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AQA FP2 2013 June Q4
7 marks Challenging +1.2
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
9 marks Standard +0.3
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
8 marks Challenging +1.2
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
12 marks Challenging +1.8
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
17 marks Challenging +1.2
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 Standard +0.3
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
8 marks Standard +0.8
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
7 marks Challenging +1.2
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
14 marks Standard +0.8
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
9 marks Challenging +1.2
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
12 marks Challenging +1.2
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 Challenging +1.2
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]
AQA FP2 2014 June Q8
11 marks Challenging +1.8
8 A curve has equation \(y = 2 \sqrt { x - 1 }\), where \(x > 1\). The length of the arc of the curve between the points on the curve where \(x = 2\) and \(x = 9\) is denoted by \(s\).
  1. Show that \(s = \int _ { 2 } ^ { 9 } \sqrt { \frac { x } { x - 1 } } \mathrm {~d} x\).
    1. Show that \(\cosh ^ { - 1 } 3 = 2 \ln ( 1 + \sqrt { 2 } )\).
    2. Use the substitution \(x = \cosh ^ { 2 } \theta\) to show that $$s = m \sqrt { 2 } + \ln ( 1 + \sqrt { 2 } )$$ where \(m\) is an integer.
      [0pt] [6 marks]
      \includegraphics[max width=\textwidth, alt={}]{5287255f-5ac4-401a-b850-758257412ff7-20_1638_1709_1069_153}
      \includegraphics[max width=\textwidth, alt={}, center]{5287255f-5ac4-401a-b850-758257412ff7-24_2489_1728_221_141}
OCR MEI Further Pure Core AS 2019 June Q1
3 marks Standard +0.3
1 In this question you must show detailed reasoning.
Find \(\sum _ { r = 1 } ^ { 100 } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\), giving your answer correct to 4 decimal places.
OCR MEI Further Pure Core AS 2019 June Q2
3 marks Standard +0.3
2 The roots of the equation \(3 x ^ { 2 } - x + 2 = 0\) are \(\alpha\) and \(\beta\).
Find a quadratic equation with integer coefficients whose roots are \(2 \alpha - 3\) and \(2 \beta - 3\).
OCR MEI Further Pure Core AS 2019 June Q3
6 marks Moderate -0.3
3 In this question you must show detailed reasoning. \(\mathbf { A }\) and \(\mathbf { B }\) are matrices such that \(\mathbf { B } ^ { - 1 } \mathbf { A } ^ { - 1 } = \left( \begin{array} { r r } 2 & 1 \\ - 1 & 1 \end{array} \right)\).
  1. Find \(\mathbf { A B }\).
  2. Given that \(\mathbf { A } = \left( \begin{array} { l l } \frac { 1 } { 3 } & 1 \\ 0 & 1 \end{array} \right)\), find \(\mathbf { B }\).
OCR MEI Further Pure Core AS 2019 June Q4
8 marks Standard +0.3
4
  1. Find \(\mathbf { M } ^ { - 1 }\), where \(\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & 3 \\ - 1 & 1 & 2 \\ - 2 & 1 & 2 \end{array} \right)\).
  2. Hence find, in terms of the constant \(k\), the point of intersection of the planes $$\begin{aligned} x + 2 y + 3 z & = 19 \\ - x + y + 2 z & = 4 \\ - 2 x + y + 2 z & = k \end{aligned}$$
  3. In this question you must show detailed reasoning. Find the acute angle between the planes \(x + 2 y + 3 z = 19\) and \(- x + y + 2 z = 4\).
OCR MEI Further Pure Core AS 2019 June Q5
6 marks Moderate -0.3
5 Prove by induction that, for all positive integers \(n , \sum _ { r = 1 } ^ { n } \frac { 1 } { 3 ^ { r } } = \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 3 ^ { n } } \right)\).
OCR MEI Further Pure Core AS 2019 June Q6
11 marks Standard +0.8
6 A linear transformation \(T\) of the \(x - y\) plane has an associated matrix \(\mathbf { M }\), where \(\mathbf { M } = \left( \begin{array} { c c } \lambda & k \\ 1 & \lambda - k \end{array} \right)\), and \(\lambda\) and \(k\) are real constants. and \(k\) are real constants.
  1. You are given that \(\operatorname { det } \mathbf { M } > 0\) for all values of \(\lambda\).
    1. Find the range of possible values of \(k\).
    2. What is the significance of the condition \(\operatorname { det } \mathbf { M } > 0\) for the transformation T? For the remainder of this question, take \(k = - 2\).
  2. Determine whether there are any lines through the origin that are invariant lines for the transformation T.
  3. The transformation T is applied to a triangle with area 3 units \({ } ^ { 2 }\). The area of the resulting image triangle is 15 units \({ } ^ { 2 }\).
    Find the possible values of \(\lambda\).
OCR MEI Further Pure Core AS 2019 June Q7
12 marks Standard +0.8
7
  1. Sketch on a single Argand diagram
    1. the set of points for which \(| z - 1 - 3 i | = 3\),
    2. the set of points for which \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
  2. Find, in exact form, the two values of \(z\) for which \(| z - 1 - 3 i | = 3\) and \(\arg ( z + 4 ) = \frac { 1 } { 4 } \pi\).
OCR MEI Further Pure Core AS 2022 June Q1
6 marks Moderate -0.3
1
    1. Write the following simultaneous equations as a matrix equation. $$\begin{aligned} x + y + 2 z & = 7 \\ 2 x - 4 y - 3 z & = - 5 \\ - 5 x + 3 y + 5 z & = 13 \end{aligned}$$
    2. Hence solve the equations.
  2. Determine the set of values of the constant \(k\) for which the matrix equation $$\left( \begin{array} { c c } k + 1 & 1 \\ 2 & k \end{array} \right) \binom { x } { y } = \binom { 23 } { - 17 }$$ has a unique solution.
OCR MEI Further Pure Core AS 2022 June Q2
7 marks Standard +0.3
2
  1. Show that the vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\) is parallel to the plane \(2 \mathrm { x } + \mathrm { y } - 3 \mathrm { z } = 10\).
  2. Determine the acute angle between the planes \(2 x + y - 3 z = 10\) and \(x - y - 3 z = 3\).
OCR MEI Further Pure Core AS 2022 June Q3
5 marks Moderate -0.3
3 The complex number \(z\) satisfies the equation \(5 ( z - \mathrm { i } ) = ( - 1 + 2 \mathrm { i } ) z ^ { * }\).
Determine \(z\), giving your answer in the form \(\mathrm { a } + \mathrm { bi }\), where \(a\) and \(b\) are real.
OCR MEI Further Pure Core AS 2022 June Q5
5 marks Standard +0.3
5 An Argand diagram is shown below. The circle has centre at the point representing \(1 + 3 i\), and the half line intersects the circle at the origin and at the point representing \(4 + 4 \mathrm { i }\). \includegraphics[max width=\textwidth, alt={}, center]{c4484913-14bf-4bf4-a290-0301586333ce-3_748_917_351_242} State the two conditions that define the set of complex numbers represented by points in the shaded segment, including its boundaries.
OCR MEI Further Pure Core AS 2022 June Q6
10 marks Moderate -0.3
6
  1. Using standard summation formulae, show that \(\sum _ { r = 1 } ^ { n } r ( r + 2 ) = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 7 )\).
  2. Use induction to prove the result in part (a).