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OCR D2 2015 June Q6
20 marks Easy -1.8
6 At the final battle in a war game, the two opposing armies, led by General Rose, \(R\), and Colonel Cole, \(C\), are facing each other across a wide river. Each army consists of four divisions. The commander of each army needs to send some of his troops North and send the rest South. Each commander has to decide how many divisions (1,2 or 3) to send North. Intelligence information is available on the number of thousands of soldiers that each army can expect to have remaining with each combination of strategies. Thousands of soldiers remaining in \(R\) 's army \(C\) 's choice \(R\) 's choice
123
1152530
2205015
3303515
Thousands of soldiers remaining in \(C\) 's army \(C\) 's choice \(R\) 's choice
123
1203510
2155020
3102540
  1. Construct a table to show the number of thousands of soldiers remaining in \(R\) 's army minus the number of thousands of soldiers remaining in \(C\) 's army (the excess for \(R\) 's army) for each combination of strategies. The commander whose army has the greatest positive excess of soldiers remaining at the end of the game will be declared the winner.
  2. Explain the meaning of the value in the top left cell of your table from part (i) (where each commander chooses strategy 1). Hence explain why this table may be regarded as representing a zero-sum game.
  3. Find the play-safe strategy for \(R\) and the play-safe strategy for \(C\). If \(C\) knows that \(R\) will choose his play-safe strategy, which strategy should \(C\) choose? One of the strategies is redundant for one of the commanders, because of dominance.
  4. Draw a table for the reduced game, once the redundant strategy has been removed. Label the rows and columns to show how many divisions have been sent North. A mixed strategy is to be employed on the resulting reduced game. This leads to the following LP problem:
    Maximise \(\quad M = m - 25\) Subject to \(\quad m \leqslant 15 x + 25 y + 35 z\) \(m \leqslant 45 x + 20 y\) \(x + y + z \leqslant 1\) and
  5. Interpret what \(x , y\) and \(z\) represent and show how \(m \leqslant 15 x + 25 y + 35 z\) was formed. A computer runs the Simplex algorithm to solve this problem. It gives \(x = 0.5385 , y = 0\) and \(z = 0.4615\).
  6. Describe how this solution should be interpreted, in terms of how General Rose chooses where to send his troops. Calculate the optimal value for \(M\) and explain its meaning. Elizabeth does not have access to a computer. She says that at the solution to the LP problem \(15 x + 25 y + 35 z\) must equal \(45 x + 20 y\) and \(x + y + z\) must equal 1 . This simplifies to give \(y + 7 z = 6 x\) and \(x + y + z = 1\).
  7. Explain why there can be no valid solution of \(y + 7 z = 6 x\) and \(x + y + z = 1\) with \(x = 0\). Elizabeth tries \(z = 0\) and gets the solution \(x = \frac { 1 } { 7 }\) and \(y = \frac { 6 } { 7 }\).
  8. Explain why this is not a solution to the LP problem.
OCR D2 2016 June Q1
7 marks Standard +0.3
1 Josh is making a calendar. He has chosen six pictures, each of which will represent two months in the calendar. He needs to choose which picture to use for each two-month period. The bipartite graph in Fig. 1 shows for which months each picture is suitable. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{490ff276-6639-40a1-bffb-dc6967f3ab21-2_497_1246_488_415} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure} Initially Josh chooses the sailing ships for March/April, the sunset for July/August, the snow scene for November/December and the swans for May/June. This incomplete matching is shown in Fig. 2 below. \begin{table}[h]
Sailing ships(1)January/February
Seascape(2)• ◯(MA)March/April
Snow scene(3)\includegraphics[max width=\textwidth, alt={}]{490ff276-6639-40a1-bffb-dc6967f3ab21-2_54_381_1451_716}(MJ)May/June
Street scene\includegraphics[max width=\textwidth, alt={}]{490ff276-6639-40a1-bffb-dc6967f3ab21-2_59_38_1536_712}(JA)July/August
Sunset(5)(SO)September/October
Swans(6)(ND)November/December
\captionsetup{labelformat=empty} \caption{Fig. 2}
\end{table}
  1. Write down the shortest possible alternating path that starts at (JF) and finishes at either (2) or (4). Hence write down a matching that only excludes (SO) and one of the pictures.
  2. Working from the incomplete matching found in part (i), write down the shortest possible alternating path that starts at (SO) and finishes at whichever of (2) and (4) has still not been matched. Hence write down a complete matching between the pictures and the months.
  3. Explain why three of the arcs in Fig. 1 must appear in the graph of any complete matching. Hence find a second complete matching.
OCR D2 2016 June Q2
10 marks Moderate -0.8
2 Water flows through pipes from an underground spring to a tap. The size of each pipe limits the amount that can flow. Valves restrict the direction of flow. The pipes are modelled as a network of directed arcs connecting a source at \(S\) to a sink at \(T\). Arc weights represent the (upper) capacities, in litres per second. Pipes may be empty. Fig. 3 shows a flow of 5 litres per second from \(S\) to \(T\). Fig. 4 shows the result of applying the labelling procedure to the network with this flow. The arrows in the direction of potential flow show excess capacities (how much more could flow in the arc, in that direction) and the arrows in the opposite direction show potential backflows (how much less could flow in the arc). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{490ff276-6639-40a1-bffb-dc6967f3ab21-3_524_876_717_141} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{490ff276-6639-40a1-bffb-dc6967f3ab21-3_524_878_717_1046} \captionsetup{labelformat=empty} \caption{Fig. 4}
\end{figure}
  1. Write down the capacity of \(\operatorname { arc } F T\) and of \(\operatorname { arc } D T\). Find the value of the cut that separates the vertices \(S , A , B , C , D , E , F\) from the sink at \(T\).
  2. (a) Update the excess capacities and the potential backflows to show an additional flow of 2 litres per second along \(S - C - B - F - T\).
    (b) Write down a further flow augmenting route and the amount by which the flow can be augmented. You do not need to update the excess capacities and potential backflows a second time.
  3. Show the flow that results from part (ii)(b) and find a cut that has the same value as your flow.
OCR D2 2016 June Q3
13 marks Standard +0.8
3 A theatre company needs to employ three technicians for a performance. One will operate the lights, one the sound system and one the flying trapeze mechanism. Four technicians have applied for these tasks. The table shows how much it will cost the theatre company, in £, to employ each technician for each task.
\multirow{2}{*}{}Task
LightingSoundTrapeze
\multirow{4}{*}{Technician}Amir868890
Bex929495
Caz889294
Dee9810098
The theatre company wants to employ the three technicians for whom the total cost is least.
The Hungarian algorithm is to be used to find the minimum cost allocation, but before this can be done the table needs to be modified.
  1. Make the necessary modification to the table. Working from your modified table, construct a reduced cost matrix by first reducing rows and then reducing columns. You should show the amount by which each row has been reduced in the row reductions and the amount by which each column has been reduced in the column reductions. Cross through the 0 's in your reduced cost matrix using the least possible number of horizontal or vertical lines. [You must ensure that the values in your table can still be read.]
  2. Complete the application of the Hungarian algorithm to find a minimum cost allocation. Write a list showing which technician should be employed for each task. Calculate the total cost to the theatre company. Although Amir put in the lowest cost for operating the lighting, you should have found that he has not been allocated this task. Amir is particularly keen to be employed to operate the lights so is prepared to reduce his cost for this task.
  3. Find a way to use two of Bex, Caz and Dee to operate the sound effects and the flying trapeze mechanism at the lowest cost. Hence find what Amir's new cost should be for the minimum total cost to the theatre company to be exactly \(\pounds 1\) less than your answer from part (ii).
OCR D2 2016 June Q4
10 marks Easy -1.2
4 Rowan and Colin are playing a game of 'scissors-paper-rock'. In each round of this game, each player chooses one of scissors ( \(\$$ ), paper ( \)\square\( ) or rock ( \)\bullet$ ). The players reveal their choices simultaneously, using coded hand signals. Rowan and Colin will play a large number of rounds. At the end of the game the player with the greater total score is the winner. The rules of the game are that scissors wins over paper, paper wins over rock and rock wins over scissors. In this version of the game, if a player chooses scissors they will score \(+ 1,0\) or - 1 points, according to whether they win, draw or lose, but if they choose paper or rock they will score \(+ 2,0\) or - 2 points. This gives the following pay-off tables. \includegraphics[max width=\textwidth, alt={}, center]{490ff276-6639-40a1-bffb-dc6967f3ab21-5_476_773_667_239} \includegraphics[max width=\textwidth, alt={}, center]{490ff276-6639-40a1-bffb-dc6967f3ab21-5_478_780_667_1071}
  1. Use an example to show that this is not a zero-sum game.
  2. Write down the minimum number of points that Rowan can win using each strategy. Hence find the strategy that maximises the minimum number of points that Rowan can win. Rowan decides to use random numbers to choose between the three strategies, choosing scissors with probability \(p\), paper with probability \(q\) and rock with probability \(( 1 - p - q )\).
  3. Find and simplify, in terms of \(p\) and \(q\), expressions for the expected number of points won by Rowan for each of Colin's possible choices. Rowan wants his expected winnings to be the same for all three of Colin's possible choices.
  4. Calculate the probability with which Rowan should play each strategy.
OCR D2 2016 June Q5
16 marks Standard +0.3
5 The network below represents a project using activity on arc. The durations of the activities are not yet shown. \includegraphics[max width=\textwidth, alt={}, center]{490ff276-6639-40a1-bffb-dc6967f3ab21-6_597_1257_340_386}
  1. If \(C\) were to turn out to be a critical activity, which two other activities would be forced to be critical?
  2. Complete the table, in the Answer Book, to show the immediate predecessor(s) for each activity. In fact, \(C\) is not a critical activity. Table 1 lists the activities and their durations, in minutes. \begin{table}[h]
    Activity\(A\)\(B\)\(C\)\(D\)\(E\)\(F\)\(G\)\(H\)\(I\)\(J\)
    Duration10151051551015515
    \captionsetup{labelformat=empty} \caption{Table 1}
    \end{table}
  3. Carry out a forward pass and a backward pass through the activity network, showing the early event time and late event time at each vertex of the network. State the minimum project completion time and list the critical activities. Each activity requires one person.
  4. Draw a schedule to show how three people can complete the project in the minimum time, with each activity starting at its earliest possible time. Each box in the Answer Book represents 5 minutes. For each person, write the letter of the activity they are doing in each box, or leave the box blank if the person is resting for those 5 minutes.
  5. Show how two people can complete the project in the minimum time. It is required to reduce the project completion time by 10 minutes. Table 2 lists those activities for which the duration could be reduced by 5 minutes, and the cost of making each reduction. \begin{table}[h]
    Activity\(A\)\(B\)\(C\)\(E\)\(G\)\(H\)\(J\)
    Cost \(( \pounds )\)200400100600100500500
    New duration51051051010
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  6. Explain why the cost of saving 5 minutes by reducing activity \(A\) is more than \(\pounds 200\). Find the cheapest way to complete the project in a time that is 10 minutes less than the original minimum project completion time. State which activities are reduced and the total cost of doing this.
OCR D2 2016 June Q6
16 marks Standard +0.3
6 The table below shows an incomplete dynamic programming tabulation to solve a maximum path problem.
StageStateActionWorkingSuboptimal maximum
\multirow[t]{2}{*}{3}0011
1022
\multirow[t]{4}{*}{2}\multirow[t]{2}{*}{0}0\(1 + 1 = 2\)\multirow[b]{2}{*}{3}
1\(1 + 2 = 3\)
\multirow[t]{2}{*}{1}0\(3 + 1 = 4\)\multirow[t]{2}{*}{4}
1\(1 + 2 = 3\)
\multirow[t]{4}{*}{1}\multirow[t]{2}{*}{0}0\(1 + =\)\multirow{4}{*}{}
1\(0 + =\)
\multirow[t]{2}{*}{1}0\(0 + =\)
1\(1 + =\)
\multirow[t]{2}{*}{0}\multirow[t]{2}{*}{0}0\(2 + =\)\multirow{2}{*}{}
1\(2 + =\)
  1. Complete the working and suboptimal maximum columns on the copy of the table in your Answer Book. Write down the weight of the maximum path and the corresponding route. Give your route using (stage; state) variables. Ken has entered a cake-making competition. The actions in the dynamic programming tabulation above represent the different types of cake that Ken could make. Each competitor must make one cake in each stage of the competition. The rules of the competition mean that, for each competitor, the actions representing their four cakes must form a route from \(( 0 ; 0 )\) to \(( 4 ; 0 )\). The weights in the tabulation are the number of points that Ken can expect to get by making each of the cakes. Each cake is also judged for how well it has been decorated. The number of points that Ken expects to get for decorating each cake is shown below. Ken is not very good at decorating the cakes. He expects to get 0 points for decorating for the cakes that are not listed below.
    Cake(0; 0) to (1; 0)(1; 0) to (2; 1)(1; 1) to (2; 0)(2; 0) to (3; 0)(2; 0) to (3; 1)(2; 1) to (3; 0)(2; 1) to (3; 1)
    Decorating points1121111
  2. Calculate the number of decorating points that Ken can expect if he makes the cakes given in the solution to part (i). When Ken meets the other competitors he realises that he is not good enough to win the competition, so he decides instead to try to win the judges' special award. For each cake, the absolute difference between the score for cake-making and the score for decorating is calculated. The award is given to the person whose biggest absolute difference is least. (Note: to find the absolute difference, calculate larger number-smaller number, or 0 if they are the same.)
  3. Draw the graph that the dynamic programming tabulation represents. Label the vertices using (stage; state) labels with \(( 0 ; 0 )\) at the left hand side and \(( 4 ; 0 )\) at the right hand side. Make the graph into a network by weighting the arcs with the absolute differences.
  4. Use a dynamic programming tabulation to find the minimax route for the absolute differences.
OCR D2 Specimen Q1
9 marks Moderate -0.8
1 [Answer this question on the insert provided.]
Six neighbours have decided to paint their houses in bright colours. They will each use a different colour.
  • Arthur wants to use lavender, orange or tangerine.
  • Bridget wants to use lavender, mauve or pink.
  • Carlos wants to use pink or scarlet.
  • Davinder wants to use mauve or pink.
  • Eric wants to use lavender or orange.
  • Ffion wants to use mauve.
Arthur chooses lavender, Bridget chooses mauve, Carlos chooses pink and Eric chooses orange. This leaves Davinder and Ffion with colours that they do not want.
  1. Draw a bipartite graph on the insert, showing which neighbours ( \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm { F }\) ) want which colours (L, M, O, P, S, T). On a separate diagram on the insert, show the incomplete matching described above.
  2. By constructing alternating paths obtain the complete matching between the neighbours and the colours. Give your paths and show your matching on the insert.
  3. Fill in the table on the insert to show how the Hungarian algorithm could have been used to find the complete matching. (You do not need to carry out the Hungarian algorithm.)
OCR D2 Specimen Q2
9 marks Moderate -0.3
2 A company has organised four regional training sessions to take place at the same time in four different cities. The company has to choose four of its five trainers, one to lead each session. The cost ( \(\pounds 1000\) 's) of using each trainer in each city is given in the table.
\multirow{7}{*}{Trainer}\multirow{2}{*}{}City
LondonGlasgowManchesterSwansea
Adam4324
Betty3542
Clive3633
Dave2643
Eleanor2534
  1. Convert this into a square matrix and then apply the Hungarian algorithm, reducing rows first, to allocate the trainers to the cities at minimum cost.
  2. Betty discovers that she is not available on the date set for the training. Find the new minimum cost allocation of trainers to cities.
OCR D2 Specimen Q3
10 marks Standard +0.3
3 [Answer this question on the insert provided.]
A flying doctor travels between islands using small planes. Each flight has a weight limit that restricts how much he can carry. A plague has broken out on Farr Island and the doctor needs to take several crates of medical supplies to the island. The crates must be carried on the same planes as the doctor. The diagram shows a network with (stage; state) variables at the vertices representing the islands, arcs representing flight routes that can be used, and weights on the arcs representing the number of crates that the doctor can carry on each flight. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{09279013-7088-4db2-99dd-098b32fbcad7-03_506_1084_671_477} \captionsetup{labelformat=empty} \caption{Stage 0}
\end{figure} Stage 1 Stage 2
  1. It is required to find the route from ( \(0 ; 0\) ) to ( \(3 ; 0\) ) for which the minimum number of crates that can be carried on any stage is a maximum (the maximin route). The insert gives a dynamic programming tabulation showing stages, states and actions, together with columns for working out the route minimum at each stage and for indicating the current maximin. Complete the table on the insert sheet and hence find the maximin route and the maximum number of crates that can be carried.
  2. It is later found that the number of crates that can be carried on the route from ( \(2 ; 0\) ) to ( \(3 ; 0\) ) has been recorded incorrectly and should be 15 instead of 5 . What is the maximin route now, and how many crates can be carried?
OCR D2 Specimen Q4
13 marks Moderate -0.5
4 Henry is planning a surprise party for Lucinda. He has left the arrangements until the last moment, so he will hold the party at their home. The table below lists the activities involved, the expected durations, the immediate predecessors and the number of people needed for each activity. Henry has some friends who will help him, so more than one activity can be done at a time.
ActivityDuration (hours)Preceded byNumber of people
A: Telephone other friends2-3
\(B\) : Buy food1A2
C: Prepare food4B5
D: Make decorations3A3
\(E\) : Put up decorations1D3
\(F\) : Guests arrive1C, E1
  1. Draw an activity network to represent these activities and the precedences. Carry out forward and reverse passes to determine the minimum completion time and the critical activities. If Lucinda is expected home at 7.00 p.m., what is the latest time that Henry or his friends can begin telephoning the other friends?
  2. Draw a resource histogram showing time on the horizontal axis and number of people needed on the vertical axis, assuming that each activity starts at its earliest possible start time. What is the maximum number of people needed at any one time?
  3. Now suppose that Henry’s friends can start buying the food and making the decorations as soon as the telephoning begins. Construct a timetable, with a column for 'time' and a column for each person, showing who should do which activity when, in order than the party can be organised in the minimum time using a total of only six people (Henry and five friends). When should the telephoning begin with this schedule?
OCR D2 Specimen Q5
14 marks Standard +0.3
5 [Answer this question on the insert provided.]
Fig. 1 shows a directed flow network. The weight on each arc shows the capacity in litres per second. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{09279013-7088-4db2-99dd-098b32fbcad7-05_620_1082_424_502} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Find the capacity of the cut \(\mathscr { C }\) shown.
  2. Deduce that there is no possible flow from \(S\) to \(T\) in which both arcs leading into \(T\) are saturated. Explain your reasoning clearly. Fig. 2 shows a possible flow of 160 litres per second through the network. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{09279013-7088-4db2-99dd-098b32fbcad7-05_499_1084_1471_500} \captionsetup{labelformat=empty} \caption{Fig. 2}
    \end{figure}
  3. On the diagram in the insert, show the excess capacities and potential backflows for this flow.
  4. Use the labelling procedure to augment the flow as much as possible. Show your working clearly, but do not obscure your answer to part (iii).
  5. Show the final flow that results from part (iv). Explain clearly how you know that this flow is maximal.
OCR D2 Specimen Q6
17 marks Standard +0.8
6 Rose is playing a game against a computer. Rose aims a laser beam along a row, \(A , B\) or \(C\), and, at the same time, the computer aims a laser beam down a column, \(X , Y\) or \(Z\). The number of points won by Rose is determined by where the two laser beams cross. These values are given in the table. The computer loses whatever Rose wins.
Computer
\cline { 2 - 5 }\(X\)\(Y\)\(Z\)
\cline { 2 - 5 } Rose\(A\)134
\(B\)432
\(C\)321
\cline { 2 - 5 }
  1. Find Rose's play-safe strategy and show that the computer's play-safe strategy is \(Y\). How do you know that the game does not have a stable solution?
  2. Explain why Rose should never choose row \(C\) and hence reduce the game to a \(2 \times 3\) pay-off matrix.
  3. Rose intends to play the game a large number of times. She decides to use a standard six-sided die to choose between row \(A\) and row \(B\), so that row \(A\) is chosen with probability \(a\) and row \(B\) is chosen with probability \(1 - a\). Show that the expected pay-off for Rose when the computer chooses column \(X\) is \(4 - 3 a\), and find the corresponding expressions for when the computer chooses column \(Y\) and when it chooses column \(Z\). Sketch a graph showing the expected pay-offs against \(a\), and hence decide on Rose's optimal choice for \(a\). Describe how Rose could use the die to decide whether to play \(A\) or \(B\). The computer is to choose \(X , Y\) and \(Z\) with probabilities \(x , y\) and \(z\) respectively, where \(x + y + z = 1\). Graham is an AS student studying the D1 module. He wants to find the optimal choices for \(x , y\) and \(z\) and starts off by producing a pay-off matrix for the computer.
  4. Graham produces the following pay-off matrix.
    310
    012
    Write down the pay-off matrix for the computer and explain what Graham did to its entries to get the values in his pay-off matrix.
  5. Graham then sets up the linear programming problem: $$\begin{array} { l l } \text { maximise } & P = p - 4 , \\ \text { subject to } & p - 3 x - y \leqslant 0 , \\ & p - y - 2 z \leqslant 0 , \\ & x + y + z \leqslant 1 , \\ \text { and } & p \geqslant 0 , x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$ The Simplex algorithm is applied to the problem and gives \(x = 0.4\) and \(y = 0\). Find the values of \(z , p\) and \(P\) and interpret the solution in the context of the game. \href{http://physicsandmathstutor.com}{physicsandmathstutor.com}
OCR H240/01 2018 June Q1
4 marks Moderate -0.8
1 The points \(A\) and \(B\) have coordinates \(( 1,5 )\) and \(( 4,17 )\) respectively. Find the equation of the straight line which passes through the point \(( 2,8 )\) and is perpendicular to \(A B\). Give your answer in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are constants.
OCR H240/01 2018 June Q2
4 marks Moderate -0.3
2
  1. Use the trapezium rule, with four strips each of width 0.5 , to estimate the value of $$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { x ^ { 2 } } \mathrm {~d} x$$ giving your answer correct to 3 significant figures.
  2. Explain how the trapezium rule could be used to obtain a more accurate estimate.
OCR H240/01 2018 June Q3
4 marks Moderate -0.8
3 In this question you must show detailed reasoning.
Find the two real roots of the equation \(x ^ { 4 } - 5 = 4 x ^ { 2 }\). Give the roots in an exact form.
OCR H240/01 2018 June Q4
4 marks Moderate -0.8
4 Prove algebraically that \(n ^ { 3 } + 3 n - 1\) is odd for all positive integers \(n\).
OCR H240/01 2018 June Q5
8 marks Moderate -0.3
5 The equation of a circle is \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
  1. Find the centre and radius of the circle.
  2. Find the coordinates of any points where the line \(y = 2 x - 3\) meets the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
  3. State what can be deduced from the answer to part (ii) about the line \(y = 2 x - 3\) and the circle \(x ^ { 2 } + y ^ { 2 } + 6 x - 2 y - 10 = 0\).
OCR H240/01 2018 June Q6
9 marks Standard +0.3
6 The cubic polynomial \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = 2 x ^ { 3 } - 7 x ^ { 2 } + 2 x + 3\).
  1. Given that ( \(x - 3\) ) is a factor of \(\mathrm { f } ( x )\), express \(\mathrm { f } ( x )\) in a fully factorised form.
  2. Sketch the graph of \(y = \mathrm { f } ( x )\), indicating the coordinates of any points of intersection with the axes.
  3. Solve the inequality \(\mathrm { f } ( x ) < 0\), giving your answer in set notation.
  4. The graph of \(y = \mathrm { f } ( x )\) is transformed by a stretch parallel to the \(x\)-axis, scale factor \(\frac { 1 } { 2 }\). Find the equation of the transformed graph.
OCR H240/01 2018 June Q7
9 marks Moderate -0.3
7 Chris runs half marathons, and is following a training programme to improve his times. His time for his first half marathon is 150 minutes. His time for his second half marathon is 147 minutes. Chris believes that his times can be modelled by a geometric progression.
  1. Chris sets himself a target of completing a half marathon in less than 120 minutes. Show that this model predicts that Chris will achieve his target on his thirteenth half marathon.
  2. After twelve months Chris has spent a total of 2974 minutes, to the nearest minute, running half marathons. Use this model to find how many half marathons he has run.
  3. Give two reasons why this model may not be appropriate when predicting the time for a half marathon.
OCR H240/01 2018 June Q8
7 marks Standard +0.3
8
  1. Find the first three terms in the expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\).
  2. The expansion of \(\frac { a + b x } { \sqrt { 4 - x } }\) is \(16 - x \ldots\). Find the values of the constants \(a\) and \(b\).
OCR H240/01 2018 June Q9
7 marks Moderate -0.3
9 The function f is defined for all real values of \(x\) as \(\mathrm { f } ( x ) = c + 8 x - x ^ { 2 }\), where \(c\) is a constant.
  1. Given that the range of f is \(\mathrm { f } ( x ) \leqslant 19\), find the value of \(c\).
  2. Given instead that \(\mathrm { ff } ( 2 ) = 8\), find the possible values of \(c\).
OCR H240/01 2018 June Q10
10 marks Standard +0.3
10 A curve has parametric equations \(x = t + \frac { 2 } { t }\) and \(y = t - \frac { 2 } { t }\), for \(t \neq 0\).
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), giving your answer in its simplest form.
  2. Explain why the curve has no stationary points.
  3. By considering \(x + y\), or otherwise, find a cartesian equation of the curve, giving your answer in a form not involving fractions or brackets.
OCR H240/01 2018 June Q11
11 marks Standard +0.8
11 In a science experiment a substance is decaying exponentially. Its mass, \(M\) grams, at time \(t\) minutes is given by \(M = 300 e ^ { - 0.05 t }\).
  1. Find the time taken for the mass to decrease to half of its original value. A second substance is also decaying exponentially. Initially its mass was 400 grams and, after 10 minutes, its mass was 320 grams.
  2. Find the time at which both substances are decaying at the same rate.
OCR H240/01 2018 June Q12
10 marks Challenging +1.8
12 In this question you must show detailed reasoning. \includegraphics[max width=\textwidth, alt={}, center]{1ba9fa5f-310f-4429-9bd1-4004852d5b3e-6_716_479_292_794} The diagram shows the curve \(y = \frac { 4 \cos 2 x } { 3 - \sin 2 x }\), for \(x \geqslant 0\), and the normal to the curve at the point \(\left( \frac { 1 } { 4 } \pi , 0 \right)\). Show that the exact area of the shaded region enclosed by the curve, the normal to the curve and the \(y\)-axis is \(\ln \frac { 9 } { 4 } + \frac { 1 } { 128 } \pi ^ { 2 }\).
[0pt] [10]