Questions — AQA FP2 (142 questions)

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AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 SPS SPS ASFM SPS ASFM Mechanics SPS ASFM Pure SPS ASFM Statistics SPS FM SPS FM Mechanics SPS FM Pure SPS FM Statistics SPS SM SPS SM Mechanics SPS SM Pure SPS SM Statistics WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
AQA FP2 2011 June Q6
6
  1. Show that $$( k + 1 ) \left( 4 ( k + 1 ) ^ { 2 } - 1 \right) = 4 k ^ { 3 } + 12 k ^ { 2 } + 11 k + 3$$
  2. Prove by induction that, for all integers \(n \geqslant 1\), $$1 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } + \ldots + ( 2 n - 1 ) ^ { 2 } = \frac { 1 } { 3 } n \left( 4 n ^ { 2 } - 1 \right)$$
AQA FP2 2011 June Q7
7
    1. Use de Moivre's Theorem to show that $$\cos 5 \theta = \cos ^ { 5 } \theta - 10 \cos ^ { 3 } \theta \sin ^ { 2 } \theta + 5 \cos \theta \sin ^ { 4 } \theta$$ and find a similar expression for \(\sin 5 \theta\).
    2. Deduce that $$\tan 5 \theta = \frac { \tan \theta \left( 5 - 10 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta \right) } { 1 - 10 \tan ^ { 2 } \theta + 5 \tan ^ { 4 } \theta }$$
  2. Explain why \(t = \tan \frac { \pi } { 5 }\) is a root of the equation $$t ^ { 4 } - 10 t ^ { 2 } + 5 = 0$$ and write down the three other roots of this equation in trigonometrical form.
    (3 marks)
  3. Deduce that $$\tan \frac { \pi } { 5 } \tan \frac { 2 \pi } { 5 } = \sqrt { 5 }$$
AQA FP2 2012 June Q1
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
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
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
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
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
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
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
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
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
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
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)
AQA FP2 2013 June Q4
4
  1. Given that \(\mathrm { f } ( r ) = r ^ { 2 } \left( 2 r ^ { 2 } - 1 \right)\), show that $$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = ( 2 r - 1 ) ^ { 3 }$$
  2. Use the method of differences to show that $$\sum _ { r = n + 1 } ^ { 2 n } ( 2 r - 1 ) ^ { 3 } = 3 n ^ { 2 } \left( 10 n ^ { 2 } - 1 \right)$$ (4 marks)
AQA FP2 2013 June Q5
5 The cubic equation $$z ^ { 3 } + p z ^ { 2 } + q z + 37 - 36 \mathrm { i } = 0$$ where \(p\) and \(q\) are constants, has three complex roots, \(\alpha , \beta\) and \(\gamma\). It is given that \(\beta = - 2 + 3 \mathrm { i }\) and \(\gamma = 1 + 2 \mathrm { i }\).
    1. Write down the value of \(\alpha \beta \gamma\).
    2. Hence show that \(( 8 + \mathrm { i } ) \alpha = 37 - 36 \mathrm { i }\).
    3. Hence find \(\alpha\), giving your answer in the form \(m + n \mathrm { i }\), where \(m\) and \(n\) are integers.
  2. Find the value of \(p\).
  3. Find the value of the complex number \(q\).
AQA FP2 2013 June Q6
6
  1. Show that \(\frac { 1 } { 5 \cosh x - 3 \sinh x } = \frac { \mathrm { e } ^ { x } } { m + \mathrm { e } ^ { 2 x } }\), where \(m\) is an integer.
  2. Use the substitution \(u = \mathrm { e } ^ { x }\) to show that $$\int _ { 0 } ^ { \ln 2 } \frac { 1 } { 5 \cosh x - 3 \sinh x } \mathrm {~d} x = \frac { \pi } { 8 } - \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 } \right)$$
AQA FP2 2013 June Q7
7
    1. Show that $$\frac { \mathrm { d } } { \mathrm {~d} u } \left( 2 u \sqrt { 1 + 4 u ^ { 2 } } + \sinh ^ { - 1 } 2 u \right) = k \sqrt { 1 + 4 u ^ { 2 } }$$ where \(k\) is an integer.
    2. Hence show that $$\int _ { 0 } ^ { 1 } \sqrt { 1 + 4 u ^ { 2 } } \mathrm {~d} u = p \sqrt { 5 } + q \sinh ^ { - 1 } 2$$ where \(p\) and \(q\) are rational numbers.
  2. The arc of the curve with equation \(y = \frac { 1 } { 2 } \cos 4 x\) between the points where \(x = 0\) and \(x = \frac { \pi } { 8 }\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
    1. Show that the area \(S\) of the curved surface formed is given by $$S = \pi \int _ { 0 } ^ { \frac { \pi } { 8 } } \cos 4 x \sqrt { 1 + 4 \sin ^ { 2 } 4 x } \mathrm {~d} x$$
    2. Use the substitution \(u = \sin 4 x\) to find the exact value of \(S\).
AQA FP2 2013 June Q8
8
    1. Use de Moivre's theorem to show that $$\cos 4 \theta = \cos ^ { 4 } \theta - 6 \cos ^ { 2 } \theta \sin ^ { 2 } \theta + \sin ^ { 4 } \theta$$ and find a similar expression for \(\sin 4 \theta\).
    2. Deduce that $$\tan 4 \theta = \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta }$$
  2. Explain why \(t = \tan \frac { \pi } { 16 }\) is a root of the equation $$t ^ { 4 } + 4 t ^ { 3 } - 6 t ^ { 2 } - 4 t + 1 = 0$$ and write down the three other roots in trigonometric form.
  3. Hence show that $$\tan ^ { 2 } \frac { \pi } { 16 } + \tan ^ { 2 } \frac { 3 \pi } { 16 } + \tan ^ { 2 } \frac { 5 \pi } { 16 } + \tan ^ { 2 } \frac { 7 \pi } { 16 } = 28$$
AQA FP2 2014 June Q1
7 marks
1
  1. Express - 9 i in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    [0pt] [2 marks]
  2. Solve the equation \(z ^ { 4 } + 9 \mathrm { i } = 0\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    [0pt] [5 marks]
AQA FP2 2014 June Q2
3 marks
2
  1. Sketch, on the Argand diagram below, the locus \(L\) of points satisfying $$\arg ( z - 2 \mathrm { i } ) = \frac { 2 \pi } { 3 }$$
    1. A circle \(C\), of radius 3, has its centre lying on \(L\) and touches the line \(\operatorname { Im } ( z ) = 2\). Sketch \(C\) on the Argand diagram used in part (a).
    2. Find the centre of \(C\), giving your answer in the form \(a + b \mathrm { i }\).
      [0pt] [3 marks]
AQA FP2 2014 June Q3
3
  1. Express \(( k + 1 ) ^ { 2 } + 5 ( k + 1 ) + 8\) in the form \(k ^ { 2 } + a k + b\), where \(a\) and \(b\) are constants.
  2. Prove by induction that, for all integers \(n \geqslant 1\), $$\sum _ { r = 1 } ^ { n } r ( r + 1 ) \left( \frac { 1 } { 2 } \right) ^ { r - 1 } = 16 - \left( n ^ { 2 } + 5 n + 8 \right) \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$$
AQA FP2 2014 June Q4
4 The roots of the equation $$z ^ { 3 } + 2 z ^ { 2 } + 3 z - 4 = 0$$ are \(\alpha , \beta\) and \(\gamma\).
    1. Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
    2. Hence show that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 2\).
  2. Find the value of:
    1. \(( \alpha + \beta ) ( \beta + \gamma ) + ( \beta + \gamma ) ( \gamma + \alpha ) + ( \gamma + \alpha ) ( \alpha + \beta )\);
    2. \(( \alpha + \beta ) ( \beta + \gamma ) ( \gamma + \alpha )\).
  3. Find a cubic equation whose roots are \(\alpha + \beta , \beta + \gamma\) and \(\gamma + \alpha\).
AQA FP2 2014 June Q5
2 marks
5
  1. Using the definition \(\sinh \theta = \frac { 1 } { 2 } \left( \mathrm { e } ^ { \theta } - \mathrm { e } ^ { - \theta } \right)\), prove the identity $$4 \sinh ^ { 3 } \theta + 3 \sinh \theta = \sinh 3 \theta$$
  2. Given that \(x = \sinh \theta\) and \(16 x ^ { 3 } + 12 x - 3 = 0\), find the value of \(\theta\) in terms of a natural logarithm.
  3. Hence find the real root of the equation \(16 x ^ { 3 } + 12 x - 3 = 0\), giving your answer in the form \(2 ^ { p } - 2 ^ { q }\), where \(p\) and \(q\) are rational numbers.
    [0pt] [2 marks]
AQA FP2 2014 June Q6
6
    1. Use De Moivre's Theorem to show that if \(z = \cos \theta + \mathrm { i } \sin \theta\), then $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$
    2. Write down a similar expression for \(z ^ { n } + \frac { 1 } { z ^ { n } }\).
    1. Expand \(\left( z - \frac { 1 } { z } \right) ^ { 2 } \left( z + \frac { 1 } { z } \right) ^ { 2 }\) in terms of \(z\).
    2. Hence show that $$8 \sin ^ { 2 } \theta \cos ^ { 2 } \theta = A + B \cos 4 \theta$$ where \(A\) and \(B\) are integers.
  3. Hence, by means of the substitution \(x = 2 \sin \theta\), find the exact value of $$\int _ { 1 } ^ { 2 } x ^ { 2 } \sqrt { 4 - x ^ { 2 } } \mathrm {~d} x$$ \includegraphics[max width=\textwidth, alt={}, center]{5287255f-5ac4-401a-b850-758257412ff7-14_1180_1707_1525_153}
AQA FP2 2014 June Q7
7 marks
7
  1. Given that \(y = \tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right)\) and \(x \neq 1\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }\).
    [0pt] [4 marks]
  2. Hence, given that \(x < 1\), show that \(\tan ^ { - 1 } \left( \frac { 1 + x } { 1 - x } \right) - \tan ^ { - 1 } x = \frac { \pi } { 4 }\).
    [0pt] [3 marks]