Questions — OCR MEI (4333 questions)

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OCR MEI M1 Q7
Standard +0.3
7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9a79f274-1a3f-4d11-9775-313d82075035-004_341_1107_484_498} \captionsetup{labelformat=empty} \caption{Fig. 7}
\end{figure}
  1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
  2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
  3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
  4. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)
OCR MEI FP3 2007 June Q1
24 marks Challenging +1.2
1 Three planes \(P , Q\) and \(R\) have the following equations. $$\begin{array} { l l } \text { Plane } P : & 8 x - y - 14 z = 20 \\ \text { Plane } Q : & 6 x + 2 y - 5 z = 26 \\ \text { Plane } R : & 2 x + y - z = 40 \end{array}$$ The line of intersection of the planes \(P\) and \(Q\) is \(K\).
The line of intersection of the planes \(P\) and \(R\) is \(L\).
  1. Show that \(K\) and \(L\) are parallel lines, and find the shortest distance between them.
  2. Show that the shortest distance between the line \(K\) and the plane \(R\) is \(5 \sqrt { 6 }\). The line \(M\) has equation \(\mathbf { r } = ( \mathbf { i } - 4 \mathbf { j } ) + \lambda ( 5 \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k } )\).
  3. Show that the lines \(K\) and \(M\) intersect, and find the coordinates of the point of intersection.
  4. Find the shortest distance between the lines \(L\) and \(M\).
OCR MEI FP3 2007 June Q2
24 marks Challenging +1.3
2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
  3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
  4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.
OCR MEI FP3 2007 June Q3
24 marks Challenging +1.8
3 The curve \(C\) has equation \(y = \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } \ln x\), and \(a\) is a constant with \(a \geqslant 1\).
  1. Show that the length of the arc of \(C\) for which \(1 \leqslant x \leqslant a\) is \(\frac { 1 } { 2 } a ^ { 2 } + \frac { 1 } { 4 } \ln a - \frac { 1 } { 2 }\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(\boldsymbol { y }\)-axis.
  3. Show that the radius of curvature of \(C\) at the point where \(x = a\) is \(a \left( a + \frac { 1 } { 4 a } \right) ^ { 2 }\).
  4. Find the centre of curvature corresponding to the point \(\left( 1 , \frac { 1 } { 2 } \right)\) on \(C\). \(C\) is one member of the family of curves defined by \(y = p x ^ { 2 } - p ^ { 2 } \ln x\), where \(p\) is a parameter.
  5. Find the envelope of this family of curves.
OCR MEI FP3 2007 June Q4
24 marks Challenging +1.8
4
  1. Prove that, for a group of order 10, every proper subgroup must be cyclic. The set \(M = \{ 1,2,3,4,5,6,7,8,9,10 \}\) is a group under the binary operation of multiplication modulo 11.
  2. Show that \(M\) is cyclic.
  3. List all the proper subgroups of \(M\). The group \(P\) of symmetries of a regular pentagon consists of 10 transformations $$\{ \mathrm { A } , \mathrm {~B} , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm {~F} , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm {~J} \}$$ and the binary operation is composition of transformations. The composition table for \(P\) is given below.
    ABCDEFGHIJ
    ACJGHABIFED
    BFEHGBADCJI
    CGDIFCJEBAH
    DJCBEDGFIHA
    EABCDEFGHIJ
    FHIDCFEJABG
    GIHEBGDAJCF
    HDGJAHIBEFC
    IEFAJIHCDGB
    JBAFIJCHGDE
    One of these transformations is the identity transformation, some are rotations and the rest are reflections.
  4. Identify which transformation is the identity, which are rotations and which are reflections.
  5. State, giving a reason, whether \(P\) is isomorphic to \(M\).
  6. Find the order of each element of \(P\).
  7. List all the proper subgroups of \(P\).
OCR MEI FP3 2007 June Q5
24 marks Challenging +1.2
5 A computer is programmed to generate a sequence of letters. The process is represented by a Markov chain with four states, as follows. The first letter is \(A , B , C\) or \(D\), with probabilities \(0.4,0.3,0.2\) and 0.1 respectively.
After \(A\), the next letter is either \(C\) or \(D\), with probabilities 0.8 and 0.2 respectively.
After \(B\), the next letter is either \(C\) or \(D\), with probabilities 0.1 and 0.9 respectively.
After \(C\), the next letter is either \(A\) or \(B\), with probabilities 0.4 and 0.6 respectively.
After \(D\), the next letter is either \(A\) or \(B\), with probabilities 0.3 and 0.7 respectively.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Use your calculator to find \(\mathbf { P } ^ { 4 }\) and \(\mathbf { P } ^ { 7 }\). (Give elements correct to 4 decimal places.)
  3. Find the probability that the 8th letter is \(C\).
  4. Find the probability that the 12th letter is the same as the 8th letter.
  5. By investigating the behaviour of \(\mathbf { P } ^ { n }\) when \(n\) is large, find the probability that the ( \(n + 1\) )th letter is \(A\) when
    (A) \(n\) is a large even number,
    (B) \(n\) is a large odd number. The program is now changed. The initial probabilities and the transition probabilities are the same as before, except for the following. After \(D\), the next letter is \(A , B\) or \(D\), with probabilities \(0.3,0.6\) and 0.1 respectively.
  6. Write down the new transition matrix \(\mathbf { Q }\).
  7. Verify that \(\mathbf { Q } ^ { n }\) approaches a limit as \(n\) becomes large, and hence write down the equilibrium probabilities for \(A , B , C\) and \(D\).
  8. When \(n\) is large, find the probability that the \(( n + 1 )\) th, \(( n + 2 )\) th and \(( n + 3 )\) th letters are DDD.
OCR MEI FP3 2016 June Q1
24 marks Challenging +1.2
1 Positions in space around an aerodrome are modelled by a coordinate system with a point on the runway as the origin, O . The \(x\)-axis is east, the \(y\)-axis is north and the \(z\)-axis is vertically upwards. Units of distance are kilometres. Units of time are hours.
At time \(t = 0\), an aeroplane, P , is at \(( 3,4,8 )\) and is travelling in a direction \(\left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)\) at a constant speed of \(900 \mathrm { kmh } ^ { - 1 }\).
  1. Find the least distance of the path of P from the point O . At time \(t = 0\), a second aeroplane, Q , is at \(( 80,40,10 )\). It is travelling in a straight line towards the point O . Its speed is constant at \(270 \mathrm { kmh } ^ { - 1 }\).
  2. Show that the shortest distance between the paths of the two aeroplanes is 2.24 km correct to three significant figures.
  3. By finding the points on the paths where the shortest distance occurs and the times at which the aeroplanes are at these points, show that in fact the aeroplanes are never this close.
  4. A third aeroplane, R , is at position \(( 29,19,5.5 )\) at time \(t = 0\) and is travelling at \(285 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in a direction \(\left( \begin{array} { c } 18 \\ 6 \\ 1 \end{array} \right)\). Given that Q is in the process of landing and cannot change course, show that R needs to be instructed to alter course or change speed.
OCR MEI FP3 2016 June Q2
24 marks
2 A surface, S , has equation \(z = 3 x ^ { 2 } + 6 x y + y ^ { 3 }\).
  1. Find the equation of the section where \(y = 1\) in the form \(z = \mathrm { f } ( x )\). Sketch this section. Find in three-dimensional vector form the equation of the line of symmetry of this section.
  2. Show that there are two stationary points on S , at \(\mathrm { O } ( 0,0,0 )\) and at \(\mathrm { P } ( - 2,2 , - 4 )\).
  3. Given that the point ( \(- 2 + h , 2 + k , \lambda\) ) lies on the surface, show that $$\lambda = - 4 + 3 ( h + k ) ^ { 2 } + k ^ { 2 } ( k + 3 ) .$$ By considering small values of \(h\) and \(k\), deduce that there is a local minimum at P .
  4. By considering small values of \(x\) and \(y\), show that the stationary point at O is neither a maximum nor a minimum.
  5. Given that \(18 x + 18 y - z = d\) is a tangent plane to S , find the two possible values of \(d\).
OCR MEI FP3 2016 June Q3
24 marks Challenging +1.2
3 Fig. 3 shows the curve with parametric equations \(x = t - 3 t ^ { 3 } , y = 1 + 3 t ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{07eaad51-dc00-44d2-8bff-8652d62902ec-4_634_1294_388_386} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Show that the values of \(t\) where the curve cuts the \(y\)-axis are \(t = 0 , \pm \frac { 1 } { \sqrt { 3 } }\). Write down the corresponding values of \(y\).
  2. Find the radius and centre of curvature when \(t = \frac { 1 } { \sqrt { 3 } }\). The arc of the curve given by \(0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }\) is denoted by \(C\).
  3. Find the length of \(C\).
  4. Show that the area of the curved surface generated when \(C\) is rotated about the \(y\)-axis through \(2 \pi\) radians is \(\frac { \pi } { 3 }\).
OCR MEI FP3 2016 June Q4
24 marks Challenging +1.2
4
  1. The elements of the set \(P = \{ 1,3,9,11 \}\) are combined under the binary operation, *, defined as multiplication modulo 16.
    1. Demonstrate associativity for the elements \(3,9,11\) in that order. Assuming associativity holds in general, show that \(P\) forms a group under the binary operation *.
    2. Write down the order of each element.
    3. Write down all subgroups of \(P\).
    4. Show that the group in part (i) is cyclic.
  2. Now consider a group of order 4 containing the identity element \(e\) and the two distinct elements, \(a\) and \(b\), where \(a ^ { 2 } = b ^ { 2 } = e\). Construct the composition table. Show that the group is non-cyclic.
  3. Now consider the four matrices \(\mathbf { I } , \mathbf { X } , \mathbf { Y }\) and \(\mathbf { Z }\) where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0 \\ 0 & 1 \end{array} \right) , \mathbf { X } = \left( \begin{array} { r r } 1 & 0 \\ 0 & - 1 \end{array} \right) , \mathbf { Y } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right) , \mathbf { Z } = \left( \begin{array} { r r } - 1 & 0 \\ 0 & - 1 \end{array} \right) .$$ The group G consists of the set \(\{ \mathbf { I } , \mathbf { X } , \mathbf { Y } , \mathbf { Z } \}\) with binary operation matrix multiplication. Determine which of the groups in parts (a) and (b) is isomorphic to G, and specify the isomorphism.
  4. The distinct elements \(\{ p , q , r , s \}\) are combined under the binary operation \({ } ^ { \circ }\). You are given that \(p ^ { \circ } q = r\) and \(q ^ { \circ } p = s\). By reference to the group axioms, prove that \(\{ p , q , r , s \}\) is not a group under \({ } ^ { \circ }\). Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}
OCR MEI FP3 2016 June Q5
24 marks Challenging +1.2
5 Each day that Adam is at work he carries out one of three tasks A, B or C. Each task takes a whole day. Adam chooses the task to carry out on each day according to the following set of three rules.
  1. If, on any given day, he has worked on task A then the next day he will choose task A with probability 0.75 , and tasks B and C with equal probability.
  2. If, on any given day, he has worked on task B then the next day he will choose task B or task C with equal probability but will never choose task A .
  3. If, on any given day, he has worked on task C then the next day he will choose task A with probability \(p\) and tasks B and C with equal probability.
    1. Write down the transition matrix.
    2. Over a long period Adam carries out the tasks \(\mathrm { A } , \mathrm { B }\) and C with equal frequency. Find the value of \(p\).
    3. On day 1 Adam chooses task A . Find the probability that he also chooses task A on day 5 .
    Adam decides to change rule 3 as follows.
    If, on any given day, he has worked on task C then the next day he will choose tasks \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) with probabilities \(0.4,0.3,0.3\) respectively.
  4. On day 1 Adam chooses task A. Find the probability that he chooses the same task on day 7 as he did on day 4 .
  5. On a particular day, Adam chooses task A. Find the expected number of consecutive further days on which he will choose A. Adam changes all three rules again as follows.
    • If he works on A one day then on the next day he chooses C .
    • If he works on B one day then on the next day he chooses A or C each with probability 0.5.
    • If he works on C one day then on the next day he chooses A or B each with probability 0.5 .
    • Find the long term probabilities for each task.
OCR MEI D1 Q1
14 marks Moderate -0.8
1 The bipartite graph in Fig. 1 represents a board game for two players. At each turn a player tosses a coin and moves their counter. The graph shows which square the counter is moved to if the coin shows heads, and which square if it shows tails. Each player starts with their counter on square 1. Play continues until one player gets their counter to square 9 and wins. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5d8d35b7-e4ba-4bc0-93a1-0cae58093a02-002_723_1287_569_425} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Draw a tree to show all of the possibilities for the player's first three moves.
  2. Show how a player can win in 3 turns.
  3. List all squares which it is possible for a counter to occupy after 3 turns.
  4. Show that a game can continue indefinitely.
OCR MEI D1 Q3
12 marks Moderate -0.8
3 The following algorithm finds the highest common factor of two positive integers. ("int (x)" stands for the integer part of x, e.g. int (7.8) = 7.) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5d8d35b7-e4ba-4bc0-93a1-0cae58093a02-004_888_693_422_717} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Run the algorithm with \(\mathrm { A } = 84\) and \(\mathrm { B } = 660\), showing all of your calculations.
  2. Run the algorithm with \(\mathrm { A } = 660\) and \(\mathrm { B } = 84\), showing as many calculations as are necessary.
  3. The algorithm is run with \(\mathrm { A } = 30\) and \(\mathrm { B } = 42\), and the result is shown in Table 3.2 below. \begin{table}[h]
    ABQR 1R 2
    3042112
    123026
    6
    61220
    \captionsetup{labelformat=empty} \caption{Print 6}
    \end{table} Table 3.2 The first line of the table shows that \(12 = 42 - 1 \times 30\).
    Use the second line to obtain a similar expression for 6 in terms of 30 and 12.
    Hence express 6 in the form \(\mathrm { m } \times 30 - \mathrm { n } \times 42\), where m and n are integers.
OCR MEI D1 2005 January Q1
8 marks Moderate -0.8
1 The bipartite graph in Fig. 1 represents a board game for two players. At each turn a player tosses a coin and moves their counter. The graph shows which square the counter is moved to if the coin shows heads, and which square if it shows tails. Each player starts with their counter on square 1. Play continues until one player gets their counter to square 9 and wins. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9ee9306-18ca-42b3-9f2e-b23849374b5e-2_723_1287_569_425} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Draw a tree to show all of the possibilities for the player's first three moves.
  2. Show how a player can win in 3 turns.
  3. List all squares which it is possible for a counter to occupy after 3 turns.
  4. Show that a game can continue indefinitely.
OCR MEI D1 2005 January Q3
8 marks Easy -1.2
3 The following algorithm finds the highest common factor of two positive integers. ("int (x)" stands for the integer part of x, e.g. int (7.8) = 7.) \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9ee9306-18ca-42b3-9f2e-b23849374b5e-4_888_693_422_717} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
\end{figure}
  1. Run the algorithm with \(\mathrm { A } = 84\) and \(\mathrm { B } = 660\), showing all of your calculations.
  2. Run the algorithm with \(\mathrm { A } = 660\) and \(\mathrm { B } = 84\), showing as many calculations as are necessary.
  3. The algorithm is run with \(\mathrm { A } = 30\) and \(\mathrm { B } = 42\), and the result is shown in Table 3.2 below. \begin{table}[h]
    ABQR 1R 2
    3042112
    123026
    6
    61220
    \captionsetup{labelformat=empty} \caption{Print 6}
    \end{table} Table 3.2 The first line of the table shows that \(12 = 42 - 1 \times 30\).
    Use the second line to obtain a similar expression for 6 in terms of 30 and 12.
    Hence express 6 in the form \(\mathrm { m } \times 30 - \mathrm { n } \times 42\), where m and n are integers.
OCR MEI D1 2005 January Q5
16 marks Moderate -0.8
5 There is an insert for use in parts (iii) and (iv) of this question.
This question concerns the simulation of cars passing through two sets of pedestrian controlled traffic lights. The time intervals between cars arriving at the first set of lights are distributed according to Table 5.1. \begin{table}[h]
Time interval (seconds)251525
Probability\(\frac { 3 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 1 } { 7 }\)\(\frac { 1 } { 7 }\)
\captionsetup{labelformat=empty} \caption{Table 5.1}
\end{table}
  1. Give an efficient rule for using two-digit random numbers to simulate arrival intervals.
  2. Use two-digit random numbers from the list below to simulate the arrival times of five cars at the first lights. The first car arrives at the time given by the first arrival interval. Random numbers: \(24,01,99,89,77,19,58,42\) The two sets of traffic lights are 23 seconds driving time apart. Moving cars are always at least 2 seconds apart. If there is a queue at a set of lights, then when the red light ends the first car in the queue moves off immediately, the second car 2 seconds later, the third 2 seconds after that, etc. In this simple model there is to be no consideration of accelerations or decelerations, and the lights are either red or green. Table 5.2 shows the times when the lights are red. \begin{table}[h]
    \multirow{2}{*}{
    first set
    of lights
    }    		red start time1450105155
    \cline { 2 - 6 }red end time2965120170
    \multirow{2}{*}{
    second set
    of lights
    }    		red start time1055105150
    \cline { 2 - 6 }red end time2570120165
    \captionsetup{labelformat=empty} \caption{Table 5.2}
    \end{table}
  3. Complete the table in the insert to simulate the passage of 10 cars through both sets of traffic lights. Use the arrival times given there.
  4. Find the mean delay experienced by these cars in passing through each set of lights.
  5. How could the output from this simulation model be made more reliable?
OCR MEI D1 2005 January Q6
16 marks Moderate -0.5
6 A recipe for jam states that the weight of sugar used must be between the weight of fruit used and four thirds of the weight of fruit used. Georgia has 10 kg of fruit available and 11 kg of sugar.
  1. Define two variables and formulate inequalities in those variables to model this information.
  2. Draw a graph to represent your inequalities.
  3. Find the vertices of your feasible region and identify the points which would represent the best mix of ingredients under each of the following circumstances.
    (A) There is to be as much jam as possible, given that the weight of jam produced is the sum of the weights of the fruit and the sugar.
    (B) There is to be as much jam as possible, given that it is to have the lowest possible proportion of sugar.
    (C) There is to be as much jam as possible, given that it is to have the highest possible proportion of sugar.
    (D) Fruit costs \(\pounds 1\) per kg, sugar costs 50 p per kg and the objective is to produce as much jam as possible within a budget of \(\pounds 15\).
OCR MEI D1 2006 January Q1
8 marks Moderate -0.3
1 Table 1 shows a precedence table for a project. \begin{table}[h]
ActivityImmediate predecessorsDuration (days)
A-5
B-3
CA3
DA, B4
EA, B5
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Draw an activity-on-arc network to represent the precedences.
  2. Find the early event time and late event time for each vertex of your network, and list the critical activities.
  3. Extra resources become available which enable the durations of three activities to be reduced, each by up to two days. Which three activities should have their durations reduced so as to minimise the completion time of the project? What will be the new minimum project completion time?
OCR MEI D1 2006 January Q3
8 marks Moderate -0.5
3 Fig. 3 shows a graph representing the seven bus journeys run each day between four rural towns. Each directed arc represents a single bus journey. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ee39642f-f323-4614-a02a-4500199626de-4_317_515_392_772} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Show that if there is only one bus, which is in service at all times, then it must start at one town and end at a different town. Give the start town and the end town.
  2. Show that there is only one Hamilton cycle in the graph. Show that, if an extra journey is added from your end town to your start town, then there is still only one Hamilton cycle.
  3. A tourist is staying in town B. Give a route for her to visit every town by bus, visiting each town only once and returning to B . Section B (48 marks)
OCR MEI D1 2006 January Q4
16 marks Moderate -0.5
4 Table 4 shows the butter and sugar content in two recipes. The first recipe is for 1 kg of toffee and the second is for 1 kg of fudge. \begin{table}[h] \section*{Table 6.1} (ii) Specify an efficient rule for using one-digit random numbers to simulate the time taken at the till by customers purchasing fuel. Table 6.2 shows the distribution of time taken at the till by customers who are not buying fuel.
Time taken (mins)11.522.53
Probability\(\frac { 1 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 2 } { 7 }\)\(\frac { 1 } { 7 }\)\(\frac { 1 } { 7 }\)
\section*{Table 6.2} (iii) Specify an efficient rule for using two-digit random numbers to simulate the time taken at the till by customers not buying fuel. What is the advantage in using two-digit random numbers instead of one-digit random numbers in this part of the question? The table in the insert shows a partially completed simulation study of 10 customers arriving at the till.
(iv) Complete the table using the random numbers which are provided.
(v) Calculate the mean total time spent queuing and paying.
OCR MEI D1 2008 January Q1
8 marks Moderate -0.8
1 The graph shows routes that are available to an international lorry driver. The solid arcs represent motorways and the broken arcs represent ferry crossings. \includegraphics[max width=\textwidth, alt={}, center]{dfe6db33-33d0-4dff-95f7-fbf097e3963e-2_668_1131_587_466}
  1. Give a route from Milan to Chania involving exactly two ferry crossings. How many such routes are there?
  2. Give a route from Milan to Chania involving exactly three ferry crossings. How many such routes are there?
  3. Give a route from Milan to Chania using as many ferry crossings as possible, without repeating any arc.
    [0pt]
  4. Give a route leaving Piraeus and finishing elsewhere which uses every arc once and only once.[3]
OCR MEI D1 2008 January Q2
8 marks Moderate -0.8
2 Consider the following linear programming problem.
Maximise $$\mathrm { P } = 6 x + 7 y$$ subject to $$\begin{aligned} & 2 x + 3 y \leqslant 9 \\ & 3 x + 2 y \leqslant 12 \\ & x \geqslant 0 \\ & y \geqslant 0 \end{aligned}$$
  1. Use a graphical approach to solve the problem.
  2. Give the optimal values of \(x , y\) and P when \(x\) and \(y\) are integers.
OCR MEI D1 2008 January Q3
8 marks Easy -2.0
3 The following algorithm (J. M. Oudin, 1940) claims to compute the date of Easter Sunday in the Gregorian calendar system.
The algorithm uses the year, y, to give the month, m, and day, d, of Easter Sunday.
All variables are integers and all remainders from division are dropped. For example, 7 divided by 3 is 2 remainder 1 . The remainder is dropped, giving the answer 2. $$\begin{aligned} & c = y / 100 \\ & n = y - 19 \times ( y / 19 ) \\ & k = ( c - 17 ) / 25 \\ & i = c - ( c / 4 ) - ( c - k ) / 3 + ( 19 \times n ) + 15 \\ & i = i - 30 \times ( i / 30 ) \\ & i = i - ( i / 28 ) \times ( 1 - ( i / 28 ) \times ( 29 / ( i + 1 ) ) \times ( ( 21 - n ) / 11 ) ) \\ & j = y + ( y / 4 ) + i + 2 - c + ( c / 4 ) \\ & j = j - 7 \times ( j / 7 ) \\ & p = i - j \\ & m = 3 + ( p + 40 ) / 44 \\ & d = p + 28 - 31 \times ( m / 4 ) \end{aligned}$$ For example, for 2008: \(\mathrm { y } = 2008\) \(\mathrm { c } = 2008 / 100 = 20\) \(n = 2008 - 19 \times ( 2008 / 19 ) = 2008 - 19 \times ( 105 ) = 13\), etc.
Complete the calculation for 2008.
OCR MEI D1 2008 January Q4
16 marks Moderate -0.5
4 In a population colonizing an island 40\% of the first generation (parents) have brown eyes, \(40 \%\) have blue eyes and \(20 \%\) have green eyes. Offspring eye colour is determined according to the following rules. \section*{Eye colours of parents Eye colour of offspring} (1) both brown
(2) one brown and one blue \(50 \%\) brown and \(50 \%\) blue
(3) one brown and one green blue
(4) both blue \(25 \%\) brown, \(50 \%\) blue and \(25 \%\) green
(5) one blue and one green 50\% blue and \(50 \%\) green
(6) both green green
  1. Give an efficient rule for using 1-digit random numbers to simulate the eye colour of a parent randomly selected from the colonizing population.
  2. Give an efficient rule for using 1-digit random numbers to simulate the eye colour of offspring born of parents both of whom have blue eyes. The table in your answer book shows an incomplete simulation in which parent eye colours have been randomly selected, but in which offspring eye colours remain to be determined or simulated.
  3. Complete the table using the given random numbers where needed. (You will need your own rules for cases \(( 2 )\) and 5 .)
    Each time you use a random number, explain how you decide which eye colour for the offspring. \(\square\)
OCR MEI D1 2008 January Q5
16 marks Moderate -0.3
5 The table shows some of the activities involved in building a block of flats. The table gives their durations and their immediate predecessors.
ActivityDuration (weeks)Immediate Predecessors
ASurvey sites8-
BPurchase land22A
CSupply materials10-
DSupply machinery4-
EExcavate foundations9B, D
FLay drains11B, C, D
GBuild walls9E, F
HLay floor10E, F
IInstall roof3G
JInstall electrics5G
  1. Draw an activity on arc network for these activities.
  2. Mark on your diagram the early and late times for each event. Give the minimum completion time and the critical activities. Each of the tasks E, F, H and J can be speeded up at extra cost. The maximum number of weeks by which each task can be shortened, and the extra cost for each week that is saved, are shown in the table below.
    TaskEFHJ
    Maximum number of weeks by
    which task may be shortened
    		3313
    Cost per week of shortening task
    (in thousands of pounds)
    		3015620
  3. Find the new shortest time for the flats to be completed.
  4. List the activities which will need to be speeded up to achieve the shortest time found in part (iii), and the times by which each must be shortened.
  5. Find the total extra cost needed to achieve the new shortest time.