Questions — Edexcel (10514 questions)

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Edexcel D2 Q5
11 marks Standard +0.3
5. A travel company offers a touring holiday which stops at four locations, \(A , B , C\) and \(D\). The tour may be taken with the locations appearing in any order, but the number of days spent in each location is dependent on its position in the tour, as shown in the table below.
\multirow{2}{*}{}Stage
1234
A7856
\(B\)6965
C9857
D7766
Showing the state of the table at each stage, use the Hungarian algorithm to find the order in which to complete the tour so as to maximise the total number of days. State the maximum total number of days that can be spent in the four locations.
(11 marks)
Edexcel D2 Q6
13 marks Standard +0.3
6. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 5 } \multicolumn{2}{c|}{}\(B\)
\cline { 2 - 5 } \multicolumn{2}{c|}{}IIIIII
\multirow{3}{*}{\(A\)}I35- 2
\cline { 2 - 5 }II7- 4- 1
\cline { 2 - 5 }III9- 48
  1. Applying the dominance rule, explain, with justification, which strategy can be ignored by
    1. player \(A\),
    2. player \(B\).
  2. For the reduced table, find the optimal strategy for
    1. player \(A\),
    2. player \(B\).
  3. Find the value of the game.
Edexcel D2 Q7
17 marks Moderate -0.5
7. This question should be answered on the sheet provided. A tinned food producer delivers goods to six supermarket warehouses, \(B , C , D , E , F\) and \(G\), from its base, \(A\). The distances, in kilometres, between each location are given in the table below. \section*{Please hand this sheet in for marking}
Edexcel D2 Q1
6 marks Moderate -0.8
  1. A glazing company runs a promotion for a special type of window. As a result of this the company receives orders for 30 of these windows from business \(B _ { 1 } , 18\) from business \(B _ { 2 }\) and 22 from business \(B _ { 3 }\). The company has stocks of 20 of these windows at factory \(F _ { 1 } , 35\) at factory \(F _ { 2 }\) and 15 at factory \(F _ { 3 }\). The table below shows the profit, in pounds, that the company will make for each window it sells according to which factory supplies each business.
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(B _ { 1 }\)\(B _ { 2 }\)\(B _ { 3 }\)
\(F _ { 1 }\)201417
\(F _ { 2 }\)181919
\(F _ { 3 }\)151723
The glazing company wishes to supply the windows so that the total profit is a maximum.
Formulate this information as a linear programming problem.
  1. State your decision variables.
  2. Write down the objective function in terms of your decision variables.
  3. Write down the constraints and state what each one represents.
Edexcel D2 Q2
7 marks Moderate -0.8
2. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{726bca96-7f98-4ed5-b642-f5007a958c8b-03_492_862_301_502} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Figure 1 shows a network in which the nodes represent five major rides in a theme park and the arcs represent paths between these rides. The numbers on the arcs give the length, in metres, of the paths.
  1. By inspection, add additional arcs to make a complete network showing the shortest distances between the rides.
    (2 marks)
  2. Use the nearest neighbour algorithm, starting at \(A\), and your complete network to find an upper bound to the length of a tour visiting each ride exactly once.
  3. Interpret the tour found in part (b) in terms of the original network.
Edexcel D2 Q3
7 marks Moderate -0.3
3. Whilst Clive is in hospital, four of his friends decide to redecorate his lounge as a welcomehome surprise. They divide the work to be done into four jobs which must be completed in the following order:
  • strip the wallpaper,
  • paint the woodwork and ceiling,
  • hang the new wallpaper,
  • replace the fittings and tidy up.
The table below shows the time, in hours, that each of the friends is likely to take to complete each job.
AliceBhavinCarlDieter
Strip wallpaper5354
Paint7564
Hang wallpaper8476
Replace fittings5323
As they do not know how long Clive will be in hospital his friends wish to complete the redecoration in the shortest possible total time.
  1. Use the Hungarian method to obtain the optimal allocation of the jobs, showing the state of the table after each stage in the algorithm.
    (6 marks)
  2. Hence, find the minimum time in which the friends can redecorate the lounge.
    (1 mark)
Edexcel D2 Q4
10 marks Standard +0.3
4. This question should be answered on the sheet provided. The owner of a small plane is planning a journey from her local airport, \(A\) to the airport nearest her parents, \(K\). On the journey she will make three refuelling stops, the first at \(B , C\) or \(D\), the second at \(E , F\) or \(G\) and the third at \(H , I\) or \(J\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{726bca96-7f98-4ed5-b642-f5007a958c8b-05_727_1303_523_356} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Figure 2 shows all the possible flights that can be made on the journey with the number by each arc indicating the distance of each flight in hundreds of miles. As her plane does not have a large fuel tank, the owner wishes to choose a route that minimises the maximum distance of any one flight. Find the route that she should use and state the maximum distance of any one stage on this route.
Edexcel D2 Q5
10 marks Moderate -0.3
5. A car-hire firm has six branches in a region. Three of the branches, \(A , B\) and \(C\), have spare cars, whereas the other three, \(D , E\) and \(F\), require cars. The total number of cars required is equal to the number of cars available. The table below shows the cost in pounds of sending one car from each branch with spares to each branch needing more cars and the number of cars available or required by each branch.
\backslashbox{Branches with spare cars}{Branches needing cars}\(D\)\(E\)\(F\)Available
\(A\)6477
B8538
C4425
Required596
  1. Use the north-west corner method to obtain a possible pattern of moving cars and find its cost. The firm wishes to minimise the cost of redistributing the cars.
  2. Calculate shadow costs for the pattern found in part (a) and improvement indices for each unoccupied cell.
  3. State, with a reason, whether or not the pattern found in part (a) is optimal.
Edexcel D2 Q6
14 marks Moderate -0.3
6. This question should be answered on the sheet provided. A furniture company in Leeds is considering opening outlets in six other cities.
The table below shows the distances, in miles, between all seven cities.
LeedsLiverpoolManchesterNewcastleNottinghamOxfordYork
Leeds-7140967116528
Liverpool71-311559215593
Manchester4031-1366214167
Newcastle96155136-15625078
Nottingham719262156-9478
Oxford16515514125094-172
York2893677878172-
  1. Starting with Leeds, obtain and draw a minimum spanning tree for this network of cities showing your method clearly.
    (4 marks)
    A representative of the company is to visit each of the areas being considered. He wishes to plan a journey of minimum length starting and ending in Leeds and visiting each of the other cities in the table once. Assuming that the network satisfies the triangle inequality,
  2. find an initial upper bound for the length of his journey,
  3. improve this upper bound to find an upper bound of less than 575 miles.
  4. By deleting Leeds, find a lower bound for his journey.
Edexcel D2 Q7
21 marks Challenging +1.2
7. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
Edexcel D2 Q1
7 marks Moderate -0.8
  1. A team of gardeners is called in to attend to the grounds of a stately home. The three gardeners will each be assigned to one of three areas, the lawns, the hedgerows and the flower beds. The table below shows the estimated time, in hours, it will take each gardener to do each job.
\cline { 2 - 4 } \multicolumn{1}{c|}{}LawnsHedgerowsFlower Beds
Alan44.56
Beth345
Colin3.556
The team wishes to complete the tasks in the least total time.
Formulate this information as a linear programming problem.
  1. State your decision variables.
  2. Write down the objective function in terms of your decision variables.
  3. Write down the constraints and explain what each one represents.
Edexcel D2 Q2
10 marks Moderate -0.3
2. This question should be answered on the sheet provided. A pool player is to play in four tournaments. Some of the tournaments take place simultaneously and the player has to choose one of each of the following: $$\begin{array} { l l } 1 ^ { \text {st } } \text { tournament: } & A , B \text { or } C , \\ 2 ^ { \text {nd } } \text { tournament: } & D , E \text { or } F , \\ 3 ^ { \text {rd } } \text { tournament: } & G , H \text { or } I , \\ 4 ^ { \text {th } } \text { tournament: } & J , K \text { or } L \end{array}$$ Each tournament has six rounds and the player estimates how well he will do in each tournament based on which tournament he plays before it. The table below shows his expectations with each number indicating the round he expects to reach and a "7" indicating he expects to win the tournament.
\multirow{2}{*}{}Expected performance in tournament
ABC\(D\)E\(F\)\(G\)\(H\)IJ\(K\)\(L\)
\multirow{10}{*}{Previous tournament}None533
A637
B554
C755
D533
E356
\(F\)365
G241
H322
\(I\)253
He wishes to choose the tournaments such that his worst performance is as good as possible. Use dynamic programming to find which tournaments he should play.
(10 marks)
Edexcel D2 Q3
10 marks Moderate -0.5
3. Four people are contributing to the entertainment section of an email magazine. For one issue reviews are required for a film, a musical, a ballet and a concert such that each person reviews one show. The people in charge of the magazine will pay each person's expenses and the cost, in pounds, for each reviewer to attend each show are given below.
FilmMusicalBalletConcert
Andrew5201218
Betty6181516
Carlos421915
Davina5161113
Use the Hungarian algorithm to find an optimal assignment which minimises the total cost. State the total cost of this allocation.
(10 marks)
Edexcel D2 Q4
15 marks Standard +0.8
4. The payoff matrix for player \(A\) in a two-person zero-sum game is shown below.
\cline { 3 - 4 } \multicolumn{2}{c|}{}\(B\)
\cline { 3 - 4 }III
\multirow{2}{*}{\(A\)}I4\({ } ^ { - } 8\)
\cline { 2 - 4 }II2\({ } ^ { - } 4\)
\cline { 2 - 4 }III\({ } ^ { - } 8\)2
  1. Explain why the game does not have a saddle point.
  2. Using a graphical method, find the optimal strategy for player \(B\).
  3. Find the optimal strategy for player \(A\).
  4. Find the value of the game.
Edexcel D2 Q5
16 marks Moderate -0.5
5. A carpet manufacturer has two warehouses, \(W _ { 1 }\) and \(W _ { 2 }\), which supply carpets for three sales outlets, \(S _ { 1 } , S _ { 2 }\) and \(S _ { 3 }\). At one point \(S _ { 1 }\) requires 40 rolls of carpet, \(S _ { 2 }\) requires 23 rolls of carpet and \(S _ { 3 }\) requires 37 rolls of carpet. At this point \(W _ { 1 }\) has 45 rolls in stock and \(W _ { 2 }\) has 40 rolls in stock. The following table shows the cost, in pounds, of transporting one roll from each warehouse to each sales outlet:
\cline { 2 - 4 } \multicolumn{1}{c|}{}\(S _ { 1 }\)\(S _ { 2 }\)\(S _ { 3 }\)
\(W _ { 1 }\)8711
\(W _ { 2 }\)91011
The company's manager wishes to supply the 85 rolls that are in stock such that transportation costs are kept to a minimum.
  1. Use the north-west corner rule to obtain an initial solution to the problem.
  2. Calculate improvement indices for the unused routes.
  3. Use the stepping-stone method to obtain an optimal solution.
Edexcel D2 Q6
17 marks Standard +0.8
6. This question should be answered on the sheet provided. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{073926c5-03cc-41d4-82bf-315740ead663-6_672_984_322_431} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} A band is going on tour to play gigs in six towns, including their home town, \(A\). The network in Figure 1 shows the distances, in miles, between the various towns. The band must begin and end their tour at \(A\) and visit each of the other towns once, and they wish to keep the total distance travelled as small as possible.
  1. By inspection, draw a complete network showing the shortest distances between the towns.
  2. Use your complete network and the nearest neighbour algorithm, starting at \(A\), to find an upper bound for the total distance travelled.
    1. Use your complete network to obtain and draw a minimum spanning tree and hence obtain another upper bound for the total distance travelled.
    2. Improve this upper bound using two shortcuts to find an upper bound below 225 miles.
  4. By deleting \(A\), find a lower bound for the total distance travelled.
  5. State an interval of as small a width as possible within which \(d\), the minimum distance travelled, in miles, must lie. \section*{Please hand this sheet in for marking}
    StagePrevious tournamentCurrent tournament
    \multirow[t]{3}{*}{1}G
    J
    K
    L
    \(H\)
    J
    K
    L
    I
    J
    K
    L
    \multirow[t]{3}{*}{2}D
    G
    H
    I
    \(E\)
    G
    H
    I
    \(F\)
    G
    H
    I
    \multirow[t]{3}{*}{3}A
    D
    E
    F
    \(B\)
    D
    E
    F
    C
    D
    E
    F
    4None
    A
    B
    C
    \section*{Please hand this sheet in for marking}
    1. \includegraphics[max width=\textwidth, alt={}, center]{073926c5-03cc-41d4-82bf-315740ead663-8_684_992_461_427}
    2. \section*{Sheet for answering question 6 (cont.)}
      2. \(\_\_\_\_\)
    5. \(\_\_\_\_\)
Edexcel AS Paper 1 2018 June Q1
4 marks Easy -1.3
  1. Find
$$\int \left( \frac { 2 } { 3 } x ^ { 3 } - 6 \sqrt { x } + 1 \right) \mathrm { d } x$$ giving your answer in its simplest form.
Edexcel AS Paper 1 2018 June Q2
5 marks Moderate -0.8
  1. Show that \(x ^ { 2 } - 8 x + 17 > 0\) for all real values of \(x\)
  2. "If I add 3 to a number and square the sum, the result is greater than the square of the original number." State, giving a reason, if the above statement is always true, sometimes true or never true.
Edexcel AS Paper 1 2018 June Q3
4 marks Easy -1.3
Given that the point \(A\) has position vector \(4 \mathbf { i } - 5 \mathbf { j }\) and the point \(B\) has position vector \(- 5 \mathbf { i } - 2 \mathbf { j }\), (a) find the vector \(\overrightarrow { A B }\),
(b) find \(| \overrightarrow { A B } |\). Give your answer as a simplified surd.
Edexcel AS Paper 1 2018 June Q4
4 marks Moderate -0.8
  1. The line \(l _ { 1 }\) has equation \(4 y - 3 x = 10\)
The line \(l _ { 2 }\) passes through the points \(( 5 , - 1 )\) and \(( - 1,8 )\).
Determine, giving full reasons for your answer, whether lines \(l _ { 1 }\) and \(l _ { 2 }\) are parallel, perpendicular or neither.
Edexcel AS Paper 1 2018 June Q5
5 marks Moderate -0.5
  1. A student's attempt to solve the equation \(2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3\) is shown below.
$$\begin{aligned} & 2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3 \\ & 2 \log _ { 2 } \left( \frac { x } { \sqrt { x } } \right) = 3 \\ & 2 \log _ { 2 } ( \sqrt { x } ) = 3 \\ & \log _ { 2 } x = 3 \\ & x = 3 ^ { 2 } = 9 \end{aligned}$$ using the subtraction law for logs simplifying using the power law for logs using the definition of a log
  1. Identify two errors made by this student, giving a brief explanation of each.
  2. Write out the correct solution.
Edexcel AS Paper 1 2018 June Q6
7 marks Moderate -0.8
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-12_599_1084_292_486} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A company makes a particular type of children's toy.
The annual profit made by the company is modelled by the equation $$P = 100 - 6.25 ( x - 9 ) ^ { 2 }$$ where \(P\) is the profit measured in thousands of pounds and \(x\) is the selling price of the toy in pounds. A sketch of \(P\) against \(x\) is shown in Figure 1.
Using the model,
  1. explain why \(\pounds 15\) is not a sensible selling price for the toy. Given that the company made an annual profit of more than \(\pounds 80000\)
  2. find, according to the model, the least possible selling price for the toy. The company wishes to maximise its annual profit.
    State, according to the model,
    1. the maximum possible annual profit,
    2. the selling price of the toy that maximises the annual profit.
Edexcel AS Paper 1 2018 June Q7
6 marks Standard +0.3
In a triangle \(A B C\), side \(A B\) has length 10 cm , side \(A C\) has length 5 cm , and angle \(B A C = \theta\) where \(\theta\) is measured in degrees. The area of triangle \(A B C\) is \(15 \mathrm {~cm} ^ { 2 }\)
  1. Find the two possible values of \(\cos \theta\) Given that \(B C\) is the longest side of the triangle,
  2. find the exact length of \(B C\).
Edexcel AS Paper 1 2018 June Q8
9 marks Moderate -0.3
  1. A lorry is driven between London and Newcastle.
In a simple model, the cost of the journey \(\pounds C\) when the lorry is driven at a steady speed of \(v\) kilometres per hour is $$C = \frac { 1500 } { v } + \frac { 2 v } { 11 } + 60$$
  1. Find, according to this model,
    1. the value of \(v\) that minimises the cost of the journey,
    2. the minimum cost of the journey.
      (Solutions based entirely on graphical or numerical methods are not acceptable.)
  2. Prove by using \(\frac { \mathrm { d } ^ { 2 } C } { \mathrm {~d} v ^ { 2 } }\) that the cost is minimised at the speed found in (a)(i).
  3. State one limitation of this model.
Edexcel AS Paper 1 2018 June Q9
9 marks Standard +0.3
9. $$g ( x ) = 4 x ^ { 3 } - 12 x ^ { 2 } - 15 x + 50$$
  1. Use the factor theorem to show that \(( x + 2 )\) is a factor of \(\mathrm { g } ( x )\).
  2. Hence show that \(\mathrm { g } ( x )\) can be written in the form \(\mathrm { g } ( x ) = ( x + 2 ) ( a x + b ) ^ { 2 }\), where \(a\) and \(b\) are integers to be found. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{f7935caa-6626-4ba8-87ef-e9bb59e1ac3e-22_517_807_607_621} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { g } ( x )\)
  3. Use your answer to part (b), and the sketch, to deduce the values of \(x\) for which
    1. \(\mathrm { g } ( x ) \leqslant 0\)
    2. \(\mathrm { g } ( 2 x ) = 0\)