Questions D1 (899 questions)

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OCR MEI D1 2008 June Q2
2 The following algorithm acts on a list of three or more numbers.
Step 1: Set both X and Y equal to the first number on the list.
Step 2: If there is no next number then go to Step 5.
Step 3: If the next number on the list is bigger than X then set X equal to it. If it is less than Y then set Y equal to it. Step 4: Go to Step 2.
Step 5: Delete a number equal to X from the list and delete a number equal to Y from the list.
Step 6: If there is one number left then record it as the answer and stop.
Step 7: If there are two numbers left then record their mean as the answer and stop.
Step 8: Go to Step 1.
  1. Apply the algorithm to the list \(5,14,153,6,24,2,14,15\), counting the number of comparisons which you have to make.
  2. Apply the algorithm to the list \(5,14,153,6,24,2,14\), counting the number of comparisons which you have to make.
  3. Say what the algorithm is finding.
  4. The order of the algorithm is quadratic. Explain what this means when it is applied to long lists.
OCR MEI D1 2008 June Q3
3 The graph represents four towns together with (two-way) roads connecting them.
\includegraphics[max width=\textwidth, alt={}, center]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-4_191_286_319_886} A path is a set of connected arcs linking one vertex to another. A path contains no repeated vertex. \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 2 }\) and \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 3 } \rightarrow \mathrm {~T} _ { 2 }\) are paths.
  1. There are six paths from \(\mathrm { T } _ { 1 }\). List them.
  2. List the paths from \(\mathrm { T } _ { 4 }\).
  3. How many paths are there altogether? For this question a route is defined to be a path in which the direction of the arcs is not relevant. Thus \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 2 }\) and \(\mathrm { T } _ { 2 } \rightarrow \mathrm {~T} _ { 1 }\) are the same route. Similarly \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 3 } \rightarrow \mathrm {~T} _ { 2 }\) and \(\mathrm { T } _ { 2 } \rightarrow \mathrm {~T} _ { 3 } \rightarrow \mathrm {~T} _ { 1 }\) are the same route (but note that \(\mathrm { T } _ { 1 } \rightarrow \mathrm {~T} _ { 2 } \rightarrow \mathrm {~T} _ { 3 }\) is different).
  4. How many routes are there altogether?
OCR MEI D1 2008 June Q4
2 marks
4 Joe is to catch a plane to go on holiday. He has arranged to leave his car at a car park near to the airport. There is a bus service from the car park to the airport, and the bus leaves when there are at least 15 passengers on board. Joe is delayed getting to the car park and arrives needing the bus to leave within 15 minutes if he is to catch his plane. He is the \(10 ^ { \text {th } }\) passenger to board the bus, so he has to wait for another 5 passengers to arrive. The distribution of the time intervals between car arrivals and the distribution of the number of passengers per car are given below.
Time interval between cars (minutes)12345
Probability\(\frac { 1 } { 10 }\)\(\frac { 3 } { 10 }\)\(\frac { 2 } { 5 }\)\(\frac { 1 } { 10 }\)\(\frac { 1 } { 10 }\)
Number of passengers per car123456
Probability\(\frac { 1 } { 6 }\)\(\frac { 1 } { 3 }\)\(\frac { 1 } { 12 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 12 }\)\(\frac { 1 } { 12 }\)
  1. Give an efficient rule for using 2-digit random numbers to simulate the intervals between car arrivals.
  2. Give an efficient rule for using 2-digit random numbers to simulate the number of passengers in a car.
  3. The incomplete table in your answer book shows the results of nine simulations of the situation. Complete the table, showing in each case whether or not Joe catches his plane.
  4. Use the random numbers provided in your answer book to run a tenth simulation.
    [0pt]
  5. Estimate the probability of Joe catching his plane. State how you could improve your estimate. [2]
OCR MEI D1 2008 June Q5
5
  1. The graphs below illustrate the precedences involved in running two projects, each consisting of the same activities \(\mathrm { A } , \mathrm { B } , \mathrm { C } , \mathrm { D }\) and E . \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Project 1} \includegraphics[alt={},max width=\textwidth]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-6_280_385_429_495}
    \end{figure} \begin{figure}[h]
    \captionsetup{labelformat=empty} \caption{Project 2} \includegraphics[alt={},max width=\textwidth]{8eba759f-38bc-4d14-ac65-9a0ee6c79741-6_255_392_429_1187}
    \end{figure}
    1. For one activity the precedences in the two projects are different. State which activity and describe the difference.
    2. The table below shows the durations of the five activities.
      ActivityABCDE
      Duration21\(x\)32
      Give the total time for project 1 for all possible values of \(x\).
      Give the total time for project 2 for all possible values of \(x\).
  2. The durations and precedences for the activities in a project are shown in the table.
    ActivityDurationImmediate predecessors
    R2-
    S1-
    T5-
    w3R, S
    X2R, S, T
    Y3R
    Z1W, Y
    1. Draw an activity on arc network to represent this information.
    2. Find the early time and the late time for each event. Give the project duration and list the critical activities.
OCR MEI D1 2008 June Q6
6 The matrix gives the lengths of the arcs of a network.
ABCDEF
A-107-95
B10--1-4
C7---3-
D-1--2-
E9-32--
F54----
  1. Using the copy of the matrix in your answer book, apply the tabular form of Prim's algorithm to find a minimum connector for the network. Start by choosing vertex A and show the order in which you include vertices.
    List the arcs in your connector and give its total length. Serena takes a different approach to find a minimum connector. She first uses Dijkstra's algorithm to find shortest paths from A to each of the other vertices. She then uses the arcs in those paths to construct a connector.
  2. Draw the network using the vertices printed in your answer book.
  3. Apply Serena's method to produce a connector. List the arcs in the connector and give its total length. Serena adapts her method by starting from each vertex in turn, producing six connectors, from which she chooses the best.
  4. Serena's approach will not find the minimum connector in all networks, but it is an algorithm. What is its algorithmic complexity? Justify your answer.
OCR MEI D1 2009 June Q1
1 The numbers on opposite faces of the die shown (a standard die) add up to 7. The adjacency graph for the die is a graph which has vertices representing faces. In the adjacency graph two vertices are joined with an arc if they share an edge of the die. For example, vertices 2 and 6 are joined by an arc because they share an edge of the die.
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-2_246_213_488_1608}
  1. List the pairs of numbers which are opposite each other.
  2. Draw the adjacency graph.
  3. Identify and sketch a solid which has the following adjacency graph.
    \includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-2_287_307_1027_879}
OCR MEI D1 2009 June Q2
2 In this question INT( \(m\) ) means the integer part of \(m\). Thus INT(3.5) \(= 3\) and INT(4) \(= 4\).
A game for two players starts with a number, \(n\), of counters. Players alternately pick up a number of counters, at least 1 and not more than half of those left. The player forced to pick up the last counter is the loser. Arif programs his computer to play the game, using the rule "pick up INT(half of the remaining counters), or the last counter if forced".
  1. You are to play against Arif's computer with \(n = 5\) and with Arif's computer going first. What happens at each turn?
  2. You are to play against Arif's computer with \(n = 6\) and with Arif's computer going first. What happens at each turn?
  3. Now play against Arif's computer with \(n = 7\) and with Arif's computer going first. Describe what happens.
OCR MEI D1 2009 June Q3
3 Consider the following linear programming problem:
Maximise \(\quad 3 x + 4 y\)
subject to \(\quad 2 x + 5 y \leqslant 60\)
\(x + 2 y \leqslant 25\)
\(x + y \leqslant 18\)
  1. Graph the inequalities and hence solve the LP.
  2. The right-hand side of the second inequality is increased from 25 . At what new value will this inequality become redundant?
OCR MEI D1 2009 June Q4
4 The diagram represents a very simple maze with two vertices, A and B. At each vertex a rat either exits the maze or runs to the other vertex, each with probability 0.5 . The rat starts at vertex A .
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_79_930_534_571}
  1. Describe how to use 1-digit random numbers to simulate this situation.
  2. Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex, and calculate the mean number of times it runs between vertices before exiting. The second diagram represents a maze with three vertices, A, B and C. At each vertex there are three possibilities, and the rat chooses one, each with probability \(1 / 3\). The rat starts at vertex A.
    \includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-4_566_889_1082_589}
  3. Describe how to use 1-digit random numbers to simulate this situation.
  4. Use the random digits provided in your answer book to run 10 simulations, each starting at vertex A. Hence estimate the probability of the rat exiting at each vertex.
OCR MEI D1 2009 June Q5
5 The diagram represents canals connecting five cities. Canal lengths (shown on the arcs) are in km.
\includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-5_410_990_306_539}
  1. Draw a network in your answer book with nodes representing the five cities and arcs representing direct canal connections, i.e. canal connections which do not involve passing through another city. The company operating the canal system wishes to close some canals to save money, whilst preserving the connectivity.
  2. Starting at A, use Prim's algorithm on your answer to part (i) to find a minimum connector for the network. Give the order in which you include arcs. Draw your minimum connector and give its total length. Consider the original network together with an extra vertex, X , at the junction of four arcs.
    \includegraphics[max width=\textwidth, alt={}, center]{dab87ac5-eda4-433f-b07a-0a609aca2f65-5_405_987_1361_539}
  3. Draw the minimum connector which results from applying Prim's algorithm, starting at A , to this network. Give the length of that minimum connector. Hence advise the company on which canals to close.
  4. Give a reason why the company might face objections to such closures.
OCR MEI D1 2009 June Q6
6 Joan and Keith have to clear and tidy their garden. The table shows the jobs that have to be completed, their durations and their precedences.
JobsDuration (mins)Immediate predecessors
Aprune bushes100-
Bweed borders60A
Ccut hedges150-
Dhoe vegetable patch60-
Emow lawns40B
Fedge lawns20D, E
Gclean up cuttings30B, C
Hclean tools10F, G
  1. Draw an activity on arc network for these activities.
  2. Mark on your diagram the early time and the late time for each event. Give the minimum completion time and the critical activities.
  3. Each job is to be done by one person only. Joan and Keith are equally able to do all jobs. Draw a cascade chart indicating how to organise the jobs so that Joan and Keith can complete the project in the least time. Give that least time and explain why the minimum project completion time is shorter.
OCR MEI D1 2010 June Q1
1
  1. Use Dijkstra's algorithm to find the shortest distances and corresponding routes from A to each of the other vertices in the given network.
    \includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-2_588_792_632_632}
  2. If the shortest distances and routes between every pair of vertices are required how many applications of Dijkstra will be needed?
OCR MEI D1 2010 June Q2
2 The following steps define an algorithm which acts on two numbers.
STEP 1 Write down the two numbers side by side on the same row.
STEP 2 Beneath the left-hand number write down double that number. Beneath the right-hand number write down half of that number, ignoring any remainder. STEP 3 Repeat STEP 2 until the right-hand number is 1.
STEP 4 Delete those rows where the right-hand number is even. Add up the remaining left-hand numbers. This is the result.
  1. Apply the algorithm to the left-hand number 3 and the right-hand number 8.
  2. Apply the algorithm to the left-hand number 26 and the right-hand number 42.
  3. Use your results from parts (i) and (ii), together with any other examples you may choose, to write down what the algorithm achieves.
OCR MEI D1 2010 June Q3
3 Traffic flows in and out of a junction of three roads as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-3_296_337_333_863} Assuming that no traffic leaves the junction by the same road as it entered, then the digraph shows traffic flows across the junction.
\includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-3_362_511_852_776}
  1. Redraw the digraph to show that it is planar.
  2. Draw a digraph to show the traffic flow across the junction of 4 roads, assuming that no traffic leaves the junction by the same road as it entered.
    \includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-3_366_366_1512_854}
    (Note that the resulting digraph is also planar, but you are not required to show this.)
  3. The digraphs showing flows across the junctions omit an important aspect in their modelling of the road junctions. What is it that they omit?
OCR MEI D1 2010 June Q4
4 A wall 4 metres long and 3 metres high is to be tiled. Two sizes of tile are available, 10 cm by 10 cm and 30 cm by 30 cm .
  1. If \(x\) is the number of boxes of ten small tiles used, and \(y\) is the number of large tiles used, explain why \(10 x + 9 y \geqslant 1200\). There are only 100 of the large tiles available.
    The tiler advises that the area tiled with the small tiles should not exceed the area tiled with the large tiles.
  2. Express these two constraints in terms of \(x\) and \(y\). The smaller tiles cost 15 p each and the larger tiles cost \(\pounds 2\) each.
  3. Given that the objective is to minimise the cost of tiling the wall, state the objective function. Use the graphical approach to solve the problem.
  4. Give two other factors which would have to be taken into account in deciding how many of each tile to purchase.
OCR MEI D1 2010 June Q5
5 The diagram shows the progress of a drunkard towards his home on one particular night. For every step which he takes towards his home, he staggers one step diagonally to his left or one step diagonally to his right, randomly and with equal probability. There is a canal three steps to the right of his starting point, and no constraint to the left. On this particular occasion he falls into the canal after 5 steps.
\includegraphics[max width=\textwidth, alt={}, center]{839adc96-1bea-44ef-917e-f03e396a3061-5_723_622_488_724}
  1. Explain how you would simulate the drunkard's walk, making efficient use of one-digit random numbers.
  2. Using the random digits in the Printed Answer Book simulate the drunkard's walk and show his progress on the grid. Stop your simulation either when he falls into the canal or when he has staggered 6 steps, whichever happens first.
  3. How could you estimate the probability of him falling into the canal within 6 steps? On another occasion the drunkard sets off carrying a briefcase in his right hand. This changes the probabilities of him staggering to the right to \(\frac { 2 } { 3 }\), and to the left to \(\frac { 1 } { 3 }\).
  4. Explain how you would now simulate this situation.
  5. Simulate the drunkard's walk (with briefcase) 10 times, and hence estimate the probability of him falling into the canal within 6 steps. (In your simulations you are not required to show his progress on a grid. You only need to record his steps to the right or left.)
OCR MEI D1 2010 June Q6
6 The table shows the tasks that have to be completed in building a stadium for a sporting event, their durations and their precedences. The stadium has to be ready within two years.
TaskDuration (months)Immediate predecessors
A4-
B2-
C7-
D12A
E5A
F7A, B
G6D, J
H3C
I12E, F, H
J7E, F, H
K12C
  1. Draw an activity on arc network for these activities.
  2. Mark on your diagram the early time and the late time for each event. Give the project duration and the critical activities. In the later stages of planning the project it is discovered that task J will actually take 9 months to complete. However, other tasks can have their durations shortened by employing extra resources. The costs of "crashing" tasks (i.e. the costs of employing extra resources to complete them more quickly) are given in the table.
    Tasks which can be completed more quickly by employing extra resourcesNumber of months which can be savedCost per month of employing extra resources (£m)
    A23
    D11
    C33
    F22
    G24
  3. Find the cheapest way of completing the project within two years.
  4. If the delay in completing task J is not discovered until it is started, how can the project be completed in time, and how much extra will it cost?
OCR MEI D1 2011 June Q1
1 Two students draw graphs to represent the numbers of pairs of shoes owned by members of their class. Andrew produces a bipartite graph, but gets it wrong. Barbara produces a completely correct frequency graph. Their graphs are shown below. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e03f6fb-69db-438a-a79e-3e04fab0d08a-2_652_593_575_278} \captionsetup{labelformat=empty} \caption{Andrew's graph}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2e03f6fb-69db-438a-a79e-3e04fab0d08a-2_663_652_667_1142} \captionsetup{labelformat=empty} \caption{Barbara's graph}
\end{figure}
  1. Draw a correct bipartite graph.
  2. How many people are in the class?
  3. How many pairs of shoes in total are owned by members of the class?
  4. Which points on Barbara's graph may be deleted without losing any information? Charles produces the same frequency graph as Barbara, but joins consecutive points with straight lines.
  5. Criticise Charles's graph.
OCR MEI D1 2011 June Q2
2 The algorithm gives a method for drawing two straight lines, if certain conditions are met. Start with the equations of the two straight lines
Line 1 is \(a x + b y = c , \quad a , b , c > 0\)
Line 2 is \(d x + e y = f , \quad d , e , f > 0\)
Let \(X =\) minimum of \(\frac { c } { a }\) and \(\frac { f } { d }\)
Let \(Y =\) minimum of \(\frac { c } { b }\) and \(\frac { f } { e }\) If \(X = \frac { c } { a }\) then \(X ^ { * } = \frac { c - b Y } { a }\) and \(Y ^ { * } = \frac { f - d X } { e }\) If \(X = \frac { f } { d }\) then \(X ^ { * } = \frac { f - e Y } { d }\) and \(Y ^ { * } = \frac { c - a X } { b }\)
Draw an \(x\)-axis labelled from 0 to \(X\), and a \(y\)-axis labelled from 0 to \(Y\)
Join ( \(0 , Y\) ) to ( \(X , Y ^ { * }\) ) with a straight line
Join ( \(X ^ { * } , Y\) ) to ( \(X , 0\) ) with a straight line
  1. Apply the algorithm with \(a = 1 , b = 5 , c = 25 , d = 10 , e = 2 , f = 85\).
  2. Why might this algorithm be useful in an LP question?
OCR MEI D1 2011 June Q3
3 John has a standard die in his pocket (ie a cube with its six faces labelled from 1 to 6).
  1. Describe how John can use the die to obtain realisations of the random variable \(X\), defined below.
    \(x\)123
    \(\operatorname { Probability } ( X = x )\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 3 }\)
  2. Describe how John can use the die to obtain realisations of the random variable \(Y\), defined below.
    \(y\)123
    \(\operatorname { Probability } ( Y = y )\)\(\frac { 1 } { 2 }\)\(\frac { 1 } { 4 }\)\(\frac { 1 } { 4 }\)
  3. John attempts to use the die to obtain a realisation of a uniformly distributed 2-digit random number. He throws the die 20 times. Each time he records one less than the number showing. He then adds together his 20 recorded numbers. Criticise John's methodology.
OCR MEI D1 2011 June Q4
4 An eco-village is to be constructed consisting of large houses and standard houses.
Each large house has 4 bedrooms, needs a plot size of \(200 \mathrm {~m} ^ { 2 }\) and costs \(\pounds 60000\) to build.
Each standard house has 3 bedrooms, needs a plot size of \(120 \mathrm {~m} ^ { 2 }\) and costs \(\pounds 50000\) to build.
The area of land available for houses is \(120000 \mathrm {~m} ^ { 2 }\). The project has been allocated a construction budget of \(\pounds 42.4\) million. The market will not sustain more than half as many large houses as standard houses. So, for instance, if there are 500 standard houses then there must be no more than 250 large houses.
  1. Define two variables so that the three constraints can be formulated in terms of your variables. Formulate the three constraints in terms of your variables.
  2. Graph your three inequalities from part (i), indicating the feasible region.
  3. Find the maximum number of bedrooms which can be provided, and the corresponding numbers of each type of house.
  4. Modify your solution if the construction budget is increased to \(\pounds 45\) million.
OCR MEI D1 2011 June Q5
5 The activity network and table together show the tasks involved in constructing a house extension, their durations and precedences.
\includegraphics[max width=\textwidth, alt={}, center]{2e03f6fb-69db-438a-a79e-3e04fab0d08a-5_231_985_338_539}
ActivityDescriptionDuration (days)
AArchitect produces plans10
PlObtain planning permission14
DemoDemolish existing structure3
FoExcavate foundations4
WBuild walls3
PbInstall plumbing2
RConstruct roof3
FlLay floor2
EFit electrics2
WDInstall windows and doors1
DecoDecorate5
  1. Show the immediate predecessors for each activity.
  2. Perform a forward pass and a backward pass to find the early time and the late time for each event.
  3. Give the critical activities, the project duration, and the total float for each activity.
  4. The activity network includes one dummy activity. Explain why this dummy activity is needed. Whilst the foundations are being dug the customer negotiates the installation of a decorative corbel. This will take one day. It must be done after the walls have been built, and before the roof is constructed. The windows and doors cannot be installed until it is completed. It will not have any effect on the construction of the floor.
  5. Redraw the activity network incorporating this extra activity.
  6. Find the revised critical activities and the revised project duration.
OCR MEI D1 2011 June Q6
6 The table shows the distances in miles, where direct rail connections are possible, between 11 cities in a country. The government is proposing to construct a high-speed rail network to connect the cities.
PSFLnBrNrBmLdNcLvM
P-150-240125------
S150-15080105-135----
F-150-80-------
Ln2408080-120115120----
Br125105-120-23090----
Nr---115230-160175255--
Bm-135-12090160-120--90
Ld-----175120-21010090
Nc-----255-210-175-
Lv-------100175-35
M------9090-35-
  1. Use the tabular form of Prim's algorithm, starting at vertex P , to find a minimum connector for the network. Draw your minimum connector and give its total length.
  2. Give one advantage and two disadvantages of constructing a rail network using only the arcs of a minimum connector.
  3. Use Dijkstra's algorithm on the diagram in the Printed Answer Book, to find the shortest route and distance from P to Nr in the original network.
  4. Give the shortest distance from P to Nr using only arcs in your minimum connector.
OCR MEI D1 2012 June Q1
1 The table defines a network in which the numbers represent lengths.
ABCDEFG
A-38-5--
B3-4---6
C84-11-2
D--1---5
E5-1--4-
F----4-1
G-625-1-
  1. Draw the network.
  2. Use Dijkstra's algorithm to find the shortest route from A to G . Give the route and its length.
OCR MEI D1 2012 June Q2
2 This question concerns the following algorithm which operates on a given function, f. The algorithm finds a point between A and B at which the function has a minimum, with a maximum error of 0.05 .
Step 1Input A
Step 2Input B , where \(\mathrm { B } > \mathrm { A }\)
Step 3Let \(\mathrm { R } = \mathrm { A } + \left( \frac { \sqrt { 5 } - 1 } { 2 } \right) \times ( \mathrm { B } - \mathrm { A } )\)
Step 4Let \(\mathrm { L } = \mathrm { A } + \mathrm { B } - \mathrm { R }\)
Step 5Find \(f ( \mathrm {~L} )\) and \(f ( \mathrm { R } )\)
Step 6If \(\mathrm { f } ( \mathrm { L } ) \leqslant \mathrm { f } ( \mathrm { R } )\) then let \(\mathrm { B } = \mathrm { R }\) and go to Step 8
Step 7If \(\mathrm { f } ( \mathrm { L } ) > \mathrm { f } ( \mathrm { R } )\) then let \(\mathrm { A } = \mathrm { L }\) and go to Step 8
Step 8If \(\mathrm { B } - \mathrm { A } < 0.1\) then go to step 10
Step 9Go to step 3
Step 10Print \(\frac { ( \mathrm { A } + \mathrm { B } ) } { 2 }\) and stop
  1. Working correct to three decimal places, perform two iterations of the algorithm for \(\mathrm { f } ( x ) = 2 x ^ { 2 } - 15 x + 30\), when \(\mathrm { A } = 3\) and \(\mathrm { B } = 4\). Start at Step 1 and stop when you reach Step 8 for the second time.
  2. The \(\left( \frac { \sqrt { 5 } - 1 } { 2 } \right)\) factor in Step 3 ensures that either the new ' \(L\) ' is equal to the old ' \(R\) ', or vice versa. Why is this important?
  3. This algorithm is used when the function is not known explicitly, but where its value can be found for any given input. Give a practical example of where it might be used.