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AQA Further Paper 1 2019 June Q7
4 marks Challenging +1.2
7 Three non-singular square matrices, A, B and \(\mathbf { R }\) are such that $$A R = B$$ The matrix \(\mathbf { R }\) represents a rotation about the \(z\)-axis through an angle \(\theta\) and $$\mathbf { B } = \left[ \begin{array} { c c c } - \cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{array} \right]$$ 7
  1. Show that \(\mathbf { A }\) is independent of the value of \(\theta\).
    7
  2. Give a full description of the single transformation represented by the matrix \(\mathbf { A }\).
AQA Further Paper 1 2019 June Q8
10 marks Challenging +1.2
8
  1. If \(z = \cos \theta + \mathrm { i } \sin \theta\), use de Moivre's theorem to prove that $$z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ 8
  2. Express \(\sin ^ { 5 } \theta\) in terms of \(\sin 5 \theta , \sin 3 \theta\) and \(\sin \theta\) 8
  3. Hence show that $$\int _ { 0 } ^ { \frac { \pi } { 3 } } \sin ^ { 5 } \theta d \theta = \frac { 53 } { 480 }$$
AQA Further Paper 1 2019 June Q9
9 marks Challenging +1.2
9
  1. Solve the equation \(z ^ { 3 } = \sqrt { 2 } - \sqrt { 6 } \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 \leq \theta < 2 \pi\) 9
  2. The transformation represented by the matrix \(\mathbf { M } = \left[ \begin{array} { l l } 5 & 1 \\ 1 & 3 \end{array} \right]\) acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points.
AQA Further Paper 1 2019 June Q10
8 marks Standard +0.3
10 The points \(A ( 5 , - 4,6 )\) and \(B ( 6 , - 6,8 )\) lie on the line \(L\). The point \(C\) is \(( 15 , - 5,9 )\). 10
  1. \(D\) is the point on \(L\) that is closest to \(C\).
    Find the coordinates of \(D\).
    10
  2. Hence find, in exact form, the shortest distance from \(C\) to \(L\).
AQA Further Paper 1 2019 June Q12
8 marks Challenging +1.2
12 Three planes have equations $$\begin{aligned} 4 x - 5 y + z & = 8 \\ 3 x + 2 y - k z & = 6 \\ ( k - 2 ) x + k y - 8 z & = 6 \end{aligned}$$ where \(k\) is a real constant. The planes do not meet at a unique point. 12
  1. Find the possible values of \(k\).
    12
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system.
AQA Further Paper 1 2019 June Q13
14 marks Challenging +1.2
13 The equation \(z ^ { 3 } + k z ^ { 2 } + 9 = 0\) has roots \(\alpha , \beta\) and \(\gamma\). 13
    1. Show that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = k ^ { 2 }$$ 13
  2. (ii) Show that $$\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 } = - 18 k$$ 13
  3. The equation \(9 z ^ { 3 } - 40 z ^ { 2 } + r z + s = 0\) has roots \(\alpha \beta + \gamma , \beta \gamma + \alpha\) and \(\gamma \alpha + \beta\). 13
    1. Show that $$k = - \frac { 40 } { 9 }$$ Question 13 continues on the next page 13
  5. (ii) Without calculating the values of \(\alpha , \beta\) and \(\gamma\), find the value of \(s\). Show working to justify your answer. \includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-23_2488_1716_219_153} A light spring is attached to the base of a long tube and has a mass \(m\) attached to the other end, as shown in the diagram. The tube is filled with oil. When the compression of the spring is \(\varepsilon\) metres, the thrust in the spring is \(9 m \varepsilon\) newtons. \includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-24_506_250_721_895} The mass is held at rest in a position where the compression of the spring is \(\frac { 20 } { 9 }\) metres. The mass is then released from rest. During the subsequent motion the oil causes a resistive force of \(6 m v\) newtons to act on the mass, where \(v \mathrm {~ms} ^ { - 1 }\) is the speed of the mass. At time \(t\) seconds after the mass is released, the displacement of the mass above its starting position is \(x\) metres.
AQA Further Paper 1 2019 June Q14
11 marks Standard +0.8
14
  1. Find \(x\) in terms of \(t\).
    14
  2. State, giving a reason, the type of damping which occurs.
AQA Further Paper 1 2019 June Q15
11 marks Challenging +1.2
15 The diagram shows part of a spiral curve. The point \(P\) has polar coordinates \(( r , \theta )\) where \(0 \leq \theta \leq \frac { \pi } { 2 }\) The points \(T\) and \(S\) lie on the initial line and \(O\) is the pole. \(T P Q\) is the tangent to the curve at \(P\). \includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-26_624_730_653_653} 15
  1. Show that the gradient of \(T P Q\) is equal to $$\frac { \frac { \mathrm { d } r } { \mathrm {~d} \theta } \sin \theta + r \cos \theta } { \frac { \mathrm {~d} r } { \mathrm {~d} \theta } \cos \theta - r \sin \theta }$$ 15
  2. The curve has polar equation $$r = \mathrm { e } ^ { ( \cot b ) \theta }$$ where \(b\) is a constant such that \(0 < b < \frac { \pi } { 2 }\) Use the result of part (a) to show that the angle between the line \(O P\) and the tangent TPQ does not depend on \(\theta\). \includegraphics[max width=\textwidth, alt={}, center]{68359582-cd8b-4807-9127-eaf8fd339746-28_2488_1719_219_150} Question number Additional page, if required.
    Write the question numbers in the left-hand margin.
AQA Further Paper 1 2020 June Q1
1 marks Easy -1.8
1 Which of the integrals below is not an improper integral?
Circle your answer. \(\int _ { 0 } ^ { \infty } e ^ { - x } d x\) \(\int _ { 0 } ^ { 2 } \frac { 1 } { 1 - x ^ { 2 } } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \sqrt { x } \mathrm {~d} x\) \(\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { x } } \mathrm {~d} x\)
AQA Further Paper 1 2020 June Q2
1 marks Easy -1.2
2 Which one of the matrices below represents a rotation of \(90 ^ { \circ }\) about the \(x\)-axis? Circle your answer.
[0pt] [1 mark] \(\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{array} \right]\) \(\left[ \begin{array} { c c c } - 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right]\) \(\left[ \begin{array} { l l l } 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{array} \right]\) \(\left[ \begin{array} { c c c } 1 & 0 & 0 \\ 0 & 0 & - 1 \\ 0 & 1 & 0 \end{array} \right]\)
AQA Further Paper 1 2020 June Q3
1 marks Standard +0.3
3 The quadratic equation \(a x ^ { 2 } + b x + c = 0 ( a , b , c \in \mathbb { R } )\) has real roots \(\alpha\) and \(\beta\). One of the four statements below is incorrect. Which statement is incorrect? Tick ( \(\checkmark\) ) one box. \(c = 0 \Rightarrow \alpha = 0\) or \(\beta = 0\) â–¡ \(c = a \Rightarrow \alpha\) is the reciprocal of \(\beta\) â–¡ \(b < 0\) and \(c < 0 \Rightarrow \alpha > 0\) and \(\beta > 0\) â–¡ \(b = 0 \Rightarrow \alpha = - \beta\) â–¡
AQA Further Paper 1 2020 June Q4
6 marks Standard +0.8
4 （a）Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients．
ï¼»4 marksï¼½
4 It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
堛的 增
4 (b) Find the value of \(p\) and the value of \(r\).
AQA Further Paper 1 2020 June Q5
9 marks Standard +0.8
5
  1. Show that the equation of \(H _ { 1 }\) can be written in the form $$( x - 1 ) ^ { 2 } - \frac { y ^ { 2 } } { q } = r$$ where \(q\) and \(r\) are integers.
    5
  2. \(\quad \mathrm { H } _ { 2 }\) is the hyperbola $$x ^ { 2 } - y ^ { 2 } = 4$$ Describe fully a sequence of two transformations which maps the graph of \(H _ { 2 }\) onto the graph of \(H _ { 1 }\) [0pt] [4 marks]
AQA Further Paper 1 2020 June Q6
9 marks Standard +0.8
6 Let \(w\) be the root of the equation \(z ^ { 7 } = 1\) that has the smallest argument \(\alpha\) in the interval \(0 < \alpha < \pi\) 6
  1. Prove that \(w ^ { n }\) is also a root of the equation \(z ^ { 7 } = 1\) for any integer \(n\). 6
  2. Prove that \(1 + w + w ^ { 2 } + w ^ { 3 } + w ^ { 4 } + w ^ { 5 } + w ^ { 6 } = 0\) 6
  3. Show the positions of \(w , w ^ { 2 } , w ^ { 3 } , w ^ { 4 } , w ^ { 5 }\), and \(w ^ { 6 }\) on the Argand diagram below.
    [0pt] [2 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-08_835_898_1802_571} 6
  4. Prove that $$\cos \frac { 2 \pi } { 7 } + \cos \frac { 4 \pi } { 7 } + \cos \frac { 6 \pi } { 7 } = - \frac { 1 } { 2 }$$
AQA Further Paper 1 2020 June Q7
7 marks Challenging +1.2
7 Three planes have equations $$\begin{aligned} ( 4 k + 1 ) x - 3 y + ( k - 5 ) z & = 3 \\ ( k - 1 ) x + ( 3 - k ) y + 2 z & = 1 \\ 7 x - 3 y + 4 z & = 2 \end{aligned}$$ 7
  1. The planes do not meet at a unique point.
    Show that \(k = 4.5\) is one possible value of \(k\), and find the other possible value of \(k\).
    7
  2. For each value of \(k\) found in part (a), identify the configuration of the given planes.
    In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system.
    [4 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
AQA Further Paper 1 2020 June Q8
6 marks Standard +0.8
8 The three roots of the equation $$4 x ^ { 3 } - 12 x ^ { 2 } - 13 x + k = 0$$ where \(k\) is a constant, form an arithmetic sequence. Find the roots of the equation.
AQA Further Paper 1 2020 June Q9
13 marks Challenging +1.2
9 The function f is defined by $$f ( x ) = \frac { x ( x + 3 ) } { x + 4 } \quad ( x \in \mathbb { R } , x \neq - 4 )$$ 9
  1. Find the interval ( \(a , b\) ) in which \(\mathrm { f } ( x )\) does not take any values.
    Fully justify your answer.
    9
  2. Find the coordinates of the two stationary points of the graph of \(y = \mathrm { f } ( x )\) 9
  3. Show that the graph of \(y = \mathrm { f } ( x )\) has an oblique asymptote and find its equation.
    \section*{Question 9 continues on the next page} 9
  4. Sketch the graph of \(y = \mathrm { f } ( x )\) on the axes below.
    [0pt] [4 marks] \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-16_1100_1100_406_470} \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-17_2493_1732_214_139}
  5. Fird \(\begin{aligned} & \text { Do not write } \\ & \text { outside the } \end{aligned}\)
AQA Further Paper 1 2020 June Q10
10 marks Challenging +1.2
10
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 y } { x } = \frac { x + 3 } { x ( x - 1 ) \left( x ^ { 2 } + 3 \right) } \quad ( x > 1 )$$ 10
  2. Find the particular solution for which \(y = 0\) when \(x = 3\) Give your answer in the form \(y = \mathrm { f } ( x )\)
AQA Further Paper 1 2020 June Q11
11 marks Standard +0.8
11 The lines \(l _ { 1 } , l _ { 2 }\) and \(l _ { 3 }\) are defined as follows. $$\begin{aligned} & l _ { 1 } : \left( \mathbf { r } - \left[ \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right] \right) \times \left[ \begin{array} { c } - 2 \\ 1 \\ - 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 2 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 3 \\ 2 \\ 7 \end{array} \right] \right) \times \left[ \begin{array} { c } 2 \\ - 1 \\ 3 \end{array} \right] = \mathbf { 0 } \\ & l _ { 3 } : \left( \mathbf { r } - \left[ \begin{array} { c } - 5 \\ 12 \\ - 4 \end{array} \right] \right) \times \left[ \begin{array} { l } 4 \\ 0 \\ 9 \end{array} \right] = \mathbf { 0 } \end{aligned}$$ 11
    1. Explain how you know that two of the lines are parallel.
      11
    2. (ii)
      3. Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures.
      [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
      11
    3. Show that the lines \(l _ { 1 }\) and \(l _ { 3 }\) meet, and find the coordinates of their point of intersection. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-23_2488_1716_219_153}
AQA Further Paper 1 2020 June Q12
8 marks Standard +0.8
12
  1. Use the definition of the cosh function to prove that $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) = \ln \left( \frac { x + \sqrt { x ^ { 2 } - a ^ { 2 } } } { a } \right) \quad \text { for } a > 0$$ [6 marks]
    12
  2. The formulae booklet gives the integral of \(\frac { 1 } { \sqrt { x ^ { 2 } - a ^ { 2 } } }\) as $$\cosh ^ { - 1 } \left( \frac { x } { a } \right) \text { or } \ln \left( x + \sqrt { x ^ { 2 } - a ^ { 2 } } \right) + c$$ Ronald says that this contradicts the result given in part (a).
    Explain why Ronald is wrong.
AQA Further Paper 1 2020 June Q13
12 marks Standard +0.8
13 Two light elastic strings each have one end attached to a particle \(B\) of mass \(3 c \mathrm {~kg}\), which rests on a smooth horizontal table. The other ends of the strings are attached to the fixed points \(A\) and \(C\), which are 8 metres apart. \(A B C\) is a horizontal line. \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566} String \(A B\) has a natural length of 4 metres and a stiffness of \(5 c\) newtons per metre.
String \(B C\) has a natural length of 1 metre and a stiffness of \(c\) newtons per metre.
The particle is pulled a distance of \(\frac { 1 } { 3 }\) metre from its equilibrium position towards \(A\), and released from rest. 13
  1. Show that the particle moves with simple harmonic motion.
    13
  2. Find the speed of the particle when it is at a point \(P\), a distance \(\frac { 1 } { 4 }\) metre from the equilibrium position. Give your answer to two significant figures.
    [0pt] [4 marks]
AQA Further Paper 1 2020 June Q14
6 marks Challenging +1.2
14
  1. Given that $$\sinh ( A + B ) = \sinh A \cosh B + \cosh A \sinh B$$ express \(\sinh ( m + 1 ) x\) and \(\sinh ( m - 1 ) x\) in terms of \(\sinh m x , \cosh m x , \sinh x\) and \(\cosh x\) 14
  2. Hence find the sum of the series $$C _ { n } = \cosh x + \cosh 2 x + \cdots + \cosh n x$$ in terms of \(\sinh x , \sinh n x\) and \(\sinh ( n + 1 ) x\) Do not write \includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-30_2491_1736_219_139}
AQA Further Paper 1 2021 June Q1
1 marks Easy -1.2
1 Find $$\sum _ { r = 1 } ^ { 20 } \left( r ^ { 2 } - 2 r \right)$$ Circle your answer. 24502660532043680
AQA Further Paper 1 2021 June Q2
1 marks Moderate -0.8
2 Given that \(z = 1 - 3 \mathrm { i }\) is one root of the equation \(z ^ { 2 } + p z + r = 0\), where \(p\) and \(r\) are real, find the value of \(r\). Circle your answer. \(- 8 - 2610\)
AQA Further Paper 1 2021 June Q3
1 marks Moderate -0.5
3 The curve \(C\) has polar equation $$r ^ { 2 } \sin 2 \theta = 4$$ Find a Cartesian equation for \(C\).
Circle your answer. \(y = 2 x\) \(y = \frac { x } { 2 }\) \(y = \frac { 2 } { x }\) \(y = 4 x\)