Questions — OCR (4628 questions)

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OCR M2 2005 June Q5
10 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-3_319_650_1219_749} A uniform \(\operatorname { rod } A B\) of length 60 cm and weight 15 N is freely suspended from its end \(A\). The end \(B\) of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point \(C\) which is at the same horizontal level as \(A\). The rod is in equilibrium with the string at right angles to the rod (see diagram).
  1. Show that the tension in the string is 4.5 N .
  2. Find the magnitude and direction of the force acting on the rod at \(A\).
OCR M2 2005 June Q6
10 marks Standard +0.3
6 A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5 ^ { \circ }\) to the horizontal. At a certain point \(P\) on the hill the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is \(15 \mathrm {~ms} ^ { - 1 }\).
  1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
  2. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW , correct to 3 significant figures.
  3. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\).
OCR M2 2005 June Q7
11 marks Standard +0.3
7 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_76_243_269_365} \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_332_1427_322_360} A barrier is modelled as a uniform rectangular plank of wood, \(A B C D\), rigidly joined to a uniform square metal plate, \(D E F G\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m . The metal plate has mass 80 kg and side 0.5 m . The plank and plate are joined in such a way that \(C D E\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(C H\) is 0.25 m .
  1. Calculate the contact force at \(H\) when the barrier is in the closed position. In the open position, the centre of mass of the barrier is vertically above \(D\).
  2. Calculate the angle between \(A B\) and the horizontal when the barrier is in the open position.
OCR M2 2005 June Q8
13 marks Standard +0.3
8 A particle is projected with speed \(49 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of elevation \(\theta\) from a point \(O\) on a horizontal plane, and moves freely under gravity. The horizontal and upward vertical displacements of the particle from \(O\) at time \(t\) seconds after projection are \(x \mathrm {~m}\) and \(y \mathrm {~m}\) respectively.
  1. Express \(x\) and \(y\) in terms of \(\theta\) and \(t\), and hence show that $$y = x \tan \theta - \frac { x ^ { 2 } \left( 1 + \tan ^ { 2 } \theta \right) } { 490 } .$$
    \includegraphics[max width=\textwidth, alt={}]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_627_1249_1699_447}
    The particle passes through the point where \(x = 70\) and \(y = 30\). The two possible values of \(\theta\) are \(\theta _ { 1 }\) and \(\theta _ { 2 }\), and the corresponding points where the particle returns to the plane are \(A _ { 1 }\) and \(A _ { 2 }\) respectively (see diagram).
  2. Find \(\theta _ { 1 }\) and \(\theta _ { 2 }\).
  3. Calculate the distance between \(A _ { 1 }\) and \(A _ { 2 }\).
OCR M2 2006 June Q1
4 marks Easy -1.8
1 A child of mass 35 kg runs up a flight of stairs in 10 seconds. The vertical distance climbed is 4 m . Assuming that the child's speed is constant, calculate the power output.
OCR M2 2006 June Q2
5 marks Moderate -0.8
2 A small sphere of mass 0.3 kg is dropped from rest at a height of 2 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards, coming to instantaneous rest at a height of 1.4 m above the ground. Ignoring air resistance, calculate the magnitude of the impulse which the ground exerts on the sphere when it rebounds.
OCR M2 2006 June Q3
7 marks Challenging +1.2
3 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-2_710_572_721_788} A uniform solid hemisphere of weight 12 N and radius 6 cm is suspended by two vertical strings. One string is attached to the point \(O\), the centre of the plane face, and the other string is attached to the point \(A\) on the rim of the plane face. The hemisphere hangs in equilibrium and \(O A\) makes an angle of \(60 ^ { \circ }\) with the vertical (see diagram).
  1. Find the horizontal distance from the centre of mass of the hemisphere to the vertical through \(O\).
  2. Calculate the tensions in the strings.
OCR M2 2006 June Q4
9 marks Moderate -0.3
4 A car of mass 900 kg is travelling at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) on a level road. The total resistance to motion is 450 N .
  1. Calculate the power output of the car's engine. A roof box of mass 50 kg is mounted on the roof of the car. The total resistance to motion of the vehicle increases to 500 N .
  2. The car's engine continues to work at the same rate. Calculate the maximum speed of the car on the level road. The power output of the car's engine increases to 15000 W . The resistance to motion of the car, with roof box, remains 500 N .
  3. Calculate the instantaneous acceleration of the car on the level road when its speed is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  4. The car climbs a hill which is at an angle of \(5 ^ { \circ }\) to the horizontal. Calculate the instantaneous retardation of the car when its speed is \(26 \mathrm {~ms} ^ { - 1 }\).
OCR M2 2006 June Q5
9 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-3_657_549_1219_799} A uniform lamina \(A B C D E\) consists of a square and an isosceles triangle. The square has sides of 18 cm and \(B C = C D = 15 \mathrm {~cm}\) (see diagram).
  1. Taking \(x\) - and \(y\)-axes along \(A E\) and \(A B\) respectively, find the coordinates of the centre of mass of the lamina.
  2. The lamina is freely suspended from \(B\). Calculate the angle that \(B D\) makes with the vertical. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d6d87705-be4b-407d-b699-69fb441d88a7-4_441_1355_265_394} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} A light inextensible string of length 1 m passes through a small smooth hole \(A\) in a fixed smooth horizontal plane. One end of the string is attached to a particle \(P\), of mass 0.5 kg , which hangs in equilibrium below the plane. The other end of the string is attached to a particle \(Q\), of mass 0.3 kg , which rotates with constant angular speed in a circle of radius 0.2 m on the surface of the plane (see Fig. 1).
OCR M2 2006 June Q7
13 marks Standard +0.3
7 A small ball is projected at an angle of \(50 ^ { \circ }\) above the horizontal, from a point \(A\), which is 2 m above ground level. The highest point of the path of the ball is 15 m above the ground, which is horizontal. Air resistance may be ignored.
  1. Find the speed with which the ball is projected from \(A\). The ball hits a net at a point \(B\) when it has travelled a horizontal distance of 45 m .
  2. Find the height of \(B\) above the ground.
  3. Find the speed of the ball immediately before it hits the net.
OCR M2 2006 June Q8
14 marks Standard +0.3
8 Two uniform smooth spheres, \(A\) and \(B\), have the same radius. The mass of \(A\) is 2 kg and the mass of \(B\) is \(m \mathrm {~kg}\). Sphere \(A\) is travelling in a straight line on a smooth horizontal surface, with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it collides directly with sphere \(B\), which is at rest. As a result of the collision, sphere \(A\) continues in the same direction with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
  1. Find the greatest possible value of \(m\). It is given that \(m = 1\).
  2. Find the coefficient of restitution between \(A\) and \(B\). On another occasion \(A\) and \(B\) are travelling towards each other, each with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when they collide directly.
  3. Find the kinetic energy lost due to the collision.
OCR D1 2005 January Q1
5 marks Easy -1.8
1 Use the shuttle sort algorithm to sort the list $$\begin{array} { l l l l l l } 6 & 5 & 9 & 4 & 5 & 2 \end{array}$$ into increasing order. Write down the list that results from each pass through the algorithm.
OCR D1 2005 January Q2
5 marks Moderate -0.8
2
  1. A graph has six vertices; two are of order 3 and the rest are of order 4. Calculate the number of arcs in the graph, showing your working.
  2. Is the graph Eulerian, semi-Eulerian or neither? Give a reason to support your answer. A simple graph is one in which any two vertices are directly connected by at most one arc and no vertex is directly connected to itself. A connected graph is one in which every vertex is connected, directly or indirectly, to every other vertex.
  3. Explain why a simple graph with six vertices, two of order 3 and the rest of order 4, must also be a connected graph.
OCR D1 2005 January Q3
7 marks Standard +0.3
3 The diagram shows a network. The weights on the arcs represent distances in miles. The direct path between any two adjacent vertices is never longer than any indirect path. \includegraphics[max width=\textwidth, alt={}, center]{197624b2-ca67-4bad-9c2c-dc68c10be0fd-02_490_748_1372_699}
  1. By deleting vertex \(U\) and all arcs connected to \(U\), find a lower bound for the length of the shortest cycle that visits every vertex of this network.
  2. Find a vertex that can be used as the start vertex for the nearest neighbour method to give a cycle that passes through every vertex of this network. Give your cycle and its length.
OCR D1 2005 January Q4
12 marks Moderate -0.3
4 [Answer this question on the insert provided.]
A competition challenges teams to hike across a moor, visiting each of eight peaks, in the quickest possible time. The teams all start at peak \(A\) and finish at peak \(H\), but other than this the peaks may be visited in any order. The estimated journey times, in hours, between peaks are shown in the table. A dash in the table means that there is no direct route between two peaks.
\(A\)\(B\)CD\(E\)\(F\)G\(H\)
A-423----
\(B\)4-1-3---
C21-2-65-
\(D\)3-2---4-
\(E\)-3---8-7
\(F\)--6-8--8
\(G\)--54---9
\(H\)----789-
  1. Use Prim's algorithm on the table in the insert to find a minimum spanning tree. Start by crossing out row \(A\). Show which entries in the table are chosen and indicate the order in which the rows are deleted. What can you deduce from this answer about the quickest possible time needed to complete the challenge?
  2. On the insert, draw a network to represent the information given in the table above. A team decides to visit each peak exactly once on the hike from peak \(A\) to peak \(H\).
  3. Explain why the team cannot use the arc \(A C\).
  4. Explain why the team must use the arc \(E F\).
  5. There are only two possible routes that the team can use. Find both routes and determine which is the quicker route.
OCR D1 2005 January Q5
13 marks Standard +0.8
5 The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics[max width=\textwidth, alt={}, center]{197624b2-ca67-4bad-9c2c-dc68c10be0fd-04_1118_816_404_662}
  1. Write down four inequalities that define the feasible region. The objective is to maximise \(P = 5 x + 3 y\).
  2. Using the graph or otherwise, obtain the coordinates of the vertices of the feasible region and hence find the values of \(x\) and \(y\) that maximise \(P\), and the corresponding maximum value of \(P\). The objective is changed to maximise \(Q = a x + 3 y\).
  3. For what set of values of \(a\) is the maximum value of \(Q\) equal to 3?
OCR D1 2005 January Q6
13 marks Standard +0.8
6 Consider the linear programming problem:
maximise\(P = 2 x - 5 y - z\),
subject to\(5 x + 3 y - 5 z \leqslant 15\),
\(2 x + 6 y + 8 z \leqslant 24\),
and\(x \geqslant 0 , y \geqslant 0 , z \geqslant 0\).
  1. Using slack variables, \(s\) and \(t\), express the non-trivial constraints as two equations.
  2. Represent the problem as an initial Simplex tableau. Perform one iteration of the Simplex algorithm.
  3. Use the Simplex algorithm to find the values of \(x , y\) and \(z\) for which \(P\) is maximised, subject to the constraints above.
  4. The value 15 in the first constraint is increased to a new value \(k\). As a result the pivot for the first iteration changes. Show what effect this has on the final value of \(y\).
OCR D1 2005 January Q7
17 marks Moderate -0.8
7 [Answer this question on the insert provided.]
The network below represents a simplified map of the centre of a small town. The arcs represent roads and the weights on the arcs represent distances, in units of 100 metres. \includegraphics[max width=\textwidth, alt={}, center]{197624b2-ca67-4bad-9c2c-dc68c10be0fd-06_725_1259_495_443}
    1. Use Dijkstra's algorithm on the diagram in the insert to find the length of the shortest route from \(A\) to each of the other vertices. You must show your working, including temporary labels, permanent labels and the order in which the permanent labels were assigned. State the shortest route from \(A\) to \(E\) and the shortest route from \(A\) to \(J\), and give their lengths. [7]
    2. The shortest route from \(E\) to \(J\) that passes through every vertex can be treated as being made up of two parts, one from \(E\) to \(A\) and the other from \(A\) to \(J\). Use your answers to part (i) to write down the length of the shortest such route. List the vertices in the order that they are visited in travelling from \(E\) to \(J\) using this route.
    3. Explain why a similar approach to that used in parts (a)(i) and (a)(ii) would not give the shortest route between \(G\) and \(H\) that passes through every vertex.
  2. By considering pairings of odd nodes, find the length of the shortest route that starts at \(A\) and ends at \(E\) and uses every arc at least once. \section*{OXFORD CAMBRIDGE AND RSA EXAMINATIONS} Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education MATHEMATICS
    4736
    Decision Mathematics 1
    INSERT for Questions 4 and 7
    Wednesday
OCR D1 2005 January Q12
Moderate -1.0
12 JANUARY 2005
Afternoon
1 hour 30 minutes
  • This insert should be used to answer Questions 4 and 7.
  • Write your name, centre number and candidate number in the spaces provided at the top of this page.
  • Write your answers to Questions 4 and 7 in the spaces provided in this insert, and attach it to your answer booklet.
4
  1. \(A\)\(B\)CD\(E\)\(F\)G\(H\)
    A-423----
    \(B\)4-1-3---
    C21-2-65-
    \(D\)3-2---4-
    E-3---8-7
    \(F\)--6-8--8
    \(G\)--54---9
    \(H\)----789-
  2. B \(E\) \(C\) F
    • \(H\) \(A\) •
    • \({ } ^ { \text {F } }\)
    H D
    G
  3. \(\_\_\_\_\)
  4. \(\_\_\_\_\)
  5. \(\_\_\_\_\) 7 (a) (i) \includegraphics[max width=\textwidth, alt={}, center]{197624b2-ca67-4bad-9c2c-dc68c10be0fd-11_191_1179_269_482} Do not cross out your working values (temporary labels) \includegraphics[max width=\textwidth, alt={}, center]{197624b2-ca67-4bad-9c2c-dc68c10be0fd-11_871_1557_612_335} Shortest route from \(A\) to \(E =\) \(\_\_\_\_\) Length = \(\_\_\_\_\) Shortest route from \(A\) to \(J =\) \(\_\_\_\_\) Length = \(\_\_\_\_\)
  6. Length of route \(=\) \(\_\_\_\_\) Vertices visited in order \(\_\_\_\_\)
  7. Explanation \(\_\_\_\_\) (b) \(\_\_\_\_\) Length = \(\_\_\_\_\)
OCR D1 2006 January Q3
6 marks Moderate -0.8
3 Consider the following algorithm.
Step 1: Input a positive integer \(n\).
Step 2 : Draw a graph consisting of \(n\) vertices and no arcs.
Step 3 : Create a new vertex and join it directly to every vertex of order 0.
Step 4: Create a new vertex and join it directly to every vertex of odd order.
Step 5 : Stop.
  1. Draw separate diagrams to show the result of applying this algorithm in the following cases.
    (a) \(n = 1\) (b) \(n = 2\) (c) \(n = 3\) (d) \(n = 4\)
  2. State how many arcs are in the graph that results when the algorithm is applied to a set of \(n\) vertices.
OCR D1 2006 January Q4
8 marks Moderate -0.5
4
  1. Represent the linear programming problem below by an initial Simplex tableau. $$\begin{array} { l l } \text { Maximise } & P = 5 x - 4 y - 3 z , \\ \text { subject to } & 2 x - 3 y + 4 z \leqslant 10 , \\ & 6 x + 5 y + 4 z \leqslant 60 , \\ \text { and } & x \geqslant 0 , y \geqslant 0 , z \geqslant 0 . \end{array}$$
  2. Perform one iteration of the Simplex algorithm and write down the values of \(x , y , z\) and \(P\) that result from this iteration.
OCR D1 2006 January Q5
13 marks Moderate -0.8
5 Findlay is trying to get into his local swimming team. The coach will watch him swim and will then make his decision. Findlay must swim at least two lengths using each stroke and must swim at least 8 lengths in total, taking at most 10 minutes. Findlay needs to put together a routine that includes breaststroke, backstroke and butterfly. The table shows how Findlay expects to perform with each stroke.
StrokeStyle marksTime taken
Breaststroke2 marks per length2 minutes per length
Backstroke1 mark per length0.5 minutes per length
Butterfly5 marks per length1 minute per length
Findlay needs to work out how many lengths to swim using each stroke to maximise his expected total number of style marks.
  1. Identify appropriate variables for Findlay's problem and write down the objective function, to be maximised, in terms of these variables.
  2. Formulate a constraint for the total number of lengths swum, a constraint for the time spent swimming and constraints on the number of lengths swum using each stroke. Findlay decides that he will swim two lengths using butterfly. This reduces his problem to the following LP formulation: $$\begin{array} { l c } \text { maximise } & P = 2 x + y , \\ \text { subject to } & x + y \geqslant 6 , \\ & 4 x + y \leqslant 16 , \\ & x \geqslant 2 , y \geqslant 2 , \end{array}$$ with \(x\) and \(y\) both integers.
  3. Use a graphical method to identify the feasible region for this problem. Write down the coordinates of the vertices of the feasible region and hence find the integer values of \(x\) and \(y\) that maximise \(P\).
  4. Interpret your solution for Findlay.
OCR D1 2006 January Q6
16 marks Standard +0.3
6 The network represents a railway system. The vertices represent the stations and the arcs represent the tracks. The weights on the arcs represent journey times between stations, in minutes. The sum of all the weights is 105 minutes. \includegraphics[max width=\textwidth, alt={}, center]{8f17020a-14bf-4459-9241-1807b954a629-5_981_1215_468_477} Norah wants to travel around the system visiting every station. She wants to start and end at \(A\) and she wants to complete her journey in the shortest possible time.
  1. Apply the nearest neighbour method starting at \(A\) to find two suitable tours and calculate the journey time for each of these tours. Which of these answers gives the better upper bound for Norah's journey time?
  2. Construct a minimum spanning tree by using Prim's algorithm on the reduced network formed by deleting vertex \(G\) and all the arcs that are directly joined to \(G\). Draw a diagram to show the arcs in your tree. Hence calculate a lower bound for Norah's journey time. Norah now decides that she wants to use every section of track in her journey. She still wants to start and end at \(A\) and to complete her journey in the shortest possible time.
  3. Calculate the journey time for Norah's new problem. Show your working; quickest times between stations may be found by inspection. State which arcs Norah will have to travel twice and how many times she will pass through station \(D\).
OCR D1 2006 January Q7
18 marks Easy -1.2
7 Mr Rank and Miss File need to sort a pile of examination scripts into increasing order of mark. Mr Rank first goes through the pile of scripts and puts each script into one of two piles, depending on whether the mark is below 50 or not. He then sorts the scripts in the 'below 50 ' pile and Miss File sorts the scripts in the '50 and above' pile. At the end they put the two sorted piles together again.
  1. The scripts in the 'below 50' pile have the following marks, starting from the top of the pile. $$\begin{array} { l l l l l l l l } 34 & 42 & 27 & 31 & 12 & 48 & 24 & 37 \end{array}$$ Use bubble sort to sort this list into increasing order. Clearly indicate the list that results at the end of each pass through the algorithm. Give the number of swaps and the number of comparisons that were used in sorting this list.
  2. The scripts in the '50 and above' pile have the following marks, starting from the top of the pile. $$\begin{array} { l l l l l l l l } 95 & 74 & 61 & 87 & 71 & 82 & 53 & 57 \end{array}$$ Use shuttle sort to sort this list into increasing order. Clearly indicate the list that results at the end of each pass through the algorithm. List the number of swaps and number of comparisons that were used in sorting this list.
  3. Explain why splitting the original list into two piles is a linear order algorithm.
  4. Both bubble sort and shuttle sort are quadratic order algorithms. Mr Rank and Miss File use their method to sort a pile of 100 scripts. It takes about 50 seconds to split the pile and about 250 seconds to do each sort. As the sorts are done at the same time, this gives a total time taken of about 300 seconds, or 6 minutes. Approximately how long would Mr Rank and Miss File take to split a pile of 500 scripts into two roughly equal piles and sort the piles? Show all your working.
    [0pt] [4]
OCR D1 2007 January Q1
7 marks Easy -1.3
1 An airline allows each passenger to carry a maximum of 25 kg in luggage. The four members of the Adams family have bags of the following weights (to the nearest kg ):
Mr Adams:1042
Mrs Adams:1337524
Sarah Adams:5825
Tim Adams:105353
The bags need to be grouped into bundles of 25 kg maximum so that each member of the family can carry a bundle of bags.
  1. Use the first-fit method to group the bags into bundles of 25 kg maximum. Start with the bags belonging to Mr Adams, then those of Mrs Adams, followed by Sarah and finally Tim.
  2. Use the first-fit decreasing method to group the same bags into bundles of 25 kg maximum.
  3. Suggest a reason why the grouping of the bags in part (i) might be easier for the family to carry.