) - 1 = 0\).
3. Hence show that \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).