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AQA Further Paper 3 Discrete Specimen Q5
10 marks Standard +0.8
5 The binary operation * is defined as $$a * b = a + b + 4 ( \bmod 6 )$$ where \(a , b \in \mathbb { Z }\). 5
  1. Show that the set \(\{ 0,1,2,3,4,5 \}\) forms a group \(G\) under *.
    5
  2. Find the proper subgroups of the group \(G\) in part (a).
    5
  3. Determine whether or not the group \(G\) in part (a) is isomorphic to the group \(K = \left( \langle 3 \rangle , \times _ { 14 } \right)\) [0pt] [3 marks]
AQA Further Paper 3 Discrete Specimen Q6
11 marks Challenging +1.2
6
The network shows a system of pipes, where \(S\) is the source and \(T\) is the sink.
The lower and upper capacities, in litres per second, of each pipe are shown on each arc. \includegraphics[max width=\textwidth, alt={}, center]{88669bc0-9d3f-431a-8939-8aef2682412b-09_649_1399_580_424} 6
  1. There is a feasible flow from \(S\) to \(T\). 6
    1. Explain why arc \(A D\) must be at its lower capacity.
      [0pt] [1 mark] 6
  3. (ii) Explain why arc \(B E\) must be at its upper capacity.
    [0pt] [1 mark] 6
  4. Explain why a flow of 11 litres per second through the network is impossible.
    [0pt] [1 mark] 6
  5. The network in Figure 2 shows a second system of pipes, where \(S\) is the source and \(T\) is the sink. The lower and upper capacities, in litres per second, of each pipe are shown on each edge. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_760_1372_680_470}
    \end{figure} Figure 3 shows a feasible flow of 17 litres per second through the system of pipes. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-10_750_1371_1811_466}
    \end{figure} 6
    1. Using Figures 2 and 3, indicate on Figure 4 potential increases and decreases in the flow along each arc.
      [0pt] [2 marks] \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{88669bc0-9d3f-431a-8939-8aef2682412b-11_749_1384_457_426}
      \end{figure} 6
  7. (ii) Use flow augmentation on Figure 4 to find the maximum flow from \(S\) to \(T\).
    You should indicate any flow augmenting paths clearly in the table below and modify the potential increases and decreases of the flow on Figure 4.
    [0pt] [3 marks]
    Augmenting PathFlow
    6
  8. (iii) Prove the flow found in part (d) (ii) is maximum.
    6
  9. (iv) Due to maintenance work, the flow through node \(E\) is restricted to 9 litres per second.
    [0pt] Interpret the impact of this restriction on the maximum flow through the system of pipes. [2 marks]
AQA Further Paper 3 Discrete Specimen Q7
11 marks Challenging +1.2
7 A company repairs and sells computer hardware, including monitors, hard drives and keyboards. Each monitor takes 3 hours to repair and the cost of components is \(\pounds 40\). Each hard drive takes 2 hours to repair and the cost of components is \(\pounds 20\). Each keyboard takes 1 hour to repair and the cost of components is \(\pounds 5\). Each month, the business has 360 hours available for repairs and \(\pounds 2500\) available to buy components. Each month, the company sells all of its repaired hardware to a local computer shop. Each monitor, hard drive and keyboard sold gives the company a profit of \(\pounds 80 , \pounds 35\) and \(\pounds 15\) respectively. The company repairs and sells \(x\) monitors, \(y\) hard drives and \(z\) keyboards each month. The company wishes to maximise its total profit. 7
  1. Find five inequalities involving \(x , y\) and \(z\) for the company's problem.
    [0pt] [3 marks]
    7
    1. Find how many of each type of computer hardware the company should repair and sell each month.
      7
  3. (ii) Explain how you know that you had reached the optimal solution in part (b) (i).
    7
  4. (iii) The local computer shop complains that they are not receiving one of the types of computer hardware that the company repairs and sells. Using your answer to part (b) (i), suggest a way in which the company's problem can be modified to address the complaint.
    [0pt] [1 mark]
AQA Further Paper 3 Discrete Specimen Q8
6 marks Challenging +1.2
8 John and Danielle play a zero-sum game which does not have a stable solution. The game is represented by the following pay-off matrix for John.
\multirow{2}{*}{}Danielle
Strategy\(\boldsymbol { X }\)\(Y\)\(\boldsymbol { Z }\)
\multirow{3}{*}{John}\(A\)21-1
B-3-22
\(\boldsymbol { C }\)-3-41
Find the optimal mixed strategy for John.
Edexcel CP AS 2018 June Q1
5 marks Standard +0.3
1. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & - 3 \\ 4 & - 2 & 1 \\ 3 & 5 & - 2 \end{array} \right)$$
  1. Find \(\mathbf { M } ^ { - 1 }\) giving each element in exact form.
  2. Solve the simultaneous equations $$\begin{array} { r } 2 x + y - 3 z = - 4 \\ 4 x - 2 y + z = 9 \\ 3 x + 5 y - 2 z = 5 \end{array}$$
  3. Interpret the answer to part (b) geometrically.
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel CP AS 2018 June Q2
5 marks Standard +0.3
  1. The cubic equation
$$z ^ { 3 } - 3 z ^ { 2 } + z + 5 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are ( \(2 \alpha + 1\) ), ( \(2 \beta + 1\) ) and ( \(2 \gamma + 1\) ), giving your answer in the form \(w ^ { 3 } + p w ^ { 2 } + q w + r = 0\), where \(p , q\) and \(r\) are integers to be found.
VILU SIHI NI IIIUM ION OCVGHV SIHILNI IMAM ION OOVJYV SIHI NI JIIYM ION OC
Edexcel CP AS 2018 June Q3
9 marks Standard +0.3
  1. (a) Shade on an Argand diagram the set of points
$$\{ z \in \mathbb { C } : | z - 1 - \mathrm { i } | \leqslant 3 \} \cap \quad z \in \mathbb { C } : \frac { \pi } { 4 } \leqslant \arg ( z - 2 ) \leqslant \frac { 3 \pi } { 4 }$$ The complex number \(w\) satisfies $$| w - 1 - \mathrm { i } | = 3 \text { and } \arg ( w - 2 ) = \frac { \pi } { 4 }$$ (b) Find, in simplest form, the exact value of \(| w | ^ { 2 }\)
V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel CP AS 2018 June Q4
11 marks Standard +0.3
  1. Part of the mains water system for a housing estate consists of water pipes buried beneath the ground surface. The water pipes are modelled as straight line segments. One water pipe, \(W\), is buried beneath a particular road. With respect to a fixed origin \(O\), the road surface is modelled as a plane with equation \(3 x - 5 y - 18 z = 7\), and \(W\) passes through the points \(A ( - 1 , - 1 , - 3 )\) and \(B ( 1,2 , - 3 )\). The units are in metres.
    1. Use the model to calculate the acute angle between \(W\) and the road surface.
    A point \(C ( - 1 , - 2,0 )\) lies on the road. A section of water pipe needs to be connected to \(W\) from \(C\).
  2. Using the model, find, to the nearest cm, the shortest length of pipe needed to connect \(C\) to \(W\).
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel CP AS 2018 June Q5
10 marks Standard +0.3
5. $$\mathbf { A } = \left( \begin{array} { r r } - \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 } \\ \frac { \sqrt { 3 } } { 2 } & - \frac { 1 } { 2 } \end{array} \right)$$
  1. Describe fully the single geometrical transformation \(U\) represented by the matrix \(\mathbf { A }\). The transformation \(V\), represented by the \(2 \times 2\) matrix \(\mathbf { B }\), is a reflection in the line \(y = - x\)
  2. Write down the matrix \(\mathbf { B }\). Given that \(U\) followed by \(V\) is the transformation \(T\), which is represented by the matrix \(\mathbf { C }\), (c) find the matrix \(\mathbf { C }\).
  3. Show that there is a real number \(k\) for which the point \(( 1 , k )\) is invariant under \(T\).
    V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel CP AS 2018 June Q6
10 marks Standard +0.3
  1. (a) Use the standard results for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) and \(\sum _ { r = 1 } ^ { n } r\) to show that
$$\sum _ { r = 1 } ^ { n } ( 3 r - 2 ) ^ { 2 } = \frac { 1 } { 2 } n \left[ 6 n ^ { 2 } - 3 n - 1 \right]$$ for all positive integers \(n\).
(b) Hence find any values of \(n\) for which $$\sum _ { r = 5 } ^ { n } ( 3 r - 2 ) ^ { 2 } + 103 \sum _ { r = 1 } ^ { 28 } r \cos \left( \frac { r \pi } { 2 } \right) = 3 n ^ { 3 }$$
V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel CP AS 2018 June Q7
7 marks Challenging +1.8
7. $$f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q$$ where \(p\) and \(q\) are real constants.
The equation \(f ( z ) = 0\) has roots \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) form the vertices of a triangle of area 35 Given that \(z _ { 1 } = 3\), find the values of \(p\) and \(q\).
VILU SIHI NI IIIUM ION OCVGHV SIHILNI IMAM ION OOVJYV SIHI NI JIIYM ION OC
Edexcel CP AS 2018 June Q8
12 marks Standard +0.3
  1. (i) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\)
$$\left( \begin{array} { l l } 5 & - 8 \\ 2 & - 3 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 4 n + 1 & - 8 n \\ 2 n & 1 - 4 n \end{array} \right)$$ (ii) Prove by induction that for \(n \in \mathbb { Z } ^ { + }\) $$f ( n ) = 4 ^ { n + 1 } + 5 ^ { 2 n - 1 }$$ is divisible by 21
V349 SIHI NI IMIMM ION OCVJYV SIHIL NI LIIIM ION OOVJYV SIHIL NI JIIYM ION OC
Edexcel CP AS 2018 June Q9
11 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e48fac26-15a2-4a5e-9204-9d49db8a998a-32_789_452_331_497} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e48fac26-15a2-4a5e-9204-9d49db8a998a-32_681_523_424_1248} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A mathematics student is modelling the profile of a glass bottle of water. Figure 1 shows a sketch of a central vertical cross-section \(A B C D E F G H A\) of the bottle with the measurements taken by the student. The horizontal cross-section between \(C F\) and \(D E\) is a circle of diameter 8 cm and the horizontal cross-section between \(B G\) and \(A H\) is a circle of diameter 2 cm . The student thinks that the curve \(G F\) could be modelled as a curve with equation $$y = a x ^ { 2 } + b \quad 1 \leqslant x \leqslant 4$$ where \(a\) and \(b\) are constants and \(O\) is the fixed origin, as shown in Figure 2.
  1. Find the value of \(a\) and the value of \(b\) according to the model.
  2. Use the model to find the volume of water that the bottle can contain.
  3. State a limitation of the model. The label on the bottle states that the bottle holds approximately \(750 \mathrm {~cm} ^ { 3 }\) of water.
  4. Use this information and your answer to part (b) to evaluate the model, explaining your reasoning.
Edexcel CP AS 2019 June Q1
6 marks Moderate -0.3
1. $$\mathbf { M } = \left( \begin{array} { l l } 4 & - 5 \\ 2 & - 7 \end{array} \right)$$
  1. Show that the matrix \(\mathbf { M }\) is non-singular. The transformation \(T\) of the plane is represented by the matrix \(\mathbf { M }\).
    The triangle \(R\) is transformed to the triangle \(S\) by the transformation \(T\).
    Given that the area of \(S\) is 63 square units,
  2. find the area of \(R\).
  3. Show that the line \(y = 2 x\) is invariant under the transformation \(T\).
Edexcel CP AS 2019 June Q2
5 marks Standard +0.3
  1. The cubic equation
$$2 x ^ { 3 } + 6 x ^ { 2 } - 3 x + 12 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).
Without solving the equation, find the cubic equation whose roots are \(( \alpha + 3 ) , ( \beta + 3 )\) and \(( \gamma + 3 )\), giving your answer in the form \(p w ^ { 3 } + q w ^ { 2 } + r w + s = 0\), where \(p , q , r\) and \(s\) are integers to be found.
Edexcel CP AS 2019 June Q3
6 marks Standard +0.3
  1. Prove by mathematical induction that, for \(n \in \mathbb { N }\)
$$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) } = \frac { n } { 2 n + 1 }$$
Edexcel CP AS 2019 June Q4
5 marks Standard +0.3
  1. The line \(l\) has equation
$$\frac { x + 2 } { 1 } = \frac { y - 5 } { - 1 } = \frac { z - 4 } { - 3 }$$ The plane \(\Pi\) has equation $$\mathbf { r } . ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) = - 7$$ Determine whether the line \(l\) intersects \(\Pi\) at a single point, or lies in \(\Pi\), or is parallel to \(\Pi\) without intersecting it.
(5)
Edexcel CP AS 2019 June Q5
9 marks Standard +0.3
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-10_483_528_260_772} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The complex numbers \(z _ { 1 } = - 2 , z _ { 2 } = - 1 + 2 \mathrm { i }\) and \(z _ { 3 } = 1 + \mathrm { i }\) are plotted in Figure 1, on an Argand diagram for the complex plane with \(z = x + \mathrm { i } y\)
  1. Explain why \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) cannot all be roots of a quartic polynomial equation with real coefficients.
  2. Show that \(\arg \left( \frac { z _ { 2 } - z _ { 1 } } { z _ { 3 } - z _ { 1 } } \right) = \frac { \pi } { 4 }\)
  3. Hence show that \(\arctan ( 2 ) - \arctan \left( \frac { 1 } { 3 } \right) = \frac { \pi } { 4 }\) A copy of Figure 1, labelled Diagram 1, is given on page 12.
  4. Shade, on Diagram 1, the set of points of the complex plane that satisfy the inequality $$| z + 2 | \leqslant | z - 1 - \mathrm { i } |$$
    \includegraphics[max width=\textwidth, alt={}]{9312b91c-bca7-4427-a1f7-cb03065ee5e5-12_479_524_296_776}
    \section*{Diagram 1}
Edexcel CP AS 2019 June Q6
9 marks Moderate -0.8
  1. An art display consists of an arrangement of \(n\) marbles.
When arranged in ascending order of mass, the mass of the first marble is 10 grams. The mass of each subsequent marble is 3 grams more than the mass of the previous one, so that the \(r\) th marble has mass \(( 7 + 3 r )\) grams.
  1. Show that the mean mass, in grams, of the marbles in the display is given by $$\frac { 1 } { 2 } ( 3 n + 17 )$$ Given that there are 85 marbles in the display,
  2. use the standard summation formulae to find the standard deviation of the mass of the marbles in the display, giving your answer, in grams, to one decimal place.
Edexcel CP AS 2019 June Q7
8 marks Challenging +1.2
7. $$\mathrm { f } ( z ) = z ^ { 3 } - 8 z ^ { 2 } + p z - 24$$ where \(p\) is a real constant.
Given that the equation \(\mathrm { f } ( z ) = 0\) has distinct roots $$\alpha , \beta \text { and } \left( \alpha + \frac { 12 } { \alpha } - \beta \right)$$
  1. solve completely the equation \(\mathrm { f } ( z ) = 0\)
  2. Hence find the value of \(p\).
Edexcel CP AS 2019 June Q8
12 marks Standard +0.3
  1. A gas company maintains a straight pipeline that passes under a mountain.
The pipeline is modelled as a straight line and one side of the mountain is modelled as a plane. There are accessways from a control centre to two access points on the pipeline.
Modelling the control centre as the origin \(O\), the two access points on the pipeline have coordinates \(P ( - 300,400 , - 150 )\) and \(Q ( 300,300 , - 50 )\), where the units are metres.
  1. Find a vector equation for the line \(P Q\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\lambda\) is a scalar parameter. The equation of the plane modelling the side of the mountain is \(2 x + 3 y - 5 z = 300\) The company wants to create a new accessway from this side of the mountain to the pipeline. The accessway will consist of a tunnel of shortest possible length between the pipeline and the point \(M ( 100 , k , 100 )\) on this side of the mountain, where \(k\) is a constant.
  2. Using the model, find
    1. the coordinates of the point at which this tunnel will meet the pipeline,
    2. the length of this tunnel. It is only practical to construct the new accessway if it will be significantly shorter than both of the existing accessways, \(O P\) and \(O Q\).
  3. Determine whether the company should build the new accessway.
  4. Suggest one limitation of the model.
Edexcel CP AS 2019 June Q9
8 marks Standard +0.8
9. $$\mathrm { f } ( x ) = 2 x ^ { \frac { 1 } { 3 } } + x ^ { - \frac { 2 } { 3 } } \quad x > 0$$ The finite region bounded by the curve \(y = \mathrm { f } ( x )\), the line \(x = \frac { 1 } { 8 }\), the \(x\)-axis and the line \(x = 8\) is rotated through \(\theta\) radians about the \(x\)-axis to form a solid of revolution. Given that the volume of the solid formed is \(\frac { 461 } { 2 }\) units cubed, use algebraic integration to find the angle \(\theta\) through which the region is rotated.
Edexcel CP AS 2019 June Q10
12 marks Standard +0.8
  1. The population of chimpanzees in a particular country consists of juveniles and adults. Juvenile chimpanzees do not reproduce.
In a study, the numbers of juvenile and adult chimpanzees were estimated at the start of each year. A model for the population satisfies the matrix system $$\binom { J _ { n + 1 } } { A _ { n + 1 } } = \left( \begin{array} { c c } a & 0.15 \\ 0.08 & 0.82 \end{array} \right) \binom { J _ { n } } { A _ { n } } \quad n = 0,1,2 , \ldots$$ where \(a\) is a constant, and \(J _ { n }\) and \(A _ { n }\) are the respective numbers of juvenile and adult chimpanzees \(n\) years after the start of the study.
  1. Interpret the meaning of the constant \(a\) in the context of the model. At the start of the study, the total number of chimpanzees in the country was estimated to be 64000 According to the model, after one year the number of juvenile chimpanzees is 15360 and the number of adult chimpanzees is 43008
    1. Find, in terms of \(a\) $$\left( \begin{array} { c c } a & 0.15 \\ 0.08 & 0.82 \end{array} \right) ^ { - 1 }$$
    2. Hence, or otherwise, find the value of \(a\).
    3. Calculate the change in the number of juvenile chimpanzees in the first year of the study, according to this model. Given that the number of juvenile chimpanzees is known to be in decline in the country,
  3. comment on the short-term suitability of this model. A study of the population revealed that adult chimpanzees stop reproducing at the age of 40 years.
  4. Refine the matrix system for the model to reflect this information, giving a reason for your answer.
    (There is no need to estimate any unknown values for the refined model, but any known values should be made clear.)
Edexcel CP AS 2020 June Q1
6 marks Standard +0.8
  1. A system of three equations is defined by
$$\begin{aligned} k x + 3 y - z & = 3 \\ 3 x - y + z & = - k \\ - 16 x - k y - k z & = k \end{aligned}$$ where \(k\) is a positive constant.
Given that there is no unique solution to all three equations,
  1. show that \(k = 2\) Using \(k = 2\)
  2. determine whether the three equations are consistent, justifying your answer.
  3. Interpret the answer to part (b) geometrically.
Edexcel CP AS 2020 June Q2
8 marks Moderate -0.8
  1. Given that
$$\begin{aligned} z _ { 1 } & = 2 + 3 \\ \left| z _ { 1 } z _ { 2 } \right| & = 39 \sqrt { 2 } \\ \arg \left( z _ { 1 } z _ { 2 } \right) & = \frac { \pi } { 4 } \end{aligned}$$ where \(z _ { 1 }\) and \(z _ { 2 }\) are complex numbers,
  1. write \(z _ { 1 }\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\) Give the exact value of \(r\) and give the value of \(\theta\) in radians to 4 significant figures.
  2. Find \(z _ { 2 }\) giving your answer in the form \(a + \mathrm { i } b\) where \(a\) and \(b\) are integers.