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AQA Further AS Paper 2 Discrete 2024 June Q3
1 marks Easy -1.8
Which one of the graphs shown below is semi-Eulerian? Tick (\(\checkmark\)) one box. [1 mark] \includegraphics{figure_3}
AQA Further AS Paper 2 Discrete 2024 June Q4
4 marks Moderate -0.8
The set \(S\) is defined as \(S = \{1, 2, 3, 4\}\)
  1. Complete the Cayley Table shown below for \(S\) under the binary operation multiplication modulo 5 [2 marks]
    \(\times_5\)1234
    1
    2
    3
    4
  2. State the identity element for \(S\) under multiplication modulo 5 [1 mark]
  3. State the self-inverse elements of \(S\) under multiplication modulo 5 [1 mark]
AQA Further AS Paper 2 Discrete 2024 June Q5
4 marks Moderate -0.3
A network of roads connects the villages \(A\), \(B\), \(C\), \(D\), \(E\), \(F\) and \(G\) The weight on each arc in the network represents the distance, in miles, between adjacent villages. The network is shown in the diagram below. \includegraphics{figure_5}
  1. Draw, in the space below, the spanning tree of minimum total length for this road network. [3 marks]
  2. Find the total length of the spanning tree drawn in part (a). [1 mark]
AQA Further AS Paper 2 Discrete 2024 June Q6
4 marks Easy -1.2
A Young Enterprise Company decides to sell two types of cakes at a breakfast club. The two types of cakes are blueberry and chocolate. From its initial market research, the company knows that it will: • sell at most 200 cakes in total • sell at least twice as many blueberry cakes as they will chocolate cakes • make 20p profit on each blueberry cake they sell • make 15p profit on each chocolate cake they sell. The company's objective is to maximise its profit. Formulate the Young Enterprise Company's situation as a linear programming problem. [4 marks]
AQA Further AS Paper 2 Discrete 2024 June Q7
5 marks Standard +0.3
The binary operation \(\nabla\) is defined as \(a \nabla b = a + b + ab\) where \(a, b \in \mathbb{R}\)
  1. Determine if \(\nabla\) is commutative on \(\mathbb{R}\) Fully justify your answer. [2 marks]
  2. Prove that \(\nabla\) is associative on \(\mathbb{R}\) [3 marks]
AQA Further AS Paper 2 Discrete 2024 June Q8
7 marks Standard +0.3
The diagram below shows a network of pipes. \includegraphics{figure_8} The uncircled numbers on each arc represent the capacity of each pipe in m³ s⁻¹ The circled numbers on each arc represent an initial feasible flow, in m³ s⁻¹, through the network. The initial flow through pipe \(SD\) is \(x\) m³ s⁻¹ The initial flow through pipe \(DC\) is \(y\) m³ s⁻¹ The initial flow through pipe \(CB\) is \(z\) m³ s⁻¹
  1. By considering the flows at the source and the sink, explain why \(x = 7\) [3 marks]
    1. State the value of \(y\) [1 mark]
    2. State the value of \(z\) [1 mark]
  3. Prove that the maximum flow through the network is at most 27 m³ s⁻¹ [2 marks]
AQA Further AS Paper 2 Discrete 2024 June Q9
6 marks Standard +0.3
Robert, a project manager, and his team of builders are working on a small building project. Robert has divided the project into ten activities labelled \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\), \(H\), \(I\) and \(J\) as shown in the precedence table below:
ActivityImmediate Predecessor(s)Duration (Days)
\(A\)None1
\(B\)None1
\(C\)\(A\)10
\(D\)\(A\)2
\(E\)\(B, D\)5
\(F\)\(E\)6
\(G\)\(E\)1
\(H\)\(F\)1
\(I\)\(F\)2
\(J\)\(C, G, H, I\)4
  1. On the opposite page, construct an activity network for the project and fill in the earliest start time and latest finish time for each activity. [4 marks]
  2. Robert claims that the project can be completed in 20 days. Comment on the validity of Robert's claim. [2 marks]
AQA Further AS Paper 2 Discrete 2024 June Q10
7 marks Challenging +1.2
Bilal and Mayon play a zero-sum game. The game is represented by the following pay-off matrix for Bilal, where \(x\) is an integer.
Mayon
\(\mathbf{M_1}\)\(\mathbf{M_2}\)\(\mathbf{M_3}\)
\(\mathbf{B_1}\)\(-2\)\(-1\)\(1\)
Bilal \quad \(\mathbf{B_2}\)\(4\)\(-3\)\(1\)
\(\mathbf{B_3}\)\(-1\)\(x\)\(0\)
The game has a stable solution.
  1. Show that there is only one possible value for \(x\) Fully justify your answer. [6 marks]
  2. State the value of the game for Bilal. [1 mark]
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]
AQA Further Paper 1 2019 June Q12
8 marks Challenging +1.8
Three planes have equations \begin{align} 4x - 5y + z &= 8
3x + 2y - kz &= 6
(k - 2)x + ky - 8z &= 6 \end{align} where \(k\) is a real constant. The planes do not meet at a unique point.
  1. Find the possible values of \(k\). [3 marks]
  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. [5 marks]
AQA Further Paper 1 2019 June Q13
14 marks Challenging +1.8
The equation \(z^3 + kz^2 + 9 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\).
    1. Show that $$\alpha^2 + \beta^2 + \gamma^2 = k^2$$ [3 marks]
    2. Show that $$\alpha^2\beta^2 + \beta^2\gamma^2 + \gamma^2\alpha^2 = -18k$$ [4 marks]
  2. The equation \(9z^3 - 40z^2 + rz + s = 0\) has roots \(\alpha\beta + \gamma\), \(\beta\gamma + \alpha\) and \(\gamma\alpha + \beta\).
    1. Show that $$k = -\frac{40}{9}$$ [1 mark]
    2. Without calculating the values of \(\alpha\), \(\beta\) and \(\gamma\), find the value of \(s\). Show working to justify your answer. [6 marks]
AQA Further Paper 1 2019 June Q14
11 marks Challenging +1.8
In this question use \(g = 10 \text{ m s}^{-2}\) 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 \(c\) metres, the thrust in the spring is \(9mc\) newtons. \includegraphics{figure_14} 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 \(6mv\) newtons to act on the mass, where \(v \text{ m s}^{-1}\) 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.
  1. Find \(x\) in terms of \(t\). [10 marks]
  2. State, giving a reason, the type of damping which occurs. [1 mark]
AQA Further Paper 1 2019 June Q15
11 marks Challenging +1.8
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. \(TPQ\) is the tangent to the curve at \(P\). \includegraphics{figure_15}
  1. Show that the gradient of \(TPQ\) is equal to $$\frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}$$ [4 marks]
  2. The curve has polar equation $$r = e^{(\cot b)\theta}$$ 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 \(OP\) and the tangent \(TPQ\) does not depend on \(\theta\). [7 marks]
AQA Further Paper 1 2021 June Q1
1 marks Easy -1.2
Find $$\sum_{r=1}^{20}(r^2 - 2r)$$ Circle your answer. [1 mark] 2450 \quad 2660 \quad 5320 \quad 43680
AQA Further Paper 1 2021 June Q2
1 marks Moderate -0.8
Given that \(z = 1 - 3\mathrm{i}\) is one root of the equation \(z^2 + pz + r = 0\), where \(p\) and \(r\) are real, find the value of \(r\). Circle your answer. [1 mark] \(-8\) \quad \(-2\) \quad \(6\) \quad \(10\)