Questions FP2 (1279 questions)

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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 Challenging +1.2
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 2008 June Q1
6 marks Standard +0.3
1
  1. Express $$5 \sinh x + \cosh x$$ in the form \(A \mathrm { e } ^ { x } + B \mathrm { e } ^ { - x }\), where \(A\) and \(B\) are integers.
  2. Solve the equation $$5 \sinh x + \cosh x + 5 = 0$$ giving your answer in the form \(\ln a\), where \(a\) is a rational number.
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. \(\frac { \mathrm { d } } { \mathrm { d } t } ( \operatorname { sech } t ) = - \operatorname { sech } t \tanh t\).
    4. 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
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      \(\_\_\_\_\)
      \(\_\_\_\_\)
      \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 \(\frac { 1 } { \gamma }\).