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Edexcel D1 2002 November Q6
10 marks Easy -1.8
6. \(\begin{array} { l l l l l l l l l l } 55 & 80 & 25 & 84 & 25 & 34 & 17 & 75 & 3 & 5 \end{array}\)
  1. The list of numbers above is to be sorted into descending order. Perform a bubble sort to obtain the sorted list, giving the state of the list after each complete pass. The numbers in the list represent weights, in grams, of objects which are to be packed into bins that hold up to 100 g .
  2. Determine the least number of bins needed.
  3. Use the first-fit decreasing algorithm to fit the objects into bins which hold up to 100 g .
Edexcel D1 2002 November Q7
14 marks Moderate -0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{438a62e6-113c-428e-85bf-4b1cbecee0de-6_523_1404_348_345}
\end{figure} The network in Fig. 4 models a drainage system. The number on each arc indicates the capacity of that arc, in litres per second.
  1. Write down the source vertices. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{438a62e6-113c-428e-85bf-4b1cbecee0de-6_525_1404_1233_345}
    \end{figure} Figure 5 shows a feasible flow through the same network.
  2. State the value of the feasible flow shown in Fig. 5. Taking the flow in Fig. 5 as your initial flow pattern,
  3. use the labelling procedure on Diagram 1 to find a maximum flow through this network. You should list each flow-augmenting route you use, together with its flow.
    (6)
  4. Show the maximal flow on Diagram 2 and state its value.
  5. Prove that your flow is maximal.
Edexcel D1 2002 November Q8
17 marks Moderate -0.5
8. T42 Co. Ltd produces three different blends of tea, Morning, Afternoon and Evening. The teas must be processed, blended and then packed for distribution. The table below shows the time taken, in hours, for each stage of the production of a tonne of tea. It also shows the profit, in hundreds of pounds, on each tonne.
\cline { 2 - 5 } \multicolumn{1}{c|}{}ProcessingBlendingPackingProfit ( \(\pounds 100\) )
Morning blend3124
Afternoon blend2345
Evening blend4233
The total times available each week for processing, blending and packing are 35, 20 and 24 hours respectively. T42 Co. Ltd wishes to maximise the weekly profit. Let \(x , y\) and \(z\) be the number of tonnes of Morning, Afternoon and Evening blend produced each week.
  1. Formulate the above situation as a linear programming problem, listing clearly the objective function, and the constraints as inequalities.
    (4) An initial Simplex tableau for the above situation is
    Basic
    variable
    		\(x\)\(y\)\(z\)\(r\)\(s\)\(t\)Value
    \(r\)32410035
    \(s\)13201020
    \(t\)24300124
    \(P\)- 4- 5- 30000
  2. Solve this linear programming problem using the Simplex algorithm. Take the most negative number in the profit row to indicate the pivot column at each stage. T42 Co. Ltd wishes to increase its profit further and is prepared to increase the time available for processing or blending or packing or any two of these three.
  3. Use your answer to part (b) to advise the company as to which stage(s) it should increase the time available.
    (2)
Edexcel D1 2003 November Q1
4 marks Moderate -0.8
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{75ea31c7-11e7-4dd9-9312-4cf32bba622b-02_992_1292_477_342}
\end{figure} A local council is responsible for maintaining pavements in a district. The roads for which it is responsible are represented by arcs in Fig. 1.The junctions are labelled \(A , B , C , \ldots , G\). The number on each arc represents the length of that road in km. The council has received a number of complaints about the condition of the pavements. In order to inspect the pavements, a council employee needs to walk along each road twice (once on each side of the road) starting and ending at the council offices at \(C\). The length of the route is to be minimal. Ignore the widths of the roads.
  1. Explain how this situation differs from the standard Route Inspection problem.
  2. Find a route of minimum length and state its length.
Edexcel D1 2003 November Q2
5 marks Standard +0.3
2. An electronics company makes a product that consists of components \(A , B , C , D , E\) and \(F\). The table shows which components must be connected together to make the product work. The components are all placed on a circuit board and connected by wires, which are not allowed to cross.
ComponentMust be connected to
\(A\)\(B , D , E , F\)
\(B\)\(C , D , E\)
\(C\)\(D , E\)
\(D\)\(E\)
\(E\)\(F\)
\(F\)\(B\)
  1. On the diagram in the answer book draw straight lines to show which components need to be connected.
    (1)
  2. Starting with the Hamiltonian cycle \(A B C D E F A\), use the planarity algorithm to determine whether it is possible to build this product on a circuit board.
    (4)
Edexcel D1 2003 November Q3
6 marks Moderate -0.8
3. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{75ea31c7-11e7-4dd9-9312-4cf32bba622b-04_1488_677_342_612}
\end{figure} The bipartite graph in Fig. 2 shows the possible allocations of people \(A , B , C , D , E\) and \(F\) to tasks \(1,2,3,4,5\) and 6. An initial matching is obtained by matching the following pairs \(A\) to \(3 , \quad B\) to \(4 , \quad C\) to \(1 , \quad F\) to 5 .
  1. Show this matching in a distinctive way on the diagram in the answer book.
  2. Use an appropriate algorithm to find a maximal matching. You should state any alternating paths you have used.
    (5)
Edexcel D1 2003 November Q4
7 marks Standard +0.3
4. (a) Draw an activity network described in this precedence table, using as few dummies as possible.
ActivityMust be preceded by:
A-
BA
CA
DA
EC
FC
GB, \(D , E , F\)
H\(B , D , E , F\)
IF, \(D\)
JG, H, I
K\(F , D\)
L\(K\)
  1. A different project is represented by the activity network shown in Fig. 3. The duration of each activity is shown in brackets. \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{75ea31c7-11e7-4dd9-9312-4cf32bba622b-05_710_1580_1509_239}
    \end{figure} Find the range of values of \(x\) that will make \(D\) a critical activity.
    (2)
Edexcel D1 2003 November Q5
9 marks Moderate -0.8
5. Nine pieces of wood are required to build a small cabinet. The lengths, in cm, of the pieces of wood are listed below. $$20 , \quad 20 , \quad 20 , \quad 35 , \quad 40 , \quad 50 , \quad 60 , \quad 70 , \quad 75$$ Planks, one metre in length, can be purchased at a cost of \(\pounds 3\) each.
  1. The first fit decreasing algorithm is used to determine how many of these planks are to be purchased to make this cabinet. Find the total cost and the amount of wood wasted.
    (5) Planks of wood can also be bought in 1.5 m lengths, at a cost of \(\pounds 4\) each. The cabinet can be built using a mixture of 1 m and 1.5 m planks.
  2. Find the minimum cost of making this cabinet. Justify your answer.
    (4)
Edexcel D1 2003 November Q6
11 marks Easy -1.8
6. (a) Define the terms
  1. tree,
  2. spanning tree,
  3. minimum spanning tree.
    (3)
    (b) State one difference between Kruskal's algorithm and Prim's algorithm, to find a minimum spanning tree.
    (1) \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 4} \includegraphics[alt={},max width=\textwidth]{75ea31c7-11e7-4dd9-9312-4cf32bba622b-08_894_1529_920_322}
    \end{figure} (c) Use Kruskal's algorithm to find the minimum spanning tree for the network shown in Fig. 4. State the order in which you included the arcs. Draw the minimum spanning tree in Diagram 1 in the answer book and state its length.
    (4) \section*{Figure 5}
    \includegraphics[max width=\textwidth, alt={}]{75ea31c7-11e7-4dd9-9312-4cf32bba622b-09_887_1536_342_258}
    Figure 5 models a car park. Currently there are two pay-stations, one at \(E\) and one at \(N\). These two are linked by a cable as shown. New pay-stations are to be installed at \(B , H , A , F\) and \(C\). The number on each arc represents the distance between the pay-stations in metres. All of the pay-stations need to be connected by cables, either directly or indirectly. The current cable between \(E\) and \(N\) must be included in the final network. The minimum amount of new cable is to be used.
    (d) Using your answer to part (c), or otherwise, determine the minimum amount of new cable needed. Use Diagram 2 to show where these cables should be installed. State the minimum amount of new cable needed.
    (3)
Edexcel D1 2003 November Q7
16 marks Standard +0.3
7. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 6} \includegraphics[alt={},max width=\textwidth]{75ea31c7-11e7-4dd9-9312-4cf32bba622b-10_1018_1557_342_214}
\end{figure} Figure 6 shows a capacitated, directed network of pipes flowing from two oil fields \(\mathrm { F } _ { 1 }\) and \(\mathrm { F } _ { 2 }\) to three refineries \(\mathrm { R } _ { 1 } , \mathrm { R } _ { 2 }\) and \(\mathrm { R } _ { 3 }\). The number on each arc represents the capacity of the pipe and the numbers in the circles represent a possible flow of 65.
  1. Find the value of \(x\) and the value of \(y\).
  2. On Diagram 1 in the answer book, add a supersource and a supersink, and arcs showing their minimum capacities.
  3. Taking the given flow of 65 as the initial flow pattern, use the labelling procedure on Diagram 2 to find the maximum flow. State clearly your flow augmenting routes.
  4. Show the maximum flow on Diagram 3 and write down its value.
  5. Verify that this is the maximum flow by finding a cut equal to the flow.
Edexcel D1 2003 November Q8
16 marks Moderate -0.8
8. A company makes three sizes of lamps, small, medium and large. The company is trying to determine how many of each size to make in a day, in order to maximise its profit. As part of the process the lamps need to be sanded, painted, dried and polished. A single machine carries out these tasks and is available 24 hours per day. A small lamp requires one hour on this machine, a medium lamp 2 hours and a large lamp 4 hours. Let \(x =\) number of small lamps made per day, $$\begin{aligned} & y = \text { number of medium lamps made per day, } \\ & z = \text { number of large lamps made per day, } \end{aligned}$$ where \(x \geq 0 , y \geq 0\) and \(z \geq 0\).
  1. Write the information about this machine as a constraint.
    1. Re-write your constraint from part (a) using a slack variable \(s\).
    2. Explain what \(s\) means in practical terms. Another constraint and the objective function give the following Simplex tableau. The profit \(P\) is stated in euros.
      Basic variable\(x\)\(y\)\(z\)\(r\)\(s\)Value
      \(r\)3561050
      \(s\)1240124
      \(P\)- 1- 3- 4000
  3. Write down the profit on each small lamp.
  4. Use the Simplex algorithm to solve this linear programming problem.
  5. Explain why the solution to part (d) is not practical.
  6. Find a practical solution which gives a profit of 30 euros. Verify that it is feasible.
Edexcel D1 2004 November Q1
5 marks Easy -1.2
1. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{4bbe6272-3900-42de-b287-599638ca75e4-02_753_1575_486_255}
\end{figure} Figure 1 shows a directed, capacitated network where the number on each arc is its capacity. A possible flow is shown from \(S\) to \(T\) and the value in brackets on each arc is the flow in that arc.
  1. Find the values of \(x , y\) and \(z\).
  2. Find, by inspection, the maximal flow from \(S\) to \(T\) and verify that it is maximal.
    (2)
Edexcel D1 2004 November Q2
6 marks Moderate -0.8
2. (a) Define the following terms
  1. planar graph,
  2. Hamiltonian cycle.
    (b) (i) Draw a graph of \(\mathrm { K } _ { 3,2 }\) in such a way as to show that it is planar.
  3. Explain why the planarity algorithm cannot be used when drawing \(\mathrm { K } _ { 3,2 }\) as a planar graph.
Edexcel D1 2004 November Q3
8 marks Easy -1.2
3. Six newspaper reporters Asif (A), Becky (B), Chris (C), David (D), Emma (E) and Fred (F), are to be assigned to six news stories Business (1), Crime (2), Financial (3), Foreign (4), Local (5) and Sport (6). The table shows possible allocations of reporters to news stories. For example, Chris can be assigned to any one of stories 1, 2 or 4.
123456
A\(\checkmark\)
B\(\checkmark\)\(\checkmark\)
C\(\checkmark\)\(\checkmark\)\(\checkmark\)
D\(\checkmark\)
E\(\checkmark\)\(\checkmark\)\(\checkmark\)
F\(\checkmark\)
  1. Show these possible allocations on the bipartite graph on the diagram in the answer book. A possible matching is
    A to 5,
    C to 1 ,
    E to 6,
    F to 4
  2. Show this information, in a distinctive way, on the diagram in the answer book.
    (1)
  3. Use an appropriate algorithm to find a maximal matching. You should list any alternating paths you have used.
  4. Explain why it is not possible to find a complete matching.
Edexcel D1 2004 November Q4
8 marks Easy -1.8
4. \(45 , \quad 56 , \quad 37 , \quad 79 , \quad 46 , \quad 18 , \quad 90 , \quad 81 , \quad 51\)
  1. Using the quick sort algorithm, perform one complete iteration towards sorting these numbers into ascending order.
    (2)
  2. Using the bubble sort algorithm, perform one complete pass towards sorting the original list into descending order. Another list of numbers, in ascending order, is $$7 , \quad 23 , \quad 31 , \quad 37 , \quad 41 , \quad 44 , \quad 50 , \quad 62 , \quad 71 , \quad 73 , \quad 94$$
  3. Use the binary search algorithm to locate the number 73 in this list. \section*{5.} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{4bbe6272-3900-42de-b287-599638ca75e4-06_1246_1168_294_427}
    \end{figure} Figure 2 shows a network of roads connecting villages. The length of each road, in km, is shown. Village \(B\) has only a small footbridge over the river which runs through the village. It can be accessed by two roads, from \(A\) and \(D\). The driver of a snowplough, based at \(F\), is planning a route to enable her to clear all the roads of snow. The route should be of minimum length. Each road can be cleared by driving along it once. The snowplough cannot cross the footbridge. Showing all your working and using an appropriate algorithm,
Edexcel D1 2004 November Q8
17 marks Moderate -0.3
8. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Figure 5} \includegraphics[alt={},max width=\textwidth]{4bbe6272-3900-42de-b287-599638ca75e4-10_1042_1847_335_115}
\end{figure} The network in Figure 5 shows activities that need to be undertaken in order to complete a project. Each activity is represented by an arc. The number in brackets is the duration of the activity in hours. The early and late event times are shown at each node. The project can be completed in 24 hours.
  1. Find the values of \(x , y\) and \(z\).
  2. Explain the use of the dummy activity in Figure 5.
  3. List the critical activities.
  4. Explain what effect a delay of one hour to activity \(B\) would have on the time taken to complete the whole project. The company which is to undertake this project has only two full time workers available. The project must be completed in 24 hours and in order to achieve this, the company is prepared to hire additional workers at a cost of \(\pounds 28\) per hour. The company wishes to minimise the money spent on additional workers. Any worker can undertake any task and each task requires only one worker.
  5. Explain why the company will have to hire additional workers in order to complete the project in 24 hours.
  6. Schedule the tasks to workers so that the project is completed in 24 hours and at minimum cost to the company.
  7. State the minimum extra cost to the company.
AQA C1 Q7
14 marks Moderate -0.8
7 The volume, \(V \mathrm {~m} ^ { 3 }\), of water in a tank at time \(t\) seconds is given by $$V = \frac { 1 } { 3 } t ^ { 6 } - 2 t ^ { 4 } + 3 t ^ { 2 } , \quad \text { for } t \geqslant 0$$
  1. Find:
    1. \(\frac { \mathrm { d } V } { \mathrm {~d} t }\);
      (3 marks)
    2. \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} t ^ { 2 } }\).
      (2 marks)
  2. Find the rate of change of the volume of water in the tank, in \(\mathrm { m } ^ { 3 } \mathrm {~s} ^ { - 1 }\), when \(t = 2\).
    1. Verify that \(V\) has a stationary value when \(t = 1\).
    2. Determine whether this is a maximum or minimum value.
AQA C1 Q8
6 marks Standard +0.3
8 The diagram shows the curve with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) and the line \(L\). \includegraphics[max width=\textwidth, alt={}, center]{b83c4e3a-36a6-4ca9-b44f-489676ca86d4-06_469_802_411_603} The points \(A\) and \(B\) have coordinates \(( - 1,0 )\) and \(( 2,0 )\) respectively. The curve touches the \(x\)-axis at the origin \(O\) and crosses the \(x\)-axis at the point \(( 3,0 )\). The line \(L\) cuts the curve at the point \(D\) where \(x = - 1\) and touches the curve at \(C\) where \(x = 2\).
  1. Find the area of the rectangle \(A B C D\).
    1. Find \(\int \left( 3 x ^ { 2 } - x ^ { 3 } \right) \mathrm { d } x\).
    2. Hence find the area of the shaded region bounded by the curve and the line \(L\).
  3. For the curve above with equation \(y = 3 x ^ { 2 } - x ^ { 3 }\) :
    1. find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\);
    2. hence find an equation of the tangent at the point on the curve where \(x = 1\);
    3. show that \(y\) is decreasing when \(x ^ { 2 } - 2 x > 0\).
  4. Solve the inequality \(x ^ { 2 } - 2 x > 0\).
AQA C1 2005 January Q1
7 marks Moderate -0.8
1 The point \(A\) has coordinates \(( 11,2 )\) and the point \(B\) has coordinates \(( - 1 , - 1 )\).
    1. Find the gradient of \(A B\).
    2. Hence, or otherwise, show that the line \(A B\) has equation $$x - 4 y = 3$$
  2. The line with equation \(3 x + 5 y = 26\) intersects the line \(A B\) at the point \(C\). Find the coordinates of \(C\).
AQA C1 2005 January Q2
10 marks Moderate -0.8
2 A curve has equation \(y = x ^ { 5 } - 6 x ^ { 3 } - 3 x + 25\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
  2. The point \(P\) on the curve has coordinates \(( 2,3 )\).
    1. Show that the gradient of the curve at \(P\) is 5 .
    2. Hence find an equation of the normal to the curve at \(P\), expressing your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are integers.
  3. Determine whether \(y\) is increasing or decreasing when \(x = 1\).
AQA C1 2005 January Q3
11 marks Moderate -0.8
3 A circle has equation \(x ^ { 2 } + y ^ { 2 } - 12 x - 6 y + 20 = 0\).
  1. By completing the square, express the equation in the form $$( x - a ) ^ { 2 } + ( y - b ) ^ { 2 } = r ^ { 2 }$$
  2. Write down:
    1. the coordinates of the centre of the circle;
    2. the radius of the circle.
  3. The line with equation \(y = x + 4\) intersects the circle at the points \(P\) and \(Q\).
    1. Show that the \(x\)-coordinates of \(P\) and \(Q\) satisfy the equation $$x ^ { 2 } - 5 x + 6 = 0$$
    2. Find the coordinates of \(P\) and \(Q\).
AQA C1 2005 January Q4
18 marks Moderate -0.8
4
  1. The function f is defined for all values of \(x\) by \(\mathrm { f } ( x ) = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\).
    1. Find the remainder when \(\mathrm { f } ( x )\) is divided by \(x + 1\).
    2. Given that \(\mathrm { f } ( 1 ) = 0\) and \(\mathrm { f } ( - 2 ) = 0\), write down two linear factors of \(\mathrm { f } ( x )\).
    3. Hence express \(x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) as the product of three linear factors.
  2. The curve with equation \(y = x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8\) is sketched below. \includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-3_543_796_897_623}
    1. The curve intersects the \(y\)-axis at the point \(A\). Find the \(y\)-coordinate of \(A\).
    2. The curve crosses the \(x\)-axis when \(x = - 2\), when \(x = 1\) and also at the point \(B\). Use the results from part (a) to find the \(x\)-coordinate of \(B\).
    1. Find \(\int \left( x ^ { 3 } - 3 x ^ { 2 } - 6 x + 8 \right) d x\).
    2. Hence find the area of the shaded region bounded by the curve and the \(x\)-axis.
AQA C1 2005 January Q5
7 marks Easy -1.2
5
  1. Simplify \(( \sqrt { 12 } + 2 ) ( \sqrt { 12 } - 2 )\).
  2. Express \(\sqrt { 12 }\) in the form \(m \sqrt { 3 }\), where \(m\) is an integer.
  3. Express \(\frac { \sqrt { 12 } + 2 } { \sqrt { 12 } - 2 }\) in the form \(a + b \sqrt { 3 }\), where \(a\) and \(b\) are integers.
AQA C1 2005 January Q6
15 marks Moderate -0.3
6 The diagram below shows a rectangular sheet of metal 24 cm by 9 cm . \includegraphics[max width=\textwidth, alt={}, center]{10bca9b4-5327-4b35-8b75-612b396e8a76-4_512_897_386_561} A square of side \(x \mathrm {~cm}\) is cut from each corner and the metal is then folded along the broken lines to make an open box with a rectangular base and height \(x \mathrm {~cm}\).
  1. Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of liquid the box can hold is given by $$V = 4 x ^ { 3 } - 66 x ^ { 2 } + 216 x$$
    1. Find \(\frac { \mathrm { d } V } { \mathrm {~d} x }\).
    2. Show that any stationary values of \(V\) must occur when \(x ^ { 2 } - 11 x + 18 = 0\).
    3. Solve the equation \(x ^ { 2 } - 11 x + 18 = 0\).
    4. Explain why there is only one value of \(x\) for which \(V\) is stationary.
    1. Find \(\frac { \mathrm { d } ^ { 2 } V } { \mathrm {~d} x ^ { 2 } }\).
    2. Hence determine whether the stationary value is a maximum or minimum.
AQA C1 2005 January Q7
10 marks Standard +0.3
7
  1. Simplify \(( k + 5 ) ^ { 2 } - 12 k ( k + 2 )\).
  2. The quadratic equation \(3 ( k + 2 ) x ^ { 2 } + ( k + 5 ) x + k = 0\) has real roots.
    1. Show that \(( k - 1 ) ( 11 k + 25 ) \leqslant 0\).
    2. Hence find the possible values of \(k\).