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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}
AQA FP2 2016 June Q1
6 marks Standard +0.3
1
  1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { 4 r - 1 }\), show that $$\mathrm { f } ( r ) - \mathrm { f } ( r + 1 ) = \frac { A } { ( 4 r - 1 ) ( 4 r + 3 ) }$$ where \(A\) is an integer.
  2. Use the method of differences to find the value of \(\sum _ { r = 1 } ^ { 50 } \frac { 1 } { ( 4 r - 1 ) ( 4 r + 3 ) }\), giving your answer as a fraction in its simplest form.
    [0pt] [4 marks]
AQA FP2 2016 June Q2
8 marks Standard +0.3
2 The cubic equation \(3 z ^ { 3 } + p z ^ { 2 } + 17 z + q = 0\), where \(p\) and \(q\) are real, has a root \(\alpha = 1 + 2 \mathrm { i }\).
    1. Write down the value of another non-real root, \(\beta\), of this equation.
    2. Hence find the value of \(\alpha \beta\).
  2. Find the value of the third root, \(\gamma\), of this equation.
  3. Find the values of \(p\) and \(q\).
AQA FP2 2016 June Q3
10 marks Challenging +1.3
3 The arc of the curve with equation \(y = 4 - \ln \left( 1 - x ^ { 2 } \right)\) from \(x = 0\) to \(x = \frac { 3 } { 4 }\) has length \(s\).
  1. Show that \(s = \int _ { 0 } ^ { \frac { 3 } { 4 } } \left( \frac { 1 + x ^ { 2 } } { 1 - x ^ { 2 } } \right) \mathrm { d } x\).
  2. Find the value of \(s\), giving your answer in the form \(p + \ln N\), where \(p\) is a rational number and \(N\) is an integer.
    [0pt] [6 marks]
    \includegraphics[max width=\textwidth, alt={}]{a629b09d-3633-4dbd-83db-7eb89577438c-06_1870_1717_840_150}
AQA FP2 2016 June Q4
6 marks Challenging +1.2
4
  1. Given that \(y = \tan ^ { - 1 } \sqrt { ( 3 x ) }\), find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), giving your answer in terms of \(x\).
  2. Hence, or otherwise, show that \(\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 1 } { ( 1 + 3 x ) \sqrt { x } } \mathrm {~d} x = \frac { \sqrt { 3 } \pi } { n }\), where \(n\) is an integer.
    [0pt] [4 marks]
AQA FP2 2016 June Q5
12 marks Standard +0.3
5
  1. Find the modulus of the complex number \(- 4 \sqrt { 3 } + 4 \mathrm { i }\), giving your answer as an integer.
  2. The locus of points, \(L\), satisfies the equation \(| z + 4 \sqrt { 3 } - 4 \mathrm { i } | = 4\).
    1. Sketch the locus \(L\) on the Argand diagram below.
    2. The complex number \(w\) lies on \(L\) so that \(- \pi < \arg w \leqslant \pi\). Find the least possible value of arg \(w\), giving your answer in terms of \(\pi\).
  3. Solve the equation \(z ^ { 3 } = - 4 \sqrt { 3 } + 4 \mathrm { i }\), 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 2016 June Q6
14 marks Standard +0.3
6
  1. Given that \(y = \sinh x\), use the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(x = \ln \left( y + \sqrt { y ^ { 2 } + 1 } \right)\).
  2. A curve has equation \(y = 6 \cosh ^ { 2 } x + 5 \sinh x\).
    1. Show that the curve has a single stationary point and find its \(x\)-coordinate, giving your answer in the form \(\ln p\), where \(p\) is a rational number.
    2. The curve lies entirely above the \(x\)-axis. The region bounded by the curve, the coordinate axes and the line \(x = \cosh ^ { - 1 } 2\) has area \(A\). Show that $$A = a \cosh ^ { - 1 } 2 + b \sqrt { 3 } + c$$ where \(a\), \(b\) and \(c\) are integers.
      [0pt] [5 marks] \(7 \quad\) Given that \(p \geqslant - 1\), prove by induction that, for all integers \(n \geqslant 1\), $$( 1 + p ) ^ { n } \geqslant 1 + n p$$ [6 marks]
AQA FP2 2016 June Q8
13 marks Challenging +1.2
8
  1. By applying de Moivre's theorem to \(( \cos \theta + i \sin \theta ) ^ { 4 }\), where \(\cos \theta \neq 0\), show that $$( 1 + i \tan \theta ) ^ { 4 } + ( 1 - i \tan \theta ) ^ { 4 } = \frac { 2 \cos 4 \theta } { \cos ^ { 4 } \theta }$$
  2. Hence show that \(z = \mathrm { i } \tan \frac { \pi } { 8 }\) satisfies the equation \(( 1 + z ) ^ { 4 } + ( 1 - z ) ^ { 4 } = 0\), and express the three other roots of this equation in the form \(\mathrm { i } \tan \phi\), where \(0 < \phi < \pi\).
  3. Use the results from part (b) to find the values of:
    1. \(\tan ^ { 2 } \frac { \pi } { 8 } \tan ^ { 2 } \frac { 3 \pi } { 8 }\);
    2. \(\tan ^ { 2 } \frac { \pi } { 8 } + \tan ^ { 2 } \frac { 3 \pi } { 8 }\).
      \includegraphics[max width=\textwidth, alt={}]{a629b09d-3633-4dbd-83db-7eb89577438c-23_2488_1709_219_153}
      \section*{DO NOT WRITE ON THIS PAGE ANSWER IN THE SPACES PROVIDED}
OCR MEI Further Pure Core AS 2018 June Q1
4 marks Easy -1.2
1 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are defined as follows: $$\mathbf { A } = \left( \begin{array} { l } 1
OCR MEI Further Pure Core AS 2018 June Q3
5 marks Moderate -0.8
3 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r r } 2 & 0 & 3
1 & - 1 & 3 \end{array} \right) , \quad \mathbf { C } = \left( \begin{array} { l l } 1 & 3 \end{array} \right)$$ Calculate all possible products formed from two of these three matrices. 2 Find, to the nearest degree, the angle between the vectors \(\left( \begin{array} { r } 1 \\ 0 \\ - 2 \end{array} \right)\) and \(\left( \begin{array} { r } - 2 \\ 3 \\ - 3 \end{array} \right)\). 3 Find real numbers \(a\) and \(b\) such that \(( a - 3 i ) ( 5 - i ) = b - 17 i\).
OCR MEI Further Pure Core AS 2018 June Q4
5 marks Moderate -0.8
4 Find a cubic equation with real coefficients, two of whose roots are \(2 - \mathrm { i }\) and 3.
OCR MEI Further Pure Core AS 2018 June Q5
7 marks Moderate -0.3
5 A transformation of the \(x - y\) plane is represented by the matrix \(\left( \begin{array} { r r } \cos \theta & 2 \sin \theta \\ 2 \sin \theta & - \cos \theta \end{array} \right)\), where \(\theta\) is a positive acute angle.
  1. Write down the image of the point \(( 2,3 )\) under this transformation.
  2. You are given that this image is the point ( \(a , 0\) ). Find the value of \(a\).
OCR MEI Further Pure Core AS 2018 June Q6
4 marks Moderate -0.5
6 Find the invariant line of the transformation of the \(x - y\) plane represented by the matrix \(\left( \begin{array} { r r } 2 & 0 \\ 4 & - 1 \end{array} \right)\).
OCR MEI Further Pure Core AS 2018 June Q7
9 marks Standard +0.3
7
  1. Express \(\frac { 1 } { 2 r - 1 } - \frac { 1 } { 2 r + 1 }\) as a single fraction.
  2. Find how many terms of the series $$\frac { 2 } { 1 \times 3 } + \frac { 2 } { 3 \times 5 } + \frac { 2 } { 5 \times 7 } + \ldots + \frac { 2 } { ( 2 r - 1 ) ( 2 r + 1 ) } + \ldots$$ are needed for the sum to exceed 0.999999.
OCR MEI Further Pure Core AS 2018 June Q8
6 marks Standard +0.3
8 Prove by induction that \(\left( \begin{array} { l l } 1 & 1 \\ 0 & 2 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 1 & 2 ^ { n } - 1 \\ 0 & 2 ^ { n } \end{array} \right)\) for all positive integers \(n\).
OCR MEI Further Pure Core AS 2018 June Q9
9 marks Moderate -0.3
9 Fig. 9 shows a sketch of the region OPQ of the Argand diagram defined by $$\{ z : | z | \leqslant 4 \sqrt { 2 } \} \cap \left\{ z : \frac { 1 } { 4 } \pi \leqslant \arg z \leqslant \frac { 1 } { 3 } \pi \right\} .$$ \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9ef04b56-c6e5-46ea-a485-fe872932e9d8-3_549_520_397_751} \captionsetup{labelformat=empty} \caption{Fig. 9}
\end{figure}
  1. Find, in modulus-argument form, the complex number represented by the point P .
  2. Find, in the form \(a + \mathrm { i } b\), where \(a\) and \(b\) are exact real numbers, the complex number represented by the point Q .
  3. In this question you must show detailed reasoning. Determine whether the points representing the complex numbers
    • \(3 + 5 \mathrm { i }\)
    • \(5.5 ( \cos 0.8 + \mathrm { i } \sin 0.8 )\) lie within this region.
OCR MEI Further Pure Core AS 2018 June Q10
8 marks Standard +0.3
10 Three planes have equations $$\begin{aligned} - x + 2 y + z & = 0 \\ 2 x - y - z & = 0 \\ x + y & = a \end{aligned}$$ where \(a\) is a constant.
  1. Investigate the arrangement of the planes:
    • when \(a = 0\);
    • when \(a \neq 0\).
    • Chris claims that the position vectors \(- \mathbf { i } + 2 \mathbf { j } + \mathbf { k } , 2 \mathbf { i } - \mathbf { j } - \mathbf { k }\) and \(\mathbf { i } + \mathbf { j }\) lie in a plane. Determine whether or not Chris is correct.