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AQA FP2 2006 June Q4
7 marks Standard +0.3
4
  1. On one Argand diagram, sketch the locus of points satisfying:
    1. \(| z - 3 + 2 \mathrm { i } | = 4\);
    2. \(\quad \arg ( z - 1 ) = - \frac { 1 } { 4 } \pi\).
  2. Indicate on your sketch the set of points satisfying both $$| z - 3 + 2 i | \leqslant 4$$ and $$\arg ( z - 1 ) = - \frac { 1 } { 4 } \pi$$ (1 mark)
AQA FP2 2006 June Q5
13 marks Standard +0.8
5 The cubic equation $$z ^ { 3 } - 4 \mathrm { i } z ^ { 2 } + q z - ( 4 - 2 \mathrm { i } ) = 0$$ where \(q\) is a complex number, has roots \(\alpha , \beta\) and \(\gamma\).
  1. Write down the value of:
    1. \(\alpha + \beta + \gamma\);
    2. \(\alpha \beta \gamma\).
  2. Given that \(\alpha = \beta + \gamma\), show that:
    1. \(\alpha = 2 \mathrm { i }\);
    2. \(\quad \beta \gamma = - ( 1 + 2 \mathrm { i } )\);
    3. \(\quad q = - ( 5 + 2 \mathrm { i } )\).
  3. Show that \(\beta\) and \(\gamma\) are the roots of the equation $$z ^ { 2 } - 2 \mathrm { i } z - ( 1 + 2 \mathrm { i } ) = 0$$
  4. Given that \(\beta\) is real, find \(\beta\) and \(\gamma\).
AQA FP2 2006 June Q6
8 marks Standard +0.8
6
  1. The function f is given by $$\mathrm { f } ( n ) = 15 ^ { n } - 8 ^ { n - 2 }$$ Express $$\mathrm { f } ( n + 1 ) - 8 \mathrm { f } ( n )$$ in the form \(k \times 15 ^ { n }\).
  2. Prove by induction that \(15 ^ { n } - 8 ^ { n - 2 }\) is a multiple of 7 for all integers \(n \geqslant 2\).
AQA FP2 2006 June Q7
17 marks Challenging +1.2
7
  1. Find the six roots of the equation \(z ^ { 6 } = 1\), giving your answers in the form \(\mathrm { e } ^ { \mathrm { i } \phi }\), where \(- \pi < \phi \leqslant \pi\).
  2. It is given that \(w = \mathrm { e } ^ { \mathrm { i } \theta }\), where \(\theta \neq n \pi\).
    1. Show that \(\frac { w ^ { 2 } - 1 } { w } = 2 \mathrm { i } \sin \theta\).
    2. Show that \(\frac { w } { w ^ { 2 } - 1 } = - \frac { \mathrm { i } } { 2 \sin \theta }\).
    3. Show that \(\frac { 2 \mathrm { i } } { w ^ { 2 } - 1 } = \cot \theta - \mathrm { i }\).
    4. Given that \(z = \cot \theta - \mathrm { i }\), show that \(z + 2 \mathrm { i } = z w ^ { 2 }\).
    1. Explain why the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ has five roots.
    2. Find the five roots of the equation $$( z + 2 \mathrm { i } ) ^ { 6 } = z ^ { 6 }$$ giving your answers in the form \(a + \mathrm { i } b\).
AQA FP2 2007 June Q1
7 marks Standard +0.3
1
  1. Given that \(\mathrm { f } ( r ) = ( r - 1 ) r ^ { 2 }\), show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r ( 3 r + 1 )$$
  2. Use the method of differences to find the value of $$\sum _ { r = 50 } ^ { 99 } r ( 3 r + 1 )$$ (4 marks)
AQA FP2 2007 June Q2
12 marks Standard +0.8
2 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + 6 z + q = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
  1. Write down the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
  2. Given that \(p\) and \(q\) are real and that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 12\) :
    1. explain why the cubic equation has two non-real roots and one real root;
    2. find the value of \(p\).
  3. One root of the cubic equation is \(- 1 + 3 \mathrm { i }\). Find:
    1. the other two roots;
    2. the value of \(q\).
AQA FP2 2007 June Q3
5 marks Standard +0.3
3 Use De Moivre's Theorem to find the smallest positive angle \(\theta\) for which $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { 15 } = - \mathrm { i }$$ (5 marks)
AQA FP2 2007 June Q4
7 marks Standard +0.3
4
  1. Differentiate \(x \tan ^ { - 1 } x\) with respect to \(x\).
  2. Show that $$\int _ { 0 } ^ { 1 } \tan ^ { - 1 } x \mathrm {~d} x = \frac { \pi } { 4 } - \ln \sqrt { 2 }$$ (5 marks)
AQA FP2 2007 June Q5
9 marks Standard +0.3
5 The sketch shows an Argand diagram. The points \(A\) and \(B\) represent the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) respectively. The angle \(A O B = 90 ^ { \circ }\) and \(O A = O B\). \includegraphics[max width=\textwidth, alt={}, center]{847295e3-d806-43b1-8d25-688c5558bfe1-3_533_869_852_632}
  1. Explain why \(z _ { 2 } = \mathrm { i } z _ { 1 }\).
  2. On a single copy of the diagram, draw:
    1. the locus \(L _ { 1 }\) of points satisfying \(\left| z - z _ { 2 } \right| = \left| z - z _ { 1 } \right|\);
    2. the locus \(L _ { 2 }\) of points satisfying \(\arg \left( z - z _ { 2 } \right) = \arg z _ { 1 }\).
  3. Find, in terms of \(z _ { 1 }\), the complex number representing the point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
AQA FP2 2007 June Q6
7 marks Standard +0.3
6
  1. Show that $$\left( 1 - \frac { 1 } { ( k + 1 ) ^ { 2 } } \right) \times \frac { k + 1 } { 2 k } = \frac { k + 2 } { 2 ( k + 1 ) }$$
  2. Prove by induction that for all integers \(n \geqslant 2\) $$\left( 1 - \frac { 1 } { 2 ^ { 2 } } \right) \left( 1 - \frac { 1 } { 3 ^ { 2 } } \right) \left( 1 - \frac { 1 } { 4 ^ { 2 } } \right) \ldots \left( 1 - \frac { 1 } { n ^ { 2 } } \right) = \frac { n + 1 } { 2 n }$$
AQA FP2 2007 June Q7
15 marks Challenging +1.2
7 A curve has equation \(y = 4 \sqrt { x }\).
  1. Show that the length of arc \(s\) of the curve between the points where \(x = 0\) and \(x = 1\) is given by $$s = \int _ { 0 } ^ { 1 } \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x$$
    1. Use the substitution \(x = 4 \sinh ^ { 2 } \theta\) to show that $$\int \sqrt { \frac { x + 4 } { x } } \mathrm {~d} x = \int 8 \cosh ^ { 2 } \theta \mathrm {~d} \theta$$
    2. Hence show that $$s = 4 \sinh ^ { - 1 } 0.5 + \sqrt { 5 }$$
AQA FP2 2007 June Q8
13 marks Challenging +1.2
8
    1. Given that \(z ^ { 6 } - 4 z ^ { 3 } + 8 = 0\), show that \(z ^ { 3 } = 2 \pm 2 \mathrm { i }\).
    2. Hence solve the equation $$z ^ { 6 } - 4 z ^ { 3 } + 8 = 0$$ giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
  2. Show that, for any real values of \(k\) and \(\theta\), $$\left( z - k \mathrm { e } ^ { \mathrm { i } \theta } \right) \left( z - k \mathrm { e } ^ { - \mathrm { i } \theta } \right) = z ^ { 2 } - 2 k z \cos \theta + k ^ { 2 }$$
  3. Express \(z ^ { 6 } - 4 z ^ { 3 } + 8\) as the product of three quadratic factors with real coefficients.
AQA FP2 2009 June Q1
8 marks Standard +0.8
1 Given that \(z = 2 \mathrm { e } ^ { \frac { \pi \mathrm { i } } { 12 } }\) satisfies the equation $$z ^ { 4 } = a ( 1 + \sqrt { 3 } i )$$ where \(a\) is real:
  1. find the value of \(a\);
  2. find the other three roots of this equation, giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
AQA FP2 2009 June Q2
8 marks Standard +0.8
2
  1. Given that $$\frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { A } { 2 r - 1 } + \frac { B } { 2 r + 1 }$$ find the values of \(A\) and \(B\).
  2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 } = \frac { n } { 2 n + 1 }$$
  3. Find the least value of \(n\) for which \(\sum _ { r = 1 } ^ { n } \frac { 1 } { 4 r ^ { 2 } - 1 }\) differs from 0.5 by less than 0.001 .
AQA FP2 2009 June Q3
8 marks Standard +0.3
3 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + 25 z + q = 0$$ where \(p\) and \(q\) are real, has a root \(\alpha = 2 - 3 \mathrm { i }\).
  1. Write down another non-real root, \(\beta\), of this equation.
  2. Find:
    1. the value of \(\alpha \beta\);
    2. the third root, \(\gamma\), of the equation;
    3. the values of \(p\) and \(q\).
AQA FP2 2009 June Q4
15 marks Standard +0.3
4
  1. Sketch the graph of \(y = \tanh x\).
  2. Given that \(u = \tanh x\), use the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that $$x = \frac { 1 } { 2 } \ln \left( \frac { 1 + u } { 1 - u } \right)$$
    1. Show that the equation $$3 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 5$$ can be written as $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$
    2. Show that the equation $$3 \tanh ^ { 2 } x - 7 \tanh x + 2 = 0$$ has only one solution for \(x\).
      Find this solution in the form \(\frac { 1 } { 2 } \ln a\), where \(a\) is an integer.
AQA FP2 2009 June Q5
12 marks Standard +0.3
5
  1. Prove by induction that, if \(n\) is a positive integer, $$( \cos \theta + \mathrm { i } \sin \theta ) ^ { n } = \cos n \theta + \mathrm { i } \sin n \theta$$
  2. Hence, given that $$z = \cos \theta + \mathrm { i } \sin \theta$$ show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
  3. Given further that \(z + \frac { 1 } { z } = \sqrt { 2 }\), find the value of $$z ^ { 10 } + \frac { 1 } { z ^ { 10 } }$$
AQA FP2 2009 June Q6
12 marks Standard +0.8
6
  1. Two points, \(A\) and \(B\), on an Argand diagram are represented by the complex numbers \(2 + 3 \mathrm { i }\) and \(- 4 - 5 \mathrm { i }\) respectively. Given that the points \(A\) and \(B\) are at the ends of a diameter of a circle \(C _ { 1 }\), express the equation of \(C _ { 1 }\) in the form \(\left| z - z _ { 0 } \right| = k\).
  2. A second circle, \(C _ { 2 }\), is represented on the Argand diagram by the equation \(| z - 5 + 4 \mathrm { i } | = 4\). Sketch on one Argand diagram both \(C _ { 1 }\) and \(C _ { 2 }\).
  3. The points representing the complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) lie on \(C _ { 1 }\) and \(C _ { 2 }\) respectively and are such that \(\left| z _ { 1 } - z _ { 2 } \right|\) has its maximum value. Find this maximum value, giving your answer in the form \(a + b \sqrt { 5 }\).
AQA FP2 2009 June Q7
12 marks Challenging +1.8
7 The diagram shows a curve which starts from the point \(A\) with coordinates ( 0,2 ). The curve is such that, at every point \(P\) on the curve, $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 2 } s$$ where \(s\) is the length of the \(\operatorname { arc } A P\). \includegraphics[max width=\textwidth, alt={}, center]{587aac5c-fbc2-41d2-b1b3-16f3f7851d9d-4_399_764_1324_605}
    1. Show that $$\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \sqrt { 4 + s ^ { 2 } }$$ (3 marks)
    2. Hence show that $$s = 2 \sinh \frac { x } { 2 }$$
    3. Hence find the cartesian equation of the curve.
  2. Show that $$y ^ { 2 } = 4 + s ^ { 2 }$$
AQA FP2 2015 June Q1
5 marks Standard +0.8
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]
AQA FP2 2015 June Q2
11 marks Standard +0.3
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. \includegraphics[max width=\textwidth, alt={}, center]{bc3aaed2-4aef-4aec-b657-098b1e581e55-04_855_1447_920_324}
AQA FP2 2015 June Q3
9 marks Challenging +1.2
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]
AQA FP2 2015 June Q4
7 marks Standard +0.8
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 \(n \geqslant 1\).
AQA FP2 2015 June Q5
9 marks Standard +0.3
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)}
      \includegraphics[max width=\textwidth, alt={}]{bc3aaed2-4aef-4aec-b657-098b1e581e55-10_1173_1242_1217_463}
AQA FP2 2015 June Q6
8 marks Challenging +1.2
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]