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AQA Further Paper 3 Discrete 2024 June Q7
12 marks Standard +0.3
  1. By considering associativity, show that the set of integers does not form a group under the binary operation of subtraction. Fully justify your answer. [2 marks]
  2. The group \(G\) is formed by the set $$\{1, 7, 8, 11, 12, 18\}$$ under the operation of multiplication modulo 19
    1. Complete the Cayley table for \(G\) [3 marks]
      \(\times_{19}\)178111218
      1178111218
      7711
      887
      11117
      121211
      18181
    2. State the inverse of 11 in \(G\) [1 mark]
    1. State, with a reason, the possible orders of the proper subgroups of \(G\) [2 marks]
    2. Find all the proper subgroups of \(G\) Give your answers in the form \(\langle g \rangle, \times_{19}\) where \(g \in G\) [3 marks]
    3. The group \(H\) is such that \(G \cong H\) State a possible name for \(H\) [1 mark]
AQA Further Paper 3 Discrete 2024 June Q8
8 marks Standard +0.8
Figure 1 shows a network of water pipes. The number on each arc represents the upper capacity for each pipe in litres per second. The numbers in the circles represent an initial feasible flow of 103 litres per second. \includegraphics{figure_1}
  1. On Figure 1 above, add a supersource \(S\) and a supersink \(T\) to the network. [2 marks]
  2. Using flow augmentation, find the maximum flow through the network. You must indicate any flow augmenting paths clearly in the table below. You may use Figure 2, on the opposite page, in your solution. [4 marks]
    Augmenting PathExtra Flow
    Maximum Flow ______________ litres per second
  3. While the flow through the network is at its maximum value, the pipe \(EG\) develops a leak. To repair the leak, an engineer turns off the flow of water through \(EG\) The engineer claims that the maximum flow of water through the network will reduce by 31 litres per second. Comment on the validity of the engineer's claim. [2 marks]
AQA Further Paper 3 Discrete 2024 June Q9
6 marks Challenging +1.2
Janet and Samantha play a zero-sum game. The game is represented by the following pay-off matrix for Janet. Samantha
Strategy\(S_1\)\(S_2\)\(S_3\)
\multirow{4}{*}{Janet}\(J_1\)276
\(J_2\)551
\(J_3\)438
\(J_4\)164
  1. Explain why Janet should never play strategy \(J_4\) [1 mark]
  2. Janet wants to maximise her winnings from the game. She defines the following variables. \(p_1 = \) the probability of Janet playing strategy \(J_1\) \(p_2 = \) the probability of Janet playing strategy \(J_2\) \(p_3 = \) the probability of Janet playing strategy \(J_3\) \(v = \) the value of the game for Janet Janet then formulates her situation as the following linear programming problem. Maximise \(P = v\) subject to \(2p_1 + 5p_2 + 4p_3 \geq v\) \(7p_1 + 5p_2 + 3p_3 \geq v\) \(6p_1 + p_2 + 8p_3 \geq v\) and \(p_1 + p_2 + p_3 \leq 1\) \(p_1, p_2, p_3 \geq 0\)
    1. Complete the initial Simplex tableau for Janet's situation in the grid below. [2 marks]
      \(P\)\(v\)\(p_1\)\(p_2\)\(p_3\)value
    2. Hence, perform one iteration of the Simplex algorithm, showing your answer on the grid below. [2 marks]
      \(P\)\(v\)\(p_1\)\(p_2\)\(p_3\)value
  3. Further iterations of the Simplex algorithm are performed until an optimal solution is reached. The grid below shows part of the final Simplex tableau.
    \(p_1\)\(p_2\)value
    10\(\frac{1}{12}\)
    01\(\frac{1}{2}\)
    Find the probability of Janet playing strategy \(J_3\) when she is playing to maximise her winnings from the game. [1 mark]
AQA Further Paper 3 Discrete 2024 June Q10
7 marks Standard +0.3
A project is undertaken by Higton Engineering Ltd. The project is broken down into 11 separate activities \(A\), \(B\), \(\ldots\), \(K\) Figure 3 below shows a completed activity network for the project, along with the earliest start time, duration, latest finish time and the number of workers required for each activity. All times and durations are given in days. \includegraphics{figure_3}
  1. Write down the critical path. [1 mark]
  2. Using Figure 4 below, draw a resource histogram for the project to show how the project can be completed in the minimum possible time. Assume that each activity is to start as early as possible. [3 marks] \includegraphics{figure_4}
  3. Higton Engineering Ltd only has four workers available to work on the project. Find the minimum completion time for the project. Use Figure 5 below in your answer. [3 marks] \includegraphics{figure_5} Minimum completion time _____________________________________
Edexcel FD1 AS 2019 June Q1
6 marks Easy -1.2
  1. Draw the graph \(K_5\) [1]
    1. In the context of graph theory explain what is meant by 'semi-Eulerian'.
    2. Draw two semi-Eulerian subgraphs of \(K_5\), each having five vertices but with a different number of edges. [3]
  3. Explain why a graph with exactly five vertices with vertex orders 1, 2, 2, 3 and 4 cannot be a tree. [2]
Edexcel FD1 AS 2019 June Q2
7 marks Moderate -0.3
The following algorithm produces a numerical approximation for the integral $$I = \int_A^B x^4 \, dx$$
Step 1Start
Step 2Input the values of A, B and N
Step 3Let H = (B - A) / N
Step 4Let C = H / 2
Step 5Let D = 0
Step 6Let D = D + A\(^4\) + B\(^4\)
Step 7Let E = A
Step 8Let E = E + H
Step 9If E = B go to Step 12
Step 10Let D = D + 2 × E\(^4\)
Step 11Go to Step 8
Step 12Let F = C × D
Step 13Output F
Step 14Stop
For the case when A = 1, B = 3 and N = 4,
    1. complete the table in the answer book to show the results obtained at each step of the algorithm.
    2. State the final output. [4]
  2. Calculate, to 3 significant figures, the percentage error between the exact value of \(I\) and the value obtained from using the approximation to \(I\) in this case. [3]
Edexcel FD1 AS 2019 June Q3
7 marks Standard +0.3
ActivityImmediately preceding activities
A-
B-
CA
DA
EA
FB, C
GB, C
HD
ID, E, F, G
JD, E, F, G
KG
  1. Draw the activity network described in the precedence table above, using activity on arc. Your activity network must contain the minimum number of dummies. [5]
Every activity shown in the precedence table has the same duration.
  1. Explain why activity B cannot be critical. [1]
  2. State which other activities are not critical. [1]
Edexcel FD1 AS 2019 June Q4
10 marks Challenging +1.2
\includegraphics{figure_1} **Figure 1** [The total weight of the network is \(135 + 4x + 2y\)] The weights on the arcs in Figure 1 represent distances. The weights on the arcs CE and GH are given in terms of \(x\) and \(y\), where \(x\) and \(y\) are positive constants and \(7 < x + y < 20\) There are three paths from A to H that have the same minimum length.
  1. Use Dijkstra's algorithm to find \(x\) and \(y\). [7]
An inspection route starting at A and finishing at H is found. The route traverses each arc at least once and is of minimum length.
  1. State the arcs that are traversed twice. [1]
  2. State the number of times that vertex C appears in the inspection route. [1]
  3. Determine the length of the inspection route. [1]
Edexcel FD1 AS 2019 June Q5
10 marks Standard +0.3
Ben is a wedding planner. He needs to order flowers for the weddings that are taking place next month. The three types of flower he needs to order are roses, hydrangeas and peonies. Based on his experience, Ben forms the following constraints on the number of each type of flower he will need to order. • At least three-fifths of all the flowers must be roses. • For every 2 hydrangeas there must be at most 3 peonies. • The total number of flowers must be exactly 1000 The cost of each rose is £1, the cost of each hydrangea is £5 and the cost of each peony is £4 Ben wants to minimise the cost of the flowers. Let \(x\) represent the number of roses, let \(y\) represent the number of hydrangeas and let \(z\) represent the number of peonies that he will order.
  1. Formulate this as a linear programming problem in \(x\) and \(y\) only, stating the objective function and listing the constraints as simplified inequalities with integer coefficients. [7]
Ben decides to order the minimum number of roses that satisfy his constraints.
    1. Calculate the number of each type of flower that he will order to minimise the cost of the flowers.
    2. Calculate the corresponding total cost of this order. [3]
Edexcel CP1 2021 June Q1
6 marks Moderate -0.3
The transformation \(P\) is an enlargement, centre the origin, with scale factor \(k\), where \(k > 0\) The transformation \(Q\) is a rotation through angle \(\theta\) degrees anticlockwise about the origin. The transformation \(P\) followed by the transformation \(Q\) is represented by the matrix $$\mathbf{M} = \begin{pmatrix} -4 & -4\sqrt{3} \\ 4\sqrt{3} & -4 \end{pmatrix}$$
  1. Determine
    1. the value of \(k\),
    2. the smallest value of \(\theta\)
    [4] A square \(S\) has vertices at the points with coordinates \((0, 0)\), \((a, -a)\), \((2a, 0)\) and \((a, a)\) where \(a\) is a constant. The square \(S\) is transformed to the square \(S'\) by the transformation represented by \(\mathbf{M}\).
  2. Determine, in terms of \(a\), the area of \(S'\) [2]
Edexcel CP1 2021 June Q2
7 marks Standard +0.3
  1. Use the Maclaurin series expansion for \(\cos x\) to determine the series expansion of \(\cos^2\left(\frac{x}{3}\right)\) in ascending powers of \(x\), up to and including the term in \(x^4\) Give each term in simplest form. [2]
  2. Use the answer to part (a) and calculus to find an approximation, to 5 decimal places, for $$\int_{\pi/6}^{\pi/4} \left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)dx$$ [3]
  3. Use the integration function on your calculator to evaluate $$\int_{\pi/6}^{\pi/4} \left(\frac{1}{x}\cos^2\left(\frac{x}{3}\right)\right)dx$$ Give your answer to 5 decimal places. [1]
  4. Assuming that the calculator answer in part (c) is accurate to 5 decimal places, comment on the accuracy of the approximation found in part (b). [1]
Edexcel CP1 2021 June Q3
6 marks Standard +0.8
The cubic equation $$ax^3 + bx^2 - 19x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha\), \(\beta\) and \(\gamma\) The cubic equation $$w^3 - 9w^2 - 97w + c = 0$$ where \(c\) is a constant, has roots \((4\alpha - 1)\), \((4\beta - 1)\) and \((4\gamma - 1)\) Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\). [6]
Edexcel CP1 2021 June Q4
9 marks Standard +0.3
  1. \(\mathbf{A}\) is a 2 by 2 matrix and \(\mathbf{B}\) is a 2 by 3 matrix. Giving a reason for your answer, explain whether it is possible to evaluate
    1. \(\mathbf{AB}\)
    2. \(\mathbf{A} + \mathbf{B}\)
    [2]
  2. Given that $$\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix}\begin{pmatrix} 0 & 5 & 0 \\ 2 & 12 & -1 \\ -1 & -11 & 3 \end{pmatrix} = \lambda\mathbf{I}$$ where \(a\), \(b\) and \(\lambda\) are constants,
    1. determine
    2. Hence deduce the inverse of the matrix \(\begin{pmatrix} -5 & 3 & 1 \\ a & 0 & 0 \\ b & a & b \end{pmatrix}\)
    [3]
  3. Given that $$\mathbf{M} = \begin{pmatrix} 1 & 1 & 1 \\ 0 & \sin\theta & \cos\theta \\ 0 & \cos 2\theta & \sin 2\theta \end{pmatrix} \quad \text{where } 0 \leq \theta < \pi$$ determine the values of \(\theta\) for which the matrix \(\mathbf{M}\) is singular. [4]
Edexcel CP1 2021 June Q5
7 marks Moderate -0.3
  1. Evaluate the improper integral $$\int_1^{\infty} 2e^{-\frac{1}{2}x} dx$$ [3]
  2. The air temperature, \(\theta ^{\circ}C\), on a particular day in London is modelled by the equation $$\theta = 8 - 5\sin\left(\frac{\pi}{12}t\right) - \cos\left(\frac{\pi}{6}t\right) \quad 0 \leq t \leq 24$$ where \(t\) is the number of hours after midnight.
    1. Use calculus to show that the mean air temperature on this day is \(8^{\circ}C\), according to the model. [3] Given that the actual mean air temperature recorded on this day was higher than \(8^{\circ}C\),
    2. explain how the model could be refined. [1]
Edexcel CP1 2021 June Q6
12 marks Standard +0.3
A tourist decides to do a bungee jump from a bridge over a river. One end of an elastic rope is attached to the bridge and the other end of the elastic rope is attached to the tourist. The tourist jumps off the bridge. At time \(t\) seconds after the tourist reaches their lowest point, their vertical displacement is \(x\) metres above a fixed point 30 metres vertically above the river. When \(t = 0\)
  • \(x = -20\)
  • the velocity of the tourist is \(0\text{ms}^{-1}\)
  • the acceleration of the tourist is \(13.6\text{ms}^{-2}\)
In the subsequent motion, the elastic rope is assumed to remain taut so that the vertical displacement of the tourist can be modelled by the differential equation $$5k\frac{d^2x}{dt^2} + 2k\frac{dx}{dt} + 17x = 0 \quad t \geq 0$$ where \(k\) is a positive constant.
  1. Determine the value of \(k\) [2]
  2. Determine the particular solution to the differential equation. [7]
  3. Hence find, according to the model, the vertical height of the tourist above the river 15 seconds after they have reached their lowest point. [2]
  4. Give a limitation of the model. [1]
Edexcel CP1 2021 June Q7
8 marks Standard +0.8
The plane \(\Pi\) has equation $$\mathbf{r} = \begin{pmatrix} 3 \\ 3 \\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1 \\ 2 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that vector \(\mathbf{2i + 3j - 4k}\) is perpendicular to \(\Pi\). [2]
  2. Hence find a Cartesian equation of \(\Pi\). [2]
The line \(l\) has equation $$\mathbf{r} = \begin{pmatrix} 4 \\ -5 \\ 2 \end{pmatrix} + t \begin{pmatrix} 1 \\ 6 \\ -3 \end{pmatrix}$$ where \(t\) is a scalar parameter. The point \(A\) lies on \(l\). Given that the shortest distance between \(A\) and \(\Pi\) is \(2\sqrt{29}\)
  1. determine the possible coordinates of \(A\). [4]
Edexcel CP1 2021 June Q8
9 marks Standard +0.3
Two different colours of paint are being mixed together in a container. The paint is stirred continuously so that each colour is instantly dispersed evenly throughout the container. Initially the container holds a mixture of 10 litres of red paint and 20 litres of blue paint. The colour of the paint mixture is now altered by
  • adding red paint to the container at a rate of 2 litres per second
  • adding blue paint to the container at a rate of 1 litre per second
  • pumping fully mixed paint from the container at a rate of 3 litres per second.
Let \(r\) litres be the amount of red paint in the container at time \(t\) seconds after the colour of the paint mixture starts to be altered.
  1. Show that the amount of red paint in the container can be modelled by the differential equation $$\frac{dr}{dt} = 2 - \frac{r}{a}$$ where \(a\) is a positive constant to be determined. [2]
  2. By solving the differential equation, determine how long it will take for the mixture of paint in the container to consist of equal amounts of red paint and blue paint, according to the model. Give your answer to the nearest second. [6] It actually takes 9 seconds for the mixture of paint in the container to consist of equal amounts of red paint and blue paint.
  3. Use this information to evaluate the model, giving a reason for your answer. [1]
Edexcel CP1 2021 June Q9
11 marks Challenging +1.2
  1. Use a hyperbolic substitution and calculus to show that $$\int \frac{x^2}{\sqrt{x^2 - 1}} dx = \frac{1}{2}\left[x\sqrt{x^2 - 1} + \text{arcosh } x\right] + k$$ where \(k\) is an arbitrary constant. [6]
\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \frac{4}{15}x \text{ arcosh } x \quad x \geq 1$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
  1. Using algebraic integration and the result from part (a), show that the area of \(R\) is given by $$\frac{1}{15}\left[17\ln\left(3 + 2\sqrt{2}\right) - 6\sqrt{2}\right]$$ [5]
OCR Further Pure Core AS 2020 November Q1
6 marks Standard +0.8
In this question you must show detailed reasoning. Use an algebraic method to find the square roots of \(-77 - 36\text{i}\). [6]
OCR Further Pure Core AS 2020 November Q2
10 marks Moderate -0.8
P, Q and T are three transformations in 2-D. P is a reflection in the \(x\)-axis. A is the matrix that represents P.
  1. Write down the matrix A. [1]
Q is a shear in which the \(y\)-axis is invariant and the point \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) is transformed to the point \(\begin{pmatrix} 1 \\ 2 \end{pmatrix}\). B is the matrix that represents Q.
  1. Find the matrix B. [2]
T is P followed by Q. C is the matrix that represents T.
  1. Determine the matrix C. [2]
\(L\) is the line whose equation is \(y = x\).
  1. Explain whether or not \(L\) is a line of invariant points under T. [2]
An object parallelogram, \(M\), is transformed under T to an image parallelogram, \(N\).
  1. Explain what the value of the determinant of C means about
    [3]
OCR Further Pure Core AS 2020 November Q3
12 marks Moderate -0.3
In this question you must show detailed reasoning. The complex number \(7 - 4\text{i}\) is denoted by \(z\).
  1. Giving your answers in the form \(a + b\text{i}\), where \(a\) and \(b\) are rational numbers, find the following.
    1. \(3z - 4z^*\) [2]
    2. \((z + 1 - 3\text{i})^2\) [2]
    3. \(\frac{z + 1}{z - 1}\) [2]
  2. Express \(z\) in modulus-argument form giving the modulus exactly and the argument correct to 3 significant figures. [3]
  3. The complex number \(\omega\) is such that \(z\omega = \sqrt{585}(\cos(0.5) + \text{i}\sin(0.5))\). Find the following.
    [3]
OCR Further Pure Core AS 2020 November Q4
6 marks Standard +0.3
You are given the system of equations $$a^2x - 2y = 1$$ $$x + b^2y = 3$$ where \(a\) and \(b\) are real numbers.
  1. Use a matrix method to find \(x\) and \(y\) in terms of \(a\) and \(b\). [4]
  2. Explain why the method used in part (a) works for all values of \(a\) and \(b\). [2]
OCR Further Pure Core AS 2020 November Q5
7 marks Challenging +1.8
In this question you must show detailed reasoning. The cubic equation \(5x^3 + 3x^2 - 4x + 7 = 0\) has roots \(\alpha\), \(\beta\) and \(\gamma\). Find a cubic equation with integer coefficients whose roots are \(\alpha + \beta\), \(\beta + \gamma\) and \(\gamma + \alpha\). [7]
OCR Further Pure Core AS 2020 November Q6
5 marks Challenging +1.2
Prove that \(n! > 2^{2n}\) for all integers \(n \geq 9\). [5]
OCR Further Pure Core AS 2020 November Q7
6 marks Standard +0.3
The equations of two intersecting lines are $$\mathbf{r} = \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} + \lambda \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \quad \mathbf{r} = \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} -3 \\ 1 \\ -1 \end{pmatrix}$$ where \(a\) is a constant.
  1. Find a vector, \(\mathbf{b}\), which is perpendicular to both lines. [2]
  2. Show that \(\mathbf{b} \cdot \begin{pmatrix} -12 \\ a \\ -1 \end{pmatrix} = \mathbf{b} \cdot \begin{pmatrix} 2 \\ 0 \\ 5 \end{pmatrix}\). [2]
  3. Hence, or otherwise, find the value of \(a\). [2]