Questions - OCR - A-Level Maths

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OCR D1 2012 January Q1

6 marks Easy -1.8

Tom has some packages that he needs to sort into order of decreasing weight. The weights, in kg, given on the packages are as follows. 3 \quad 6 \quad 2 \quad 6 \quad 5 \quad 7 \quad 1 \quad 4 \quad 9 Use shuttle sort to put the weights into decreasing order (from largest to smallest). Show the result at the end of each pass through the algorithm and write down the number of comparisons and the number of swaps used in each pass. Write down the total number of passes, the total number of comparisons and the total number of swaps used. [6]

OCR D1 2012 January Q2

8 marks Moderate -0.8

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 joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected.

What is the minimum number of arcs that a simply connected graph with six vertices can have? Draw an example of such a graph. [2]
What is the maximum number of arcs that a simply connected graph with six vertices can have? Draw an example of such a graph. [2]
What is the maximum number of arcs that a simply connected Eulerian graph with six vertices can have? Explain your reasoning. [2]
State how you know that the graph below is semi-Eulerian and write down a semi-Eulerian trail for the graph. [2]

\includegraphics{figure_2}

OCR D1 2012 January Q3

13 marks Moderate -0.8

\includegraphics{figure_3}

Apply Dijkstra's algorithm to the copy of this network in the answer booklet to find the least weight path from A to F. State the route of the path and give its weight. [6]

In the remainder of this question, any least weight paths required may be found without using a formal algorithm.

Apply the route inspection algorithm, showing all your working, to find the weight of the least weight closed route that uses every arc at least once. [3]
Find the weight of the least weight route that uses every arc at least once, starting at A and ending at F. Explain how you reached your answer. [4]

OCR D1 2012 January Q4

18 marks Moderate -0.8

Lucy is making party bags which she will sell to raise money for charity. She has three colours of party bag: red, yellow and blue. The bags contain balloons, sweets and toys. Lucy has a stock of 40 balloons, 80 sweets and 30 toys. The table shows how many balloons, sweets and toys are needed for one party bag of each colour.

Colour of party bag	Balloons	Sweets	Toys
Red	5	3	5
Yellow	4	7	2
Blue	6	6	3

Lucy will raise £1 for each bag that she sells, irrespective of its colour. She wants to calculate how many bags of each colour she should make to maximise the total amount raised for charity. Lucy has started to model the problem as an LP formulation. Maximise \quad $P = x + y + z$, subject to \quad $3x + 7y + 6z \leq 80$.

What does the variable $x$ represent in Lucy's formulation? [1]
Explain why the constraint $3x + 7y + 6z \leq 80$ must hold and write down another two similar constraints. [3]
What other constraints and restrictions apply to the values of $x$, $y$ and $z$? [1]
What assumption is needed for the objective to be valid? [1]
Represent the problem as an initial Simplex tableau. Do not carry out any iterations yet. [3]
Perform one iteration of the Simplex algorithm, choosing a pivot from the $x$ column. Explain how the choice of pivot row was made and show how each row was calculated. [6]
Write down the values of $x$, $y$ and $z$ from the first iteration of the Simplex algorithm and hence find the number of bags of each colour that Lucy should make according to this non-optimal tableau. [2]

In the optimal solution Lucy makes 10 bags.

Without carrying out further iterations of the Simplex algorithm, find a solution in which Lucy should make 10 bags. [1]

OCR D1 2012 January Q5

18 marks Moderate -0.8

The table shows the road distances in miles between five places in Great Britain. For example, the distance between Birmingham and Cardiff is 103 miles.

	Ayr
250		Birmingham
350	103		Cardiff
235	104	209		Doncaster
446	157	121	261	Exeter

Complete the network in the answer booklet to show this information. The vertices are labelled by using the initial letter of each place. [2]
List the ten arcs by increasing order of weight. Apply Kruskal's algorithm to the list. Any entries that are crossed out should still be legible. Draw the resulting minimum spanning tree and give its total weight. [4]

A sixth vertex, F, is added to the network. The distances, in miles, between F and each of the other places are shown in the table below.

	A	B	C	D	E
Distance from F	200	50	150	59	250

Use the weight of the minimum spanning tree from part (ii) to find a lower bound for the length of the minimum tour (cycle) that visits every vertex of the extended network with six vertices. [2]
Apply the nearest neighbour method, starting from vertex A, to find an upper bound for the length of the minimum tour (cycle) through the six vertices. [2]
Use the two least weight arcs through A to form a least weight path of the form $SAT$, where $S$ and $T$ are two of $\{B, C, D, E, F\}$, and give the weight of this path. Similarly write down a least weight path of the form $UEV$, where $U$ and $V$ are two of $\{A, B, C, D, F\}$, and give the weight of this path. You should find that the two paths that you have written down use all six vertices. Now find the least weight way in which the two paths can be joined together to form a cycle through all six vertices. Hence write down a tour through the six vertices that has total weight less than the upper bound. Write down the total weight of this tour. [8]

OCR D1 2012 January Q6

9 marks Easy -1.2

The function INT($C$) gives the largest integer that is less than or equal to $C$. For example: INT(4.8) = 4, INT(7) = 7, INT(0.8) = 0, INT(−0.8) = −1, INT(−2.4) = −3. Consider the following algorithm. Line 10 \quad Input $A$ and $B$ Line 20 \quad Calculate $C = B \div A$ Line 30 \quad Let $D =$ INT($C$) Line 40 \quad Calculate $E = A \times D$ Line 50 \quad Calculate $F = B - E$ Line 60 \quad Output the value of $F$ Line 70 \quad Replace $B$ by the value of $D$ Line 80 \quad If $B = 0$ then stop, otherwise go back to line 20

Apply the algorithm using the inputs $A = 10$ and $B = 128$. Record the values of $A$, $B$, $C$, $D$, $E$, and $F$ every time they change. Record the output each time line 60 is reached. [4]
Show what happens when the input values are $A = 10$ and $B = -13$. [5]

OCR D1 2009 June Q1

8 marks Easy -1.3

The memory requirements, in KB, for eight computer files are given below. 43 \quad 172 \quad 536 \quad 17 \quad 314 \quad 462 \quad 220 \quad 231 The files are to be grouped into folders. No folder is to contain more than 1000 KB, so that the folders are small enough to transfer easily between machines.

Use the first-fit method to group the files into folders. [3]
Use the first-fit decreasing method to group the files into folders. [3]

First-fit decreasing is a quadratic order algorithm.

A computer takes 1.3 seconds to apply first-fit decreasing to a list of 500 numbers. Approximately how long will it take to apply first-fit decreasing to a list of 5000 numbers? [2]

OCR D1 2009 June Q2

9 marks Easy -1.2

Explain why it is impossible to draw a graph with four vertices in which the vertex orders are 1, 2, 3 and 3. [1]

A simple graph is one in which any two vertices are directly joined by at most one arc and no vertex is directly joined to itself. A connected graph is one in which every vertex is joined, directly or indirectly, to every other vertex. A simply connected graph is one that is both simple and connected.

1. Draw a graph with five vertices of orders 1, 1, 2, 2 and 4 that is neither simple nor connected. [2]
2. Explain why your graph from part (a) is not semi-Eulerian. [1]
3. Draw a semi-Eulerian graph with five vertices of orders 1, 1, 2, 2 and 4. [1]

Six people (Ann, Bob, Caz, Del, Eric and Fran) are represented by the vertices of a graph. Each pair of vertices is joined by an arc, forming a complete graph. If an arc joins two vertices representing people who have met it is coloured blue, but if it joins two vertices representing people who have not met it is coloured red.

1. Explain why the vertex corresponding to Ann must be joined to at least three of the others by arcs that are the same colour. [2]
2. Now assume that Ann has met Bob, Caz and Del. Bob, Caz and Del may or may not have met one another. Explain why the graph must contain at least one triangle of arcs that are all the same colour. [2]

OCR D1 2009 June Q3

11 marks Moderate -0.8

The constraints of a linear programming problem are represented by the graph below. The feasible region is the unshaded region, including its boundaries. \includegraphics{figure_3}

Write down the inequalities that define the feasible region. [4]
Write down the coordinates of the three vertices of the feasible region. [2]

The objective is to maximise $2x + 3y$.

Find the values of $x$ and $y$ at the optimal point, and the corresponding maximum value of $2x + 3y$. [3]

The objective is changed to maximise $2x + ky$, where $k$ is positive.

Find the range of values of $k$ for which the optimal point is the same as in part (iii). [2]

OCR D1 2009 June Q4

25 marks Moderate -0.3

Answer this question on the insert provided. The vertices in the network below represent the junctions between main roads near Ayton (A). The arcs represent the roads and the weights on the arcs represent distances in miles. \includegraphics{figure_4}

On the diagram in the insert, use Dijkstra's algorithm to find the shortest path from A to H. You must show your working, including temporary labels, permanent labels and the order in which permanent labels are assigned. Write down the route of the shortest path from A to H and give its length in miles. [7]

Simon is a highways surveyor. He needs to check that there are no potholes in any of the roads. He will start and end at Ayton.

Which standard network problem does Simon need to solve to find the shortest route that uses every arc? [1]

The total weight of all the arcs is 67.5 miles.

Use an appropriate algorithm to find the length of the shortest route that Simon can use. Show all your working. (You may find the lengths of shortest paths between nodes by using your answer to part (i) or by inspection.) [5]

Suppose that, instead, Simon wants to find the shortest route that uses every arc, starting from A and ending at H.

Which arcs does Simon need to travel twice? What is the length of the shortest route that he can use? [2]

There is a set of traffic lights at each junction. Simon's colleague Amber needs to check that all the traffic lights are working correctly. She will start and end at the same junction.

Show that the nearest neighbour method fails on this network if it is started from A. [1]
Apply the nearest neighbour method starting from C to find an upper bound for the distance that Amber must travel. [3]
Construct a minimum spanning tree by using Prim's algorithm on the reduced network formed by deleting node A and all the arcs that are directly joined to node A. Start building your tree at node B. (You do not need to represent the network as a matrix.) Mark the arcs in your tree on the diagram in the insert. Give the order in which nodes are added to your tree and calculate the total weight of your tree. Hence find a lower bound for the distance that Amber must travel. [6]

OCR D1 2009 June Q5

19 marks Standard +0.3

Badgers is a small company that makes badges to customers' designs. Each badge must pass through four stages in its production: printing, stamping out, fixing pin and checking. The badges can be laminated, metallic or plastic. The times taken for 100 badges of each type to pass through each of the stages and the profits that Badgers makes on every 100 badges are shown in the table below. The table also shows the total time available for each of the production stages.

	Printing (seconds)	Stamping out (seconds)	Fixing pin (seconds)	Checking (seconds)	Profit (£)
Laminated	15	5	50	100	4
Metallic	15	8	50	50	3
Plastic	30	10	50	20	1
Total time available	9000	3600	25000	10000

Suppose that the company makes $x$ hundred laminated badges, $y$ hundred metallic badges and $z$ hundred plastic badges.

Show that the printing time leads to the constraint $x + y + 2z \leq 600$. Write down and simplify constraints for the time spent on each of the other production stages. [4]
What other constraint is there on the values of $x$, $y$ and $z$? [1]

The company wants to maximise the profit from the sale of badges.

Write down an appropriate objective function, to be maximised. [1]
Represent Badgers' problem as an initial Simplex tableau. [4]
Use the Simplex algorithm, pivoting first on a value chosen from the $x$-column and then on a value chosen from the $y$-column. Interpret your solution and the values of the slack variables in the context of the original problem. [9]

OCR D2 Q1

4 marks Moderate -0.8

The payoff matrix for player $A$ in a two-person zero-sum game is shown below. \begin{array}{c|c|c|c|c} & & \multicolumn{3}{c}{B}
& & \text{I} & \text{II} & \text{III}
\hline \multirow{3}{*}{A} & \text{I} & -3 & 4 & 0
& \text{II} & 2 & 2 & 1
& \text{III} & 3 & -2 & -1
\end{array} Find the optimal strategy for each player and the value of the game. [4 marks]

OCR D2 Q2

12 marks Moderate -0.8

Activity	Time	Precedence
A	5
B	20	A
C	3	A
D	7	A
E	4	B
F	15	C
G	6	C
H	17	D
I	10	F, G
J	2	G, H
K	6	E, I
L	9	I, J
M	3	K, L

Fig. 1 Construct an activity network Use appropriate forward and backward scanning to find

the minimum number of days needed to complete the entire project, [3 marks]
the activities which lie on the critical path. [3 marks]

[6 marks]

Represent this information as a digraph. [3 marks]
Find the minimum cut, expressing it in the form $\{$ $\}|\{$ $\}$, and state its value. [2 marks]
Starting from having no flow in the system, use the labelling procedure to find a maximal flow through the system. You should list each flow-augmenting route you use, together with its flow. [5 marks]
Explain how you know that this flow is maximal. [1 mark]

[11 marks]

OCR D2 Q6

42 marks Challenging +1.2

The payoff matrix for player $A$ in a two-person zero-sum game is shown below. \begin{array}{c|c|c|c|c} & & \multicolumn{3}{c}{B}
& & \text{I} & \text{II} & \text{III}
\hline \multirow{2}{*}{A} & \text{I} & -2 & 3 & -1
& \text{II} & 4 & -5 & 2
\end{array}

Formulate this information as a linear programming problem, the solution to which will give the optimal strategy for player $B$. [7 marks]
By solving this linear programming problem, find the optimal strategy for player $B$ and the value of the game. [14 marks]

[21 marks]

OCR H240/02 2020 November Q1

9 marks Easy -1.3

Differentiate the following with respect to $x$.
1. $(2x + 3)^7$ [2]
2. $x^3 \ln x$ [3]
Find $\int \cos 5x \, dx$. [2]
Find the equation of the curve through $(1, 3)$ for which $\frac{dy}{dx} = 6x - 5$. [2]

OCR H240/02 2020 November Q2

4 marks Moderate -0.8

Simplify fully $\frac{2x^3 + x^2 - 7x - 6}{x^2 - x - 2}$. [4]

OCR H240/02 2020 November Q3

7 marks Moderate -0.3

In this question you should assume that $-1 < x < 1$.

For the binomial expansion of $(1 - x)^{-2}$
1. find and simplify the first four terms, [2]
2. write down the term in $x^n$. [1]
Write down the sum to infinity of the series $1 + x + x^2 + x^3 + \ldots$. [1]
Hence or otherwise find and simplify an expression for $2 + 3x + 4x^2 + 5x^3 + \ldots$ in the form $\frac{a - x}{(b - x)^2}$ where $a$ and $b$ are constants to be determined. [3]

OCR H240/02 2020 November Q4

5 marks Standard +0.3

In this question you must show detailed reasoning. Solve the equation $3\sin^4 \phi + \sin^2 \phi = 4$, for $0 \leq \phi < 2\pi$, where $\phi$ is measured in radians. [5]

OCR H240/02 2020 November Q5

5 marks Standard +0.3

Determine the set of values of $n$ for which $\frac{n^2 - 1}{2}$ and $\frac{n^2 + 1}{2}$ are positive integers. [3]

A 'Pythagorean triple' is a set of three positive integers $a$, $b$ and $c$ such that $a^2 + b^2 = c^2$.

Prove that, for the set of values of $n$ found in part (a), the numbers $n$, $\frac{n^2 - 1}{2}$ and $\frac{n^2 + 1}{2}$ form a Pythagorean triple. [2]