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AQA Further Paper 3 Discrete 2021 June Q6
8 marks Challenging +1.8
6
  1. A connected planar graph has \(( x + 1 ) ^ { 2 }\) vertices, \(( 25 + 2 x - 2 y )\) edges and \(( y - 1 ) ^ { 2 }\) faces, where \(x > 0\) and \(y > 0\) Find the possible values for the number of vertices, edges and faces for the graph.
    [0pt] [6 marks]
    LL
    6
  2. Explain why \(K _ { 6 }\), the complete graph with 6 vertices, is not planar. Fully justify your answer.
AQA Further Paper 3 Discrete 2021 June Q7
14 marks Standard +0.3
7 Avon and Roj play a zero-sum game. The game is represented by the following pay-off matrix for Avon. 7 (c)
  1. Find the optimal mixed strategy for Avon.
    7
  2. Find the value of the game for Avon.
7 (d) Roj thinks that his best outcome from the game is to play strategy \(\mathbf { R } _ { \mathbf { 2 } }\) each time. Avon notices that Roj always plays strategy \(\mathbf { R } _ { \mathbf { 2 } }\) and Avon wants to use this knowledge to maximise his expected pay-off from the game. Explain how your answer to part (c)(i) should change and find Avon's maximum expected pay-off from the game. \includegraphics[max width=\textwidth, alt={}, center]{59347089-ea4a-4ee6-b40e-1ab78aa7cdc3-16_2490_1735_219_139}
AQA Further Paper 3 Discrete 2023 June Q1
1 marks Easy -1.8
1 The simple-connected graph \(G\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-02_271_515_632_762} The graph \(G\) has \(n\) faces. State the value of \(n\) Circle your answer. 2345
AQA Further Paper 3 Discrete 2023 June Q2
1 marks Moderate -0.5
2 Jonathan and Hoshi play a zero-sum game.
The game is represented by the following pay-off matrix for Jonathan.
\multirow{6}{*}{Jonathan}Hoshi
Strategy\(\mathbf { H } _ { \mathbf { 1 } }\)\(\mathbf { H } _ { \mathbf { 2 } }\)\(\mathbf { H } _ { \mathbf { 3 } }\)
\(\mathbf { J } _ { \mathbf { 1 } }\)-232
\(\mathbf { J } _ { \mathbf { 2 } }\)320
\(\mathbf { J } _ { \mathbf { 3 } }\)4-13
\(\mathbf { J } _ { \mathbf { 4 } }\)310
The game does not have a stable solution.
Which strategy should Jonathan never play?
Circle your answer.
[0pt] [1 mark] \(\mathbf { J } _ { \mathbf { 1 } }\) \(\mathbf { J } _ { \mathbf { 2 } }\) \(\mathbf { J } _ { \mathbf { 3 } }\) \(\mathbf { J } _ { \mathbf { 4 } }\)
AQA Further Paper 3 Discrete 2023 June Q3
1 marks Easy -1.8
3 A student is solving a maximising linear programming problem. The graph below shows the constraints, feasible region and objective line for the student's linear programming problem. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-03_1248_1184_502_427} Which vertex is the optimal vertex? Circle your answer. \(A\) B
C
D
AQA Further Paper 3 Discrete 2023 June Q4
5 marks Standard +0.3
4 The network below represents a system of water pipes in a geothermal power station. The numbers on each arc represent the lower and upper capacity for each pipe in gallons per second. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-04_837_1413_493_312} The water is taken from a nearby river at node \(A\) The water is then pumped through the system of pipes and passes through one of three treatment facilities at nodes \(H , I\) and \(J\) before returning to the river. 4
  1. The senior management at the power station want all of the water to undergo a final quality control check at a new facility before it returns to the river. Using the language of networks, explain how the network above could be modified to include the new facility. 4
  2. Find the value of the cut \(\{ A , B , C , D , E \} \{ F , G , H , I , J \}\) 4
  3. Tim, a trainee engineer at the power station, correctly calculates the value of the cut \(\{ A , B , C , D , E , F \} \{ G , H , I , J \}\) to be 106 gallons per second. Tim then claims that the maximum flow through the network of pipes is 106 gallons per second. Comment on the validity of Tim's claim.
AQA Further Paper 3 Discrete 2023 June Q5
8 marks Standard +0.3
5 A student is solving the following linear programming problem. $$\begin{array} { l r } \text { Minimise } & Q = - 4 x - 3 y \\ \text { subject to } & x + y \leq 520 \\ & 2 x - 3 y \leq 570 \\ \text { and } & x \geq 0 , y \geq 0 \end{array}$$ 5
  1. The student wants to use the simplex algorithm to solve the linear programming problem. They modify the linear programming problem by introducing the objective function $$P = 4 x + 3 y$$ and the slack variables \(r\) and \(s\) State one further modification that must be made to the linear programming problem so that it can be solved using the simplex algorithm. 5
  2. (i) Complete the initial simplex tableau for the modified linear programming problem.
    [0pt] [2 marks]
    \(P\)\(x\)\(y\)\(r\)\(S\)value
    5 (b) (ii) Hence, perform one iteration of the simplex algorithm.
    \(P\)\(x\)\(y\)\(r\)\(s\)value
    5
  3. The student performs one further iteration of the simplex algorithm, which results in the following correct simplex tableau.
    \(P\)\(x\)\(y\)\(r\)\(s\)value
    100\(\frac { 18 } { 5 }\)\(\frac { 1 } { 5 }\)1986
    001\(\frac { 2 } { 5 }\)\(- \frac { 1 } { 5 }\)94
    010\(\frac { 3 } { 5 }\)\(\frac { 1 } { 5 }\)426
    5 (c) (i) Explain how the student can tell that the optimal solution to the modified linear programming problem can be determined from the above simplex tableau.
    5 (c) (ii) Find the optimal solution of the original linear programming problem.
AQA Further Paper 3 Discrete 2023 June Q6
8 marks Standard +0.3
6 A council wants to grit all of the roads on a housing estate. The network shows the roads on a housing estate. Each node represents a junction between two or more roads and the weight of each arc represents the length, in metres, of the road. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-08_1145_1458_539_292} The total length of all of the roads on the housing estate is 9175 metres.
In order to grit all of the roads, the council requires a gritter truck to travel along each road at least once. The gritter truck starts and finishes at the same junction. 6
  1. The gritter truck starts gritting the roads at 7:00 pm and moves with an average speed of 5 metres per second during its journey. Find the earliest time for the gritter truck to have gritted each road at least once and arrived back at the junction it started from, giving your answer to the nearest minute. Fully justify your answer.
    [0pt] [6 marks]
    6
  2. Explain how a refinement to the council's requirement, that the gritter truck must start and finish at the same junction, could reduce the time taken to grit all of the roads at least once.
    [2 marks] \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
    The planning involves producing an activity network for the project, which is shown in Figure 1 below. The duration of each activity is given in weeks. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-10_965_1600_559_221}
    \end{figure}
AQA Further Paper 3 Discrete 2023 June Q7
6 marks Moderate -0.8
7
    1. Find the earliest start time and the latest finish time for each activity and write these values on the activity network in Figure 1 7
      1. (ii) Write down the critical path. 7
    2. On Figure 2 below, draw a cascade diagram (Gantt chart) for the planned building project, assuming that each activity starts as early as possible. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-11_1127_1641_539_201}
      \end{figure} 7
    3. During further planning of the building project, Nova Merit Construction find that activity \(F\) is not necessary and they remove it from the project. Explain the effect removing activity \(F\) has on the minimum completion time of the project.
AQA Further Paper 3 Discrete 2023 June Q8
6 marks Moderate -0.3
8 The graph \(G\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-12_301_688_351_676} 8
    1. State, with a reason, whether or not \(G\) is simple. 8
      1. (ii) A student states that \(G\) is Eulerian.
        Explain why the student is correct. 8
    2. The graph \(H\) has 8 vertices with degrees 2, 2, 4, 4, 4, 4, 4 and 4 Comment on whether \(H\) is isomorphic to \(G\) 8
    3. The formula \(v - e + f = 2\), where \(v =\) number of vertices \(e =\) number of edges \(f =\) number of faces
      can be used with graphs which satisfy certain conditions. Prove that \(G\) does not satisfy the conditions for the above formula to apply.
AQA Further Paper 3 Discrete 2023 June Q9
14 marks Standard +0.3
9 The group \(\left( C , + _ { 4 } \right)\) contains the elements \(0,1,2\) and 3 9
    1. Show that \(C\) is a cyclic group.
      9
      1. (ii) State the group of symmetries of a regular polygon that is isomorphic to \(C\) 9
    2. The group ( \(V , \otimes\) ) contains the elements (1, 1), (1, -1), (-1, 1) and (-1, -1) The binary operation \(\otimes\) between elements of \(V\) is defined by $$( a , b ) \otimes ( c , d ) = ( a \times c , b \times d )$$ 9
      1. Find the element in \(V\) that is the inverse of \(( - 1,1 )\) Fully justify your answer.
        [0pt] [2 marks]
        9
    4. (ii) Determine, with a reason, whether or not \(C \cong V\) \(\mathbf { 9 }\) (c) The group \(G\) has order 16
      Rachel claims that as \(1,2,4,8\) and 16 are the only factors of 16 then, by Lagrange's theorem, the group \(G\) will have exactly 5 distinct subgroups, including the trivial subgroup and \(G\) itself. Comment on the validity of Rachel's claim. \includegraphics[max width=\textwidth, alt={}, center]{5ff6e3bb-6392-49cf-b64d-23bc595cd92e-16_2493_1721_214_150}
AQA C2 2009 June Q1
5 marks Moderate -0.8
The triangle \(ABC\), shown in the diagram, is such that \(AB = 7\) cm, \(AC = 5\) cm, \(BC = 8\) cm and angle \(ABC = \theta\). \includegraphics{figure_1}
  1. Show that \(\theta = 38.2°\), correct to the nearest \(0.1°\). [3]
  2. Calculate the area of triangle \(ABC\), giving your answer, in cm\(^2\), to three significant figures. [2]
AQA C2 2009 June Q2
8 marks Moderate -0.8
  1. Write down the value of \(n\) given that \(\frac{1}{x^3} = x^n\). [1]
  2. Expand \(\left(1 + \frac{3}{x^2}\right)^2\). [2]
  3. Hence find \(\int \left(1 + \frac{3}{x^2}\right)^2 dx\). [3]
  4. Hence find the exact value of \(\int_1^3 \left(1 + \frac{3}{x^2}\right)^2 dx\). [2]
AQA C2 2009 June Q3
7 marks Moderate -0.3
The \(n\)th term of a sequence is \(u_n\). The sequence is defined by $$u_{n+1} = ku_n + 12$$ where \(k\) is a constant. The first two terms of the sequence are given by $$u_1 = 16 \quad u_2 = 24$$
  1. Show that \(k = 0.75\). [2]
  2. Find the value of \(u_3\) and the value of \(u_4\). [2]
  3. The limit of \(u_n\) as \(n\) tends to infinity is \(L\).
    1. Write down an equation for \(L\). [1]
    2. Hence find the value of \(L\). [2]
AQA C2 2009 June Q4
6 marks Moderate -0.3
  1. Use the trapezium rule with four ordinates (three strips) to find an approximate value for \(\int_0^6 \sqrt{x^3 + 1} dx\), giving your answer to four significant figures. [4]
  2. The curve with equation \(y = \sqrt{x^3 + 1}\) is stretched parallel to the \(x\)-axis with scale factor \(\frac{1}{2}\) to give the curve with equation \(y = f(x)\). Write down an expression for \(f(x)\). [2]
AQA C2 2009 June Q5
13 marks Standard +0.3
The diagram shows part of a curve with a maximum point \(M\). \includegraphics{figure_5} The equation of the curve is $$y = 15x^{\frac{3}{2}} - x^{\frac{5}{2}}$$
  1. Find \(\frac{dy}{dx}\). [3]
  2. Hence find the coordinates of the maximum point \(M\). [4]
  3. The point \(P(1, 14)\) lies on the curve. Show that the equation of the tangent to the curve at \(P\) is \(y = 20x - 6\). [3]
  4. The tangents to the curve at the points \(P\) and \(M\) intersect at the point \(R\). Find the length of \(RM\). [3]
AQA C2 2009 June Q6
6 marks Moderate -0.3
The diagram shows a sector \(OAB\) of a circle with centre \(O\) and radius \(r\) cm. \includegraphics{figure_6} The angle \(AOB\) is \(1.2\) radians. The area of the sector is \(33.75\) cm\(^2\). Find the perimeter of the sector. [6]
AQA C2 2009 June Q7
11 marks Moderate -0.3
A geometric series has second term \(375\) and fifth term \(81\).
    1. Show that the common ratio of the series is \(0.6\). [3]
    2. Find the first term of the series. [2]
  2. Find the sum to infinity of the series. [2]
  3. The \(n\)th term of the series is \(u_n\). Find the value of \(\sum_{n=6}^{\infty} u_n\). [4]
AQA C2 2009 June Q8
9 marks Moderate -0.3
  1. Given that \(\frac{\sin \theta - \cos \theta}{\cos \theta} = 4\), prove that \(\tan \theta = 5\). [2]
    1. Use an appropriate identity to show that the equation $$2 \cos^2 x - \sin x = 1$$ can be written as $$2 \sin^2 x + \sin x - 1 = 0$$ [2]
    2. Hence solve the equation $$2 \cos^2 x - \sin x = 1$$ giving all solutions in the interval \(0° \leq x \leq 360°\). [5]
AQA C2 2009 June Q9
10 marks Moderate -0.8
    1. Find the value of \(p\) for which \(\sqrt{125} = 5^p\). [2]
    2. Hence solve the equation \(5^{2x} = \sqrt{125}\). [1]
  2. Use logarithms to solve the equation \(3^{2x-1} = 0.05\), giving your value of \(x\) to four decimal places. [3]
  3. It is given that $$\log_a x = 2(\log_a 3 + \log_a 2) - 1$$ Express \(x\) in terms of \(a\), giving your answer in a form not involving logarithms. [4]
AQA C3 2011 June Q1
7 marks Moderate -0.8
The diagram shows the curve with equation \(y = \ln(6x)\). \includegraphics{figure_1}
  1. State the \(x\)-coordinate of the point of intersection of the curve with the \(x\)-axis. [1]
  2. Find \(\frac{dy}{dx}\). [2]
  3. Use Simpson's rule with 6 strips (7 ordinates) to find an estimate for \(\int_1^7 \ln(6x) \, dx\), giving your answer to three significant figures. [4]
AQA C3 2011 June Q2
9 marks Moderate -0.3
    1. Find \(\frac{dy}{dx}\) when \(y = xe^{2x}\). [3]
    2. Find an equation of the tangent to the curve \(y = xe^{2x}\) at the point \((1, e^2)\). [2]
  2. Given that \(y = \frac{2\sin 3x}{1 + \cos 3x}\), use the quotient rule to show that $$\frac{dy}{dx} = \frac{k}{1 + \cos 3x}$$ where \(k\) is an integer. [4]
AQA C3 2011 June Q3
5 marks Standard +0.3
The curve \(y = \cos^{-1}(2x - 1)\) intersects the curve \(y = e^x\) at a single point where \(x = \alpha\).
  1. Show that \(\alpha\) lies between 0.4 and 0.5. [2]
  2. Show that the equation \(\cos^{-1}(2x - 1) = e^x\) can be written as \(x = \frac{1}{2} + \frac{1}{2}\cos(e^x)\). [1]
  3. Use the iteration \(x_{n+1} = \frac{1}{2} + \frac{1}{2}\cos(e^{x_n})\) with \(x_1 = 0.4\) to find the values of \(x_2\) and \(x_3\), giving your answers to three decimal places. [2]
AQA C3 2011 June Q4
12 marks Standard +0.3
    1. Solve the equation \(\cosec \theta = -4\) for \(0° < \theta < 360°\), giving your answers to the nearest 0.1°. [2]
    2. Solve the equation $$2\cot^2(2x + 30°) = 2 - 7\cosec(2x + 30°)$$ for \(0° < x < 180°\), giving your answers to the nearest 0.1°. [6]
  2. Describe a sequence of two geometrical transformations that maps the graph of \(y = \cosec x\) onto the graph of \(y = \cosec(2x + 30°)\). [4]
AQA C3 2011 June Q5
8 marks Moderate -0.3
The functions f and g are defined with their respective domains by $$f(x) = x^2 \quad \text{for all real values of } x$$ $$g(x) = \frac{1}{2x + 1} \quad \text{for real values of } x, \quad x \neq -0.5$$
  1. Explain why f does not have an inverse. [1]
  2. The inverse of g is \(g^{-1}\). Find \(g^{-1}(x)\). [3]
  3. State the range of \(g^{-1}\). [1]
  4. Solve the equation \(fg(x) = g(x)\). [3]