Questions — AQA FP2 (145 questions)

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AQA FP2 2008 June Q2
7 marks Standard +0.8
2
  1. Given that $$\frac { 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { A } { r ( r + 1 ) } + \frac { B } { ( r + 1 ) ( r + 2 ) }$$ show that \(A = \frac { 1 } { 2 }\) and find the value of \(B\).
  2. Use the method of differences to find $$\sum _ { r = 10 } ^ { 98 } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$ giving your answer as a rational number.
AQA FP2 2008 June Q3
12 marks Standard +0.8
3 The cubic equation $$z ^ { 3 } + q z + ( 18 - 12 i ) = 0$$ 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 \(\beta + \gamma = 2\), find the value of:
    1. \(\alpha\);
    2. \(\quad \beta \gamma\);
    3. \(q\).
  3. Given that \(\beta\) is of the form \(k \mathrm { i }\), where \(k\) is real, find \(\beta\) and \(\gamma\).
AQA FP2 2008 June Q4
12 marks Standard +0.3
4
  1. A circle \(C\) in the Argand diagram has equation $$| z + 5 - \mathrm { i } | = \sqrt { 2 }$$ Write down its radius and the complex number representing its centre.
  2. A half-line \(L\) in the Argand diagram has equation $$\arg ( z + 2 \mathrm { i } ) = \frac { 3 \pi } { 4 }$$ Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) lies on \(L\).
    1. Show that \(z _ { 1 } = - 4 + 2 \mathrm { i }\) also lies on \(C\).
    2. Hence show that \(L\) touches \(C\).
    3. Sketch \(L\) and \(C\) on one Argand diagram.
  4. The complex number \(z _ { 2 }\) lies on \(C\) and is such that \(\arg \left( z _ { 2 } + 2 \mathrm { i } \right)\) has as great a value as possible. Indicate the position of \(z _ { 2 }\) on your sketch.
AQA FP2 2008 June Q5
10 marks Challenging +1.3
5
  1. Use the definition \(\cosh x = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) to show that \(\cosh 2 x = 2 \cosh ^ { 2 } x - 1\).
    (2 marks)
    1. The arc of the curve \(y = \cosh x\) between \(x = 0\) and \(x = \ln a\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that \(S\), the surface area generated, is given by $$S = 2 \pi \int _ { 0 } ^ { \ln a } \cosh ^ { 2 } x \mathrm {~d} x$$
    2. Hence show that $$S = \pi \left( \ln a + \frac { a ^ { 4 } - 1 } { 4 a ^ { 2 } } \right)$$
AQA FP2 2008 June Q6
5 marks Standard +0.8
6 By using the substitution \(u = x - 2\), or otherwise, find the exact value of $$\int _ { - 1 } ^ { 5 } \frac { \mathrm {~d} x } { \sqrt { 32 + 4 x - x ^ { 2 } } }$$
AQA FP2 2008 June Q7
9 marks Standard +0.8
7
  1. Explain why \(n ( n + 1 )\) is a multiple of 2 when \(n\) is an integer.
    1. Given that $$\mathrm { f } ( n ) = n \left( n ^ { 2 } + 5 \right)$$ show that \(\mathrm { f } ( k + 1 ) - \mathrm { f } ( k )\), where \(k\) is a positive integer, is a multiple of 6 .
    2. Prove by induction that \(\mathrm { f } ( n )\) is a multiple of 6 for all integers \(n \geqslant 1\).
AQA FP2 2008 June Q8
14 marks Challenging +1.2
8
    1. Expand $$\left( z + \frac { 1 } { z } \right) \left( z - \frac { 1 } { z } \right)$$
    2. Hence, or otherwise, expand $$\left( z + \frac { 1 } { z } \right) ^ { 4 } \left( z - \frac { 1 } { z } \right) ^ { 2 }$$
    1. Use De Moivre's theorem to show that if \(z = \cos \theta + \mathrm { i } \sin \theta\) then $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
    2. Write down a corresponding result for \(z ^ { n } - \frac { 1 } { z ^ { n } }\).
  3. Hence express \(\cos ^ { 4 } \theta \sin ^ { 2 } \theta\) in the form $$A \cos 6 \theta + B \cos 4 \theta + C \cos 2 \theta + D$$ where \(A , B , C\) and \(D\) are rational numbers.
  4. Find \(\int \cos ^ { 4 } \theta \sin ^ { 2 } \theta d \theta\).
AQA FP2 2010 June Q1
9 marks Standard +0.3
1
  1. Show that $$9 \sinh x - \cosh x = 4 \mathrm { e } ^ { x } - 5 \mathrm { e } ^ { - x }$$
  2. Given that $$9 \sinh x - \cosh x = 8$$ find the exact value of \(\tanh x\).
AQA FP2 2010 June Q2
8 marks Standard +0.3
2
  1. Express \(\frac { 1 } { r ( r + 2 ) }\) in partial fractions.
  2. Use the method of differences to find $$\sum _ { r = 1 } ^ { 48 } \frac { 1 } { r ( r + 2 ) }$$ giving your answer as a rational number.
AQA FP2 2010 June Q3
9 marks Standard +0.3
3 Two loci, \(L _ { 1 }\) and \(L _ { 2 }\), in an Argand diagram are given by $$\begin{aligned} & L _ { 1 } : | z + 1 + 3 \mathrm { i } | = | z - 5 - 7 \mathrm { i } | \\ & L _ { 2 } : \arg z = \frac { \pi } { 4 } \end{aligned}$$
  1. Verify that the point represented by the complex number \(2 + 2 \mathrm { i }\) is a point of intersection of \(L _ { 1 }\) and \(L _ { 2 }\).
  2. Sketch \(L _ { 1 }\) and \(L _ { 2 }\) on one Argand diagram.
  3. Shade on your Argand diagram the region satisfying
    both $$| z + 1 + 3 i | \leqslant | z - 5 - 7 i |$$ and $$\frac { \pi } { 4 } \leqslant \arg z \leqslant \frac { \pi } { 2 }$$
AQA FP2 2010 June Q4
13 marks Standard +0.8
4 The roots of the cubic equation $$z ^ { 3 } - 2 z ^ { 2 } + p z + 10 = 0$$ are \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 4\).
  1. Write down the value of \(\alpha + \beta + \gamma\).
    1. Explain why \(\alpha ^ { 3 } - 2 \alpha ^ { 2 } + p \alpha + 10 = 0\).
    2. Hence show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = p + 13$$
    3. Deduce that \(p = - 3\).
    1. Find the real root \(\alpha\) of the cubic equation \(z ^ { 3 } - 2 z ^ { 2 } - 3 z + 10 = 0\).
    2. Find the values of \(\beta\) and \(\gamma\).
AQA FP2 2010 June Q5
18 marks Standard +0.8
5
  1. Using the identities $$\cosh ^ { 2 } t - \sinh ^ { 2 } t = 1 , \quad \tanh t = \frac { \sinh t } { \cosh t } \quad \text { and } \quad \operatorname { sech } t = \frac { 1 } { \cosh t }$$ show that:
    1. \(\tanh ^ { 2 } t + \operatorname { sech } ^ { 2 } t = 1\);
    2. \(\frac { \mathrm { d } } { \mathrm { d } t } ( \tanh t ) = \operatorname { sech } ^ { 2 } t\);
    3. \(\frac { \mathrm { d } } { \mathrm { d } t } ( \operatorname { sech } t ) = - \operatorname { sech } t \tanh t\).
  2. A curve \(C\) is given parametrically by $$x = \operatorname { sech } t , y = 4 - \tanh t$$
    1. Show that the arc length, \(s\), of \(C\) between the points where \(t = 0\) and \(t = \frac { 1 } { 2 } \ln 3\) is given by $$s = \int _ { 0 } ^ { \frac { 1 } { 2 } \ln 3 } \operatorname { sech } t \mathrm {~d} t$$
    2. Using the substitution \(u = \mathrm { e } ^ { t }\), find the exact value of \(s\).
      REFERENCE
      \includegraphics[max width=\textwidth, alt={}]{77a28ee7-dba2-4aea-8858-9da430383108-6_24_77_1747_166}
      \(\_\_\_\_\)
      \(\_\_\_\_\)
      \includegraphics[max width=\textwidth, alt={}]{77a28ee7-dba2-4aea-8858-9da430383108-6_91_114_2509_162}
      \includegraphics[max width=\textwidth, alt={}]{77a28ee7-dba2-4aea-8858-9da430383108-6_44_1678_2661_162
      }
AQA FP2 2010 June Q6
8 marks Challenging +1.2
6
  1. Show that \(\frac { 1 } { ( k + 2 ) ! } - \frac { k + 1 } { ( k + 3 ) ! } = \frac { 2 } { ( k + 3 ) ! }\).
  2. Prove by induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \frac { r \times 2 ^ { r } } { ( r + 2 ) ! } = 1 - \frac { 2 ^ { n + 1 } } { ( n + 2 ) ! }$$ (6 marks)
    \includegraphics[max width=\textwidth, alt={}]{77a28ee7-dba2-4aea-8858-9da430383108-7_2010_1711_693_152}
AQA FP2 2010 June Q7
10 marks Standard +0.3
7
    1. Express each of the numbers \(1 + \sqrt { 3 } \mathrm { i }\) and \(1 - \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\).
    2. Hence express $$( 1 + \sqrt { 3 } i ) ^ { 8 } ( 1 - i ) ^ { 5 }$$ in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\).
  2. Solve the equation $$z ^ { 3 } = ( 1 + \sqrt { 3 } \mathrm { i } ) ^ { 8 } ( 1 - \mathrm { i } ) ^ { 5 }$$ giving your answers in the form \(a \sqrt { 2 } \mathrm { e } ^ { \mathrm { i } \theta }\), where \(a\) is a positive integer and \(- \pi < \theta \leqslant \pi\).
    REFERENCE
    ...................................................................................................................................................
    ..........\(\_\_\_\_\)
    \end{document}
AQA FP2 2012 June Q1
7 marks Standard +0.3
1
  1. Sketch the curve \(y = \cosh x\).
  2. Solve the equation $$6 \cosh ^ { 2 } x - 7 \cosh x - 5 = 0$$ giving your answers in logarithmic form.
AQA FP2 2012 June Q2
7 marks Standard +0.3
2
  1. Draw on the Argand diagram below:
    1. the locus of points for which $$| z - 2 - 3 \mathrm { i } | = 2$$
    2. the locus of points for which $$| z + 2 - \mathrm { i } | = | z - 2 |$$
  2. Indicate on your diagram the points satisfying both $$| z - 2 - 3 \mathrm { i } | = 2$$ and $$| z + 2 - \mathrm { i } | \leqslant | z - 2 |$$ (l mark) \includegraphics[max width=\textwidth, alt={}, center]{ff63460d-0fa1-437d-bc08-3e7ce809e32b-3_1404_1431_1043_319}
AQA FP2 2012 June Q3
6 marks Standard +0.8
3
  1. Show that $$\frac { 2 ^ { r + 1 } } { r + 2 } - \frac { 2 ^ { r } } { r + 1 } = \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$
  2. Hence find $$\sum _ { r = 1 } ^ { 30 } \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$ giving your answer in the form \(2 ^ { n } - 1\), where \(n\) is an integer.
AQA FP2 2012 June Q4
13 marks Standard +0.8
4 The cubic equation $$z ^ { 3 } + p z + q = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
    1. Write down the value of \(\alpha + \beta + \gamma\).
    2. Express \(\alpha \beta \gamma\) in terms of \(q\).
  2. Show that $$\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 \alpha \beta \gamma$$
  3. Given that \(\alpha = 4 + 7 \mathrm { i }\) and that \(p\) and \(q\) are real, find the values of:
    1. \(\beta\) and \(\gamma\);
    2. \(p\) and \(q\).
  4. Find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).
AQA FP2 2012 June Q5
6 marks Standard +0.8
5 The function f , where \(\mathrm { f } ( x ) = \sec x\), has domain \(0 \leqslant x < \frac { \pi } { 2 }\) and has inverse function \(\mathrm { f } ^ { - 1 }\), where \(\mathrm { f } ^ { - 1 } ( x ) = \sec ^ { - 1 } x\).
  1. Show that $$\sec ^ { - 1 } x = \cos ^ { - 1 } \frac { 1 } { x }$$
  2. Hence show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left( \sec ^ { - 1 } x \right) = \frac { 1 } { \sqrt { x ^ { 4 } - x ^ { 2 } } }$$
AQA FP2 2012 June Q6
13 marks Challenging +1.8
6
  1. Show that $$\frac { 1 } { 4 } ( \cosh 4 x + 2 \cosh 2 x + 1 ) = \cosh ^ { 2 } x \cosh 2 x$$
  2. Show that, if \(y = \cosh ^ { 2 } x\), then $$1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \cosh ^ { 2 } 2 x$$
  3. The arc of the curve \(y = \cosh ^ { 2 } x\) between the points where \(x = 0\) and \(x = \ln 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Show that the area \(S\) of the curved surface formed is given by $$S = \frac { \pi } { 256 } ( a \ln 2 + b )$$ where \(a\) and \(b\) are integers.
AQA FP2 2012 June Q7
9 marks Standard +0.8
7
  1. Prove by induction that, for all integers \(n \geqslant 1\), $$\frac { 3 } { 1 ^ { 2 } \times 2 ^ { 2 } } + \frac { 5 } { 2 ^ { 2 } \times 3 ^ { 2 } } + \frac { 7 } { 3 ^ { 2 } \times 4 ^ { 2 } } + \ldots + \frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } } = 1 - \frac { 1 } { ( n + 1 ) ^ { 2 } }$$
  2. Find the smallest integer \(n\) for which the sum of the series differs from 1 by less than \(10 ^ { - 5 }\).
AQA FP2 2012 June Q8
14 marks Challenging +1.2
8
  1. Use De Moivre's Theorem to show that, if \(z = \cos \theta + \mathrm { i } \sin \theta\), then $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta$$
    1. Expand \(\left( z ^ { 2 } + \frac { 1 } { z ^ { 2 } } \right) ^ { 4 }\).
    2. Show that $$\cos ^ { 4 } 2 \theta = A \cos 8 \theta + B \cos 4 \theta + C$$ where \(A , B\) and \(C\) are rational numbers.
  3. Hence solve the equation $$8 \cos ^ { 4 } 2 \theta = \cos 8 \theta + 5$$ for \(0 \leqslant \theta \leqslant \pi\), giving each solution in the form \(k \pi\).
  4. Show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } \cos ^ { 4 } 2 \theta d \theta = \frac { 3 \pi } { 16 }$$
AQA FP2 2013 June Q1
7 marks Standard +0.3
1
  1. Sketch on an Argand diagram the locus of points satisfying the equation $$| z - 6 \mathrm { i } | = 3$$
  2. It is given that \(z\) satisfies the equation \(| z - 6 \mathrm { i } | = 3\).
    1. Write down the greatest possible value of \(| z |\).
    2. Find the greatest possible value of \(\arg z\), giving your answer in the form \(p \pi\), where \(- 1 < p \leqslant 1\).
AQA FP2 2013 June Q2
9 marks Standard +0.8
2
    1. Sketch on the axes below the graphs of \(y = \sinh x\) and \(y = \cosh x\).
    2. Use your graphs to explain why the equation $$( k + \sinh x ) \cosh x = 0$$ where \(k\) is a constant, has exactly one solution.
  2. A curve \(C\) has equation \(y = 6 \sinh x + \cosh ^ { 2 } x\). Show that \(C\) has only one stationary point and show that its \(y\)-coordinate is an integer. \includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_704_1416_171} \includegraphics[max width=\textwidth, alt={}, center]{53d742f4-923b-478c-8ae6-ada6c0bb4a7e-2_560_711_1416_964}
AQA FP2 2013 June Q3
6 marks Standard +0.8
3 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is defined by $$u _ { 1 } = 2 , \quad u _ { n + 1 } = \frac { 5 u _ { n } - 3 } { 3 u _ { n } - 1 }$$ Prove by induction that, for all integers \(n \geqslant 1\), $$u _ { n } = \frac { 3 n + 1 } { 3 n - 1 }$$ (6 marks)