AQA FP2 (Further Pure Mathematics 2) 2011 January

Question 1
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1
  1. Sketch on an Argand diagram the locus of points satisfying the equation $$| z - 4 + 3 \mathrm { i } | = 5$$
    1. Indicate on your diagram the point \(P\) representing \(z _ { 1 }\), where both $$\left| z _ { 1 } - 4 + 3 \mathrm { i } \right| = 5 \quad \text { and } \quad \arg z _ { 1 } = 0$$
    2. Find the value of \(\left| z _ { 1 } \right|\).
Question 2
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2
  1. Given that $$u _ { r } = \frac { 1 } { 6 } r ( r + 1 ) ( 4 r + 11 )$$ show that $$u _ { r } - u _ { r - 1 } = r ( 2 r + 3 )$$
  2. Hence find the sum of the first hundred terms of the series $$1 \times 5 + 2 \times 7 + 3 \times 9 + \ldots + r ( 2 r + 3 ) + \ldots$$
Question 3
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3
  1. Show that \(( 1 + \mathrm { i } ) ^ { 3 } = 2 \mathrm { i } - 2\).
  2. The cubic equation $$z ^ { 3 } - ( 5 + \mathrm { i } ) z ^ { 2 } + ( 9 + 4 \mathrm { i } ) z + k ( 1 + \mathrm { i } ) = 0$$ where \(k\) is a real constant, has roots \(\alpha , \beta\) and \(\gamma\).
    It is given that \(\alpha = 1 + \mathrm { i }\).
    1. Find the value of \(k\).
    2. Show that \(\beta + \gamma = 4\).
    3. Find the values of \(\beta\) and \(\gamma\).
Question 4
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4
  1. Prove that the curve $$y = 12 \cosh x - 8 \sinh x - x$$ has exactly one stationary point.
  2. Given that the coordinates of this stationary point are \(( a , b )\), show that \(a + b = 9\).
Question 5
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5
  1. Given that \(u = \sqrt { 1 - x ^ { 2 } }\), find \(\frac { \mathrm { d } u } { \mathrm {~d} x }\).
  2. Use integration by parts to show that $$\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 2 } } \sin ^ { - 1 } x \mathrm {~d} x = a \sqrt { 3 } \pi + b$$ where \(a\) and \(b\) are rational numbers.
Question 6
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6
  1. Given that $$x = \ln ( \sec t + \tan t ) - \sin t$$ show that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \sin t \tan t$$
  2. A curve is given parametrically by the equations $$x = \ln ( \sec t + \tan t ) - \sin t , \quad y = \cos t$$ The length of the arc of the curve between the points where \(t = 0\) and \(t = \frac { \pi } { 3 }\) is denoted by \(s\). Show that \(s = \ln p\), where \(p\) is an integer.
Question 7
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7
  1. Given that $$\mathrm { f } ( k ) = 12 ^ { k } + 2 \times 5 ^ { k - 1 }$$ show that $$\mathrm { f } ( k + 1 ) - 5 \mathrm { f } ( k ) = a \times 12 ^ { k }$$ where \(a\) is an integer.
  2. Prove by induction that \(12 ^ { n } + 2 \times 5 ^ { n - 1 }\) is divisible by 7 for all integers \(n \geqslant 1\).
Question 8
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8
  1. Express in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\) :
    1. \(\quad 4 ( 1 + i \sqrt { 3 } )\);
    2. \(4 ( 1 - i \sqrt { 3 } )\).
  2. The complex number \(z\) satisfies the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ Show that \(z ^ { 3 } = 4 \pm 4 \sqrt { 3 } \mathrm { i }\).
    1. Solve the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    2. Illustrate the roots on an Argand diagram.
    1. Explain why the sum of the roots of the equation $$\left( z ^ { 3 } - 4 \right) ^ { 2 } = - 48$$ is zero.
    2. Deduce that \(\cos \frac { \pi } { 9 } + \cos \frac { 3 \pi } { 9 } + \cos \frac { 5 \pi } { 9 } + \cos \frac { 7 \pi } { 9 } = \frac { 1 } { 2 }\).