Questions FP2 (1157 questions)

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AQA FP2 2012 January Q2
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
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
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
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
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
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
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
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
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
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
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
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.
AQA FP2 2013 January Q6
6 A curve is defined parametrically by $$x = t ^ { 3 } + 5 , \quad y = 6 t ^ { 2 } - 1$$ The arc length between the points where \(t = 0\) and \(t = 3\) on the curve is \(s\).
  1. Show that \(s = \int _ { 0 } ^ { 3 } 3 t \sqrt { t ^ { 2 } + A } \mathrm {~d} t\), stating the value of the constant \(A\).
  2. Hence show that \(s = 61\).
    \(7 \quad\) The polynomial \(\mathrm { p } ( n )\) is given by \(\mathrm { p } ( n ) = ( n - 1 ) ^ { 3 } + n ^ { 3 } + ( n + 1 ) ^ { 3 }\).
    1. Show that \(\mathrm { p } ( k + 1 ) - \mathrm { p } ( k )\), where \(k\) is a positive integer, is a multiple of 9 .
    2. Prove by induction that \(\mathrm { p } ( n )\) is a multiple of 9 for all integers \(n \geqslant 1\).
  4. Using the result from part (a)(ii), show that \(n \left( n ^ { 2 } + 2 \right)\) is a multiple of 3 for any positive integer \(n\).
AQA FP2 2013 January Q8
8
  1. Express \(- 4 + 4 \sqrt { 3 } \mathrm { i }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    1. Solve the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    2. The roots of the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\) are represented by the points \(P , Q\) and \(R\) on an Argand diagram. Find the area of the triangle \(P Q R\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is an integer.
  3. By considering the roots of the equation \(z ^ { 3 } = - 4 + 4 \sqrt { 3 } \mathrm { i }\), show that $$\cos \frac { 2 \pi } { 9 } + \cos \frac { 4 \pi } { 9 } + \cos \frac { 8 \pi } { 9 } = 0$$
AQA FP2 2008 June Q1
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
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
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
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
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
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
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
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
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
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
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 $$\frac { \pi } { 4 } \leqslant \arg z \leqslant \frac { \pi } { 2 }$$