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AQA FP2 2010 January Q5
8 marks Standard +0.8
5 The sum to \(r\) terms, \(S _ { r }\), of a series is given by $$S _ { r } = r ^ { 2 } ( r + 1 ) ( r + 2 )$$ Given that \(u _ { r }\) is the \(r\) th term of the series whose sum is \(S _ { r }\), show that:
    1. \(u _ { 1 } = 6\);
    2. \(u _ { 2 } = 42\);
    3. \(\quad u _ { n } = n ( n + 1 ) ( 4 n - 1 )\).
  2. Show that $$\sum _ { r = n + 1 } ^ { 2 n } u _ { r } = 3 n ^ { 2 } ( n + 1 ) ( 5 n + 2 )$$
AQA FP2 2010 January Q6
6 marks Standard +0.8
6
  1. Show that the substitution \(t = \tan \theta\) transforms the integral $$\int \frac { \mathrm { d } \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta }$$ into $$\int \frac { \mathrm { d } t } { 9 + t ^ { 2 } }$$
  2. Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \frac { d \theta } { 9 \cos ^ { 2 } \theta + \sin ^ { 2 } \theta } = \frac { \pi } { 18 }$$
AQA FP2 2010 January Q7
8 marks Standard +0.8
7 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 , \quad u _ { k + 1 } = 2 u _ { k } + 1$$
  1. Prove by induction that, for all \(n \geqslant 1\), $$u _ { n } = 3 \times 2 ^ { n - 1 } - 1$$
  2. Show that $$\sum _ { r = 1 } ^ { n } u _ { r } = u _ { n + 1 } - ( n + 2 )$$
AQA FP2 2010 January Q8
12 marks Standard +0.8
8
    1. Show that \(\omega = \mathrm { e } ^ { \frac { 2 \pi \mathrm { i } } { 7 } }\) is a root of the equation \(z ^ { 7 } = 1\).
    2. Write down the five other non-real roots in terms of \(\omega\).
  2. Show that $$1 + \omega + \omega ^ { 2 } + \omega ^ { 3 } + \omega ^ { 4 } + \omega ^ { 5 } + \omega ^ { 6 } = 0$$
  3. Show that:
    1. \(\quad \omega ^ { 2 } + \omega ^ { 5 } = 2 \cos \frac { 4 \pi } { 7 }\);
    2. \(\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }\).
AQA FP2 2011 January Q1
5 marks Standard +0.3
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 \(\left| z _ { 1 } \right|\).
AQA FP2 2011 January Q2
6 marks Standard +0.3
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 $$1 \times 5 + 2 \times 7 + 3 \times 9 + \ldots + r ( 2 r + 3 ) + \ldots$$
AQA FP2 2011 January Q3
11 marks Standard +0.8
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 \(\gamma\).
AQA FP2 2011 January Q4
11 marks Standard +0.8
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 \(a + b = 9\).
AQA FP2 2011 January Q5
8 marks Standard +0.8
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.
AQA FP2 2011 January Q6
10 marks Challenging +1.2
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.
AQA FP2 2011 January Q7
7 marks Standard +0.8
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 \(n \geqslant 1\).
AQA FP2 2011 January Q8
17 marks Challenging +1.2
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 \(\cos \frac { \pi } { 9 } + \cos \frac { 3 \pi } { 9 } + \cos \frac { 5 \pi } { 9 } + \cos \frac { 7 \pi } { 9 } = \frac { 1 } { 2 }\).
AQA FP2 2012 January Q1
8 marks Standard +0.8
1
  1. Show, by means of a sketch, that the curves with equations $$y = \sinh x$$ and $$y = \operatorname { sech } x$$ have exactly one point of intersection.
  2. Find the \(x\)-coordinate of this point of intersection, giving your answer in the form \(a \ln b\).
AQA FP2 2012 January Q2
8 marks Challenging +1.2
2
  1. Draw on an Argand diagram the locus \(L\) of points satisfying the equation \(\arg z = \frac { \pi } { 6 }\).
    (1 mark)
    1. A circle \(C\), of radius 6, has its centre lying on \(L\) and touches the line \(\operatorname { Re } ( z ) = 0\). Draw \(C\) on your Argand diagram from part (a).
    2. Find the equation of \(C\), giving your answer in the form \(\left| z - z _ { 0 } \right| = k\).
    3. The complex number \(z _ { 1 }\) lies on \(C\) and is such that \(\arg z _ { 1 }\) has its least possible value. Find \(\arg z _ { 1 }\), giving your answer in the form \(p \pi\), where \(- 1 < p \leqslant 1\).
AQA FP2 2012 January Q3
12 marks Challenging +1.2
3 A curve has cartesian equation $$y = \frac { 1 } { 2 } \ln ( \tanh x )$$
  1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sinh 2 x }$$
  2. The points \(A\) and \(B\) on the curve have \(x\)-coordinates \(\ln 2\) and \(\ln 4\) respectively. Find the arc length \(A B\), giving your answer in the form \(p \ln q\), where \(p\) and \(q\) are rational numbers.
AQA FP2 2012 January Q4
6 marks Standard +0.8
4 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = \frac { 3 } { 4 } \quad u _ { n + 1 } = \frac { 3 } { 4 - u _ { n } }$$ Prove by induction that, for all \(n \geqslant 1\), $$u _ { n } = \frac { 3 ^ { n + 1 } - 3 } { 3 ^ { n + 1 } - 1 }$$
AQA FP2 2012 January Q5
7 marks Standard +0.8
5 Find the smallest positive integer values of \(p\) and \(q\) for which $$\frac { \left( \cos \frac { \pi } { 8 } + \mathrm { i } \sin \frac { \pi } { 8 } \right) ^ { p } } { \left( \cos \frac { \pi } { 12 } - \mathrm { i } \sin \frac { \pi } { 12 } \right) ^ { q } } = \mathrm { i }$$
AQA FP2 2012 January Q6
8 marks Standard +0.3
6
  1. Express \(7 + 4 x - 2 x ^ { 2 }\) in the form \(a - b ( x - c ) ^ { 2 }\), where \(a , b\) and \(c\) are integers.
  2. By means of a suitable substitution, or otherwise, find the exact value of $$\int _ { 1 } ^ { \frac { 5 } { 2 } } \frac { \mathrm {~d} x } { \sqrt { 7 + 4 x - 2 x ^ { 2 } } }$$
AQA FP2 2012 January Q7
12 marks
7 The numbers \(\alpha , \beta\) and \(\gamma\) satisfy the equations $$\begin{aligned} & \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 10 - 12 \mathrm { i } \\ & \alpha \beta + \beta \gamma + \gamma \alpha = 5 + 6 \mathrm { i } \end{aligned}$$
  1. Show that \(\alpha + \beta + \gamma = 0\).
  2. The numbers \(\alpha , \beta\) and \(\gamma\) are also the roots of the equation $$z ^ { 3 } + p z ^ { 2 } + q z + r = 0$$ Write down the value of \(p\) and the value of \(q\).
  3. It is also given that \(\alpha = 3 \mathrm { i }\).
    1. Find the value of \(r\).
    2. Show that \(\beta\) and \(\gamma\) are the roots of the equation $$z ^ { 2 } + 3 \mathrm { i } z - 4 + 6 \mathrm { i } = 0$$
    3. Given that \(\beta\) is real, find the values of \(\beta\) and \(\gamma\).
AQA FP2 2012 January Q8
14 marks Challenging +1.2
8
  1. Write down the five roots of the equation \(z ^ { 5 } = 1\), giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\), where \(- \pi < \theta \leqslant \pi\).
  2. Hence find the four linear factors of $$z ^ { 4 } + z ^ { 3 } + z ^ { 2 } + z + 1$$
  3. Deduce that $$z ^ { 2 } + z + 1 + z ^ { - 1 } + z ^ { - 2 } = \left( z - 2 \cos \frac { 2 \pi } { 5 } + z ^ { - 1 } \right) \left( z - 2 \cos \frac { 4 \pi } { 5 } + z ^ { - 1 } \right)$$
  4. Use the substitution \(z + z ^ { - 1 } = w\) to show that \(\cos \frac { 2 \pi } { 5 } = \frac { \sqrt { 5 } - 1 } { 4 }\).
AQA FP2 2013 January Q1
7 marks Standard +0.3
1
  1. Show that $$12 \cosh x - 4 \sinh x = 4 \mathrm { e } ^ { x } + 8 \mathrm { e } ^ { - x }$$
  2. Solve the equation $$12 \cosh x - 4 \sinh x = 33$$ giving your answers in the form \(k \ln 2\).
AQA FP2 2013 January Q2
10 marks Standard +0.8
2 Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), in an Argand diagram are given by $$\begin{aligned} & L _ { 1 } : | z + 6 - 5 \mathrm { i } | = 4 \sqrt { 2 } \\ & L _ { 2 } : \quad \arg ( z + \mathrm { i } ) = \frac { 3 \pi } { 4 } \end{aligned}$$ The point \(P\) represents the complex number \(- 2 + \mathrm { i }\).
  1. Verify that the point \(P\) is a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
  2. Sketch \(L _ { 1 }\) and \(L _ { 2 }\) on one Argand diagram.
  3. The point \(Q\) is also a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\). Find the complex number that is represented by \(Q\).
AQA FP2 2013 January Q3
7 marks Standard +0.3
3
  1. Show that \(\frac { 1 } { 5 r - 2 } - \frac { 1 } { 5 r + 3 } = \frac { A } { ( 5 r - 2 ) ( 5 r + 3 ) }\), stating the value of the constant \(A\).
    (2 marks)
  2. Hence use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) } = \frac { n } { 3 ( 5 n + 3 ) }$$
  3. Find the value of $$\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) }$$ (1 mark)
AQA FP2 2013 January Q4
9 marks Standard +0.8
4 The roots of the equation $$z ^ { 3 } - 5 z ^ { 2 } + k z - 4 = 0$$ are \(\alpha , \beta\) and \(\gamma\).
    1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha \beta \gamma\).
    2. Hence find the value of \(\alpha ^ { 2 } \beta \gamma + \alpha \beta ^ { 2 } \gamma + \alpha \beta \gamma ^ { 2 }\).
  2. The value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\) is - 4 .
    1. Explain why \(\alpha , \beta\) and \(\gamma\) cannot all be real.
    2. By considering \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 }\), find the possible values of \(k\).
AQA FP2 2013 January Q5
11 marks Standard +0.8
5
  1. Using the definition \(\tanh y = \frac { \mathrm { e } ^ { y } - \mathrm { e } ^ { - y } } { \mathrm { e } ^ { y } + \mathrm { e } ^ { - y } }\), show that, for \(| x | < 1\), $$\tanh ^ { - 1 } x = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)$$
  2. Hence, or otherwise, show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \tanh ^ { - 1 } x \right) = \frac { 1 } { 1 - x ^ { 2 } }\).
  3. Use integration by parts to show that $$\int _ { 0 } ^ { \frac { 1 } { 2 } } 4 \tanh ^ { - 1 } x \mathrm {~d} x = \ln \left( \frac { 3 ^ { m } } { 2 ^ { n } } \right)$$ where \(m\) and \(n\) are positive integers.