Questions — AQA Further Paper 1 (105 questions)

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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 \(\beta = 0\) □ \(c = a \Rightarrow \alpha\) is the reciprocal of \(\beta\) □ \(b < 0\) and \(c < 0 \Rightarrow \alpha > 0\) and \(\beta > 0\) □ \(b = 0 \Rightarrow \alpha = - \beta\) □
AQA Further Paper 1 2020 June Q4
6 marks Standard +0.8
4
  1. Express \(z ^ { 4 } - 2 z ^ { 3 } + p z ^ { 2 } + r z + 80\) as the product of two quadratic factors with real coefficients.
    [4 marks]
    4 It is given that \(1 - 3 \mathrm { i }\) is one root of the quartic equation
    堛的 增
    4
  2. 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}
    1. 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
      1. (ii)
      Show that the perpendicular distance between these two parallel lines is 7.95 units, correct to three significant figures.
      [5 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
      11
  2. 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 2019 June Q1
1 marks Easy -1.2
Which one of these functions has the set \(\{x : |x| < 1\}\) as its greatest possible domain? Circle your answer. [1 mark] \(\cosh x\) \quad \(\cosh^{-1} x\) \quad \(\tanh x\) \quad \(\tanh^{-1} x\)
AQA Further Paper 1 2019 June Q2
1 marks Moderate -0.5
The first two non-zero terms of the Maclaurin series expansion of \(f(x)\) are \(x\) and \(-\frac{1}{2}x^3\) Which one of the following could be \(f(x)\)? Circle your answer. [1 mark] \(xe^{\frac{1}{2}x^2}\) \quad \(\frac{1}{2}\sin 2x\) \quad \(x \cos x\) \quad \((1 + x^3)^{-\frac{1}{2}}\)
AQA Further Paper 1 2019 June Q3
1 marks Moderate -0.8
The function \(f(x) = x^2 - 1\) Find the mean value of \(f(x)\) from \(x = -0.5\) to \(x = 1.7\) Give your answer to three significant figures. Circle your answer. [1 mark] \(-0.521\) \quad \(-0.434\) \quad \(-0.237\) \quad \(0.786\)
AQA Further Paper 1 2019 June Q4
4 marks Moderate -0.5
Solve the equation \(2z - 5iz^* = 12\) [4 marks]
AQA Further Paper 1 2019 June Q5
3 marks Standard +0.3
A plane has equation \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix} = 7\) A line has equation \(\mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\) Calculate the acute angle between the line and the plane. Give your answer to the nearest \(0.1°\) [3 marks]
AQA Further Paper 1 2019 June Q6
8 marks Standard +0.8
  1. Show that $$\cosh^3 x + \sinh^3 x = \frac{1}{4}e^{mx} + \frac{3}{4}e^{nx}$$ where \(m\) and \(n\) are integers. [3 marks]
  2. Hence find \(\cosh^6 x - \sinh^6 x\) in the form $$\frac{a \cosh(kx) + b}{8}$$ where \(a\), \(b\) and \(k\) are integers. [5 marks]
AQA Further Paper 1 2019 June Q7
4 marks Challenging +1.2
Three non-singular square matrices, \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{R}\) are such that $$\mathbf{AR} = \mathbf{B}$$ The matrix \(\mathbf{R}\) represents a rotation about the \(z\)-axis through an angle \(\theta\) and $$\mathbf{B} = \begin{pmatrix} -\cos \theta & \sin \theta & 0 \\ \sin \theta & \cos \theta & 0 \\ 0 & 0 & 1 \end{pmatrix}$$
  1. Show that \(\mathbf{A}\) is independent of the value of \(\theta\). [3 marks]
  2. Give a full description of the single transformation represented by the matrix \(\mathbf{A}\). [1 mark]
AQA Further Paper 1 2019 June Q8
10 marks Standard +0.8
  1. If \(z = \cos \theta + i \sin \theta\), use de Moivre's theorem to prove that $$z^n - \frac{1}{z^n} = 2i \sin n\theta$$ [3 marks]
  2. Express \(\sin^5 \theta\) in terms of \(\sin 5\theta\), \(\sin 3\theta\) and \(\sin \theta\) [4 marks]
  3. Hence show that $$\int_0^{\frac{\pi}{3}} \sin^5 \theta \, d\theta = \frac{53}{480}$$ [3 marks]
AQA Further Paper 1 2019 June Q9
9 marks Challenging +1.8
  1. Solve the equation \(z^3 = \sqrt{2} - \sqrt{6}i\), giving your answers in the form \(re^{i\theta}\) where \(r > 0\) and \(0 \leq \theta < 2\pi\) [5 marks]
  2. The transformation represented by the matrix \(\mathbf{M} = \begin{pmatrix} 5 & 1 \\ 1 & 3 \end{pmatrix}\) 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. [4 marks]
AQA Further Paper 1 2019 June Q10
8 marks Standard +0.8
The points \(A(5, -4, 6)\) and \(B(6, -6, 8)\) lie on the line \(L\). The point \(C\) is \((15, -5, 9)\).
  1. \(D\) is the point on \(L\) that is closest to \(C\). Find the coordinates of \(D\). [6 marks]
  2. Hence find, in exact form, the shortest distance from \(C\) to \(L\). [2 marks]
AQA Further Paper 1 2019 June Q11
7 marks Challenging +1.2
Find the general solution of the differential equation $$x \frac{dy}{dx} - 2y = \frac{x^3}{\sqrt{4 - 2x - x^2}}$$ where \(0 < x < \sqrt{5} - 1\) [7 marks]