Questions - OCR - A-Level Maths

OCR D1 2007 June Q5

16 marks Standard +0.3

5 Answer this question on the insert provided. The network below represents a simplified map of a building. The arcs represent corridors and the weights on the arcs represent the lengths of the corridors, in metres. The sum of the weights on the arcs is 765 metres. \includegraphics[max width=\textwidth, alt={}, center]{dbf782dd-879c-4f0f-b532-246a0db9f130-5_1271_1539_584_303}

Janice is the cleaning supervisor in the building. She is at the position marked as J when she is called to attend a cleaning emergency at B. On the network in the insert, use Dijkstra's algorithm, starting from vertex J and continuing until B is given a permanent label, to find the shortest path from J to B and the length of this path.
In her job J anice has to walk along each of the corridors represented on the network. This requires finding a route that covers every arc at least once, starting and ending at J. Showing all your working, find the shortest distance that J anice must walk to check all the corridors. The labelled vertices represent 'cleaning stations'. J anice wants to visit every cleaning station using the shortest possible route. She produces a simplified network with no repeated arcs and no arc that joins a vertex to itself.
On the insert, complete Janice's simplified network. Which standard network problem does Janice need to solve to find the shortest distance that she must travel?

OCR D1 2007 June Q6

13 marks Moderate -0.5

6 Answer this question on the insert provided. The table shows the distances, in miles, along the direct roads between six villages, $A$ to $F$. A dash ( - ) indicates that there is no direct road linking the villages.

	A	B	C	D	E	F
A	-	6	3	-	-	-
B	6	-	5	6	-	14
C	3	5	-	8	4	10
D	-	6	8	-	3	8
E	-	-	4	3	-	-
F	-	14	10	8	-	-

On the table in the insert, use Prim's algorithm 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. Draw your minimum spanning tree and state its total weight.
By deleting vertex B and the arcs joined to vertex B, calculate a lower bound for the length of the shortest cycle through all the vertices.
A pply the nearest neighbour method to the table above, starting from $F$, to find a cycle that passes through every vertex and use this to write down an upper bound for the length of the shortest cycle through all the vertices.
{}

OCR D2 2006 January Q1

7 marks Moderate -0.8

1 Answer this question on the insert provided. Mrs Price has bought six T shirts for her children. Each child is to have two shirts.
Amanda would like the green shirt, the pink shirt or the red shirt.
Ben would like the green shirt, the turquoise shirt, the white shirt or the yellow shirt.
Carrie would like the pink shirt, the white shirt or the yellow shirt.

On the first diagram in the insert, draw a bipartite graph to show which child would like which shirt. The children are represented as $A 1 , A 2 , B 1 , B 2 , C 1$ and $C 2$ and the shirts as $G , P , R , T , W$ and $Y$. Initially, Mrs Price puts aside the green shirt and the pink shirt for Amanda, the turquoise shirt and the white shirt for Ben and the yellow shirt for Carrie.
Show this incomplete matching on the second diagram in the insert.
Write down an alternating path consisting of three arcs to enable the matching to be improved. Use your alternating path to match the children to the shirts.
Amanda decides that she does not like the green shirt after all. Which shirts should each child have now?

OCR D2 2006 January Q2

6 marks Moderate -0.8

2 Answer this question on the insert provided. The diagram shows a directed network of paths with vertices labelled with (stage; state) labels. The weights on the arcs represent distances in km . The shortest route from $( 3 ; 0 )$ to $( 0 ; 0 )$ is required. Complete the dynamic programming tabulation on the insert, working backwards from stage 1 , to find the shortest route through the network. Give the length of this shortest route. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{9c9b1a42-8d16-446a-85a1-4c08e5e368be-2_501_1018_1741_575} \captionsetup{labelformat=empty} \caption{Stage 3 Stage 2 Stage 1}

\end{figure}

OCR D2 2006 January Q5

19 marks Moderate -0.3

5 Answer this question on the insert provided. The diagram shows an activity network for a project. The table lists the durations of the activities (in days). \includegraphics[max width=\textwidth, alt={}, center]{9c9b1a42-8d16-446a-85a1-4c08e5e368be-4_652_867_429_393}

Activity	Duration
$A$	5
$B$	3
$C$	4
$D$	2
$E$	1
$F$	3
$G$	5
$H$	2
$I$	4
$J$	3

Explain why each of the dummy activities is needed.
Complete the blank column of the table in the insert to show the immediate predecessors for each activity.
Carry out a forward pass to find the early start times for the events. Record these at the eight vertices on the copy of the network on the insert. Also calculate the late start times for the events and record these at the vertices. Find the minimum completion time for the project and list the critical activities.
By how much would the duration of activity $C$ need to increase for $C$ to become a critical activity? Assume that each activity requires one worker and that each worker is able to do any of the activities. The activities may not be split. The duration of $C$ is 4 days.
Draw a resource histogram, assuming that each activity starts at its earliest possible time. How many workers are needed with this schedule?
Describe how, by delaying the start of activity $E$ (and other activities, to be determined), the project can be completed in the minimum time by just three workers.

OCR D2 2008 January Q5

15 marks Moderate -0.8

5 Answer this question on the insert provided. The diagram shows an activity network for a project. The figures in brackets show the durations of the activities in days. \includegraphics[max width=\textwidth, alt={}, center]{95fbb09b-0301-4fc1-b694-838b8d0b64a6-06_956_921_495_612}

Complete the table in the insert to show the precedences for the activities.
Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Find the minimum project duration and list the critical activities. The number of people required for each activity is shown in the table below. The workers are all equally skilled at all of the activities.
Activity $A$ $B$ $C$ $D$ $E$ $F$ $G$ $H$ $I$ $J$
Number of workers 4 1 2 2 3 2 3 3 1 2
On graph paper, draw a resource histogram for the project with each activity starting at its earliest possible time.
Describe how the project can be completed in 21 days using just six workers.

OCR D2 2009 January Q1

9 marks Easy -1.2

1 Answer this question on the insert provided. The table shows a partially completed dynamic programming tabulation for solving a maximin problem.

Stage	State	Action	Working	Maximin
\multirow{4}{*}{1}	0	0	10
	1	0	11
	2	0	14
	3	0	15
\multirow{10}{*}{2}	\multirow{2}{*}{0}	0	(12, ) =	\multirow{2}{*}{}
		2	$( 10 , \quad ) =$
	\multirow{3}{*}{1}	0	$( 13 , \quad ) =$	\multirow{3}{*}{}
		1	$( 10 , \quad ) =$
		2	(11, ) =
	\multirow{3}{*}{2}	1	( 9, ) =	\multirow{3}{*}{}
		2	(10, ) =
		3	( 7, ) =
	\multirow{2}{*}{3}	1	( 8, ) =	\multirow{2}{*}{}
		3	(12, ) =
\multirow{4}{*}{3}	\multirow{4}{*}{0}	0	$( 15 , \quad ) =$	\multirow{4}{*}{}
		1	$( 14 , \quad ) =$
		2	(16, ) =
		3	(13, ) =

Complete the last two columns of the table in the insert.
State the maximin value and write down the maximin route.

OCR D2 2009 January Q2

15 marks Moderate -0.3

2 Answer this question on the insert provided. The diagram shows an activity network for a project. The figures in brackets show the durations of the activities in days. \includegraphics[max width=\textwidth, alt={}, center]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-3_497_1230_493_459}

Complete the table in the insert to show the precedences for the activities.
Use the boxes on the diagram in the insert to carry out a forward pass and a backward pass. Show that the minimum project completion time is 28 days and list the critical activities. The resource histogram below shows the number of workers required each day when the activities each begin at their earliest possible start time. Once an activity has been started it runs for its duration without a break. \includegraphics[max width=\textwidth, alt={}, center]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-3_457_1543_1503_299}
By considering which activities are happening each day, complete the table in the insert to show the number of workers required for each activity. You are advised to start at day 28 and work back through the days towards day 1 . Only five workers are actually available, but they are all equally skilled at each of the activities. The project can still be completed in 28 days by delaying the start of activity $E$.
Find the minimum possible delay and the maximum possible delay on activity $E$ in this case.

OCR D2 2009 January Q3

12 marks Standard +0.8

3 Answer this question on the insert provided. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-4_625_1100_358_520} \captionsetup{labelformat=empty} \caption{Fig. 1}

\end{figure} Fig. 1 represents a system of pipes through which fluid can flow from a source, $S$, to a sink, $T$. It also shows a cut $\alpha$. The weights on the arcs show the lower and upper capacities of the pipes in litres per second.

Calculate the capacity of the cut $\alpha$.
By considering vertex $B$, explain why arc $S B$ must be at its lower capacity. Then by considering vertex $E$, explain why arc $C E$ must be at its upper capacity, and hence explain why arc $H T$ must be at its lower capacity.
On the diagram in the insert, show a flow through the network of 15 litres per second. Write down one flow augmenting route that allows another 1 litre per second to flow through the network. Show that the maximum flow is 16 litres per second by finding a cut of 16 litres per second. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c5bfbe78-64c4-4254-ad83-0c90f4a54b18-4_602_1086_1809_568} \captionsetup{labelformat=empty} \caption{Fig. 2}
\end{figure} Fig. 2 represents the same system, but with pipe $E B$ installed the wrong way round.
Explain why there can be no feasible flow through this network.

OCR D2 2011 January Q6

13 marks Moderate -0.5

6 Answer this question on the insert provided. Four friends have decided to sponsor four birds at a bird sanctuary. They want to construct a route through the bird sanctuary, starting and ending at the entrance/exit, that enables them to visit the four birds in the shortest possible time. The table below shows the times, in minutes, that it takes to get between the different birds and the entrance/exit. The friends will spend the same amount of time with each bird, so this does not need to be included in the calculation.

	Entrance/exit	Kite	Lark	Moorhen	Nightjar
Entrance/exit	-	10	14	12	17
Kite	10	-	3	2	6
Lark	14	3	-	2	4
Moorhen	12	2	2	-	3
Nightjar	17	6	4	3	-

Let the stages be $0,1,2,3,4,5$. Stage 0 represents arriving at the sanctuary entrance. Stage 1 represents visiting the first bird, stage 2 the second bird, and so on, with stage 5 representing leaving the sanctuary. Let the states be $0,1,2,3,4$ representing the entrance/exit, kite, lark, moorhen and nightjar respectively.

Calculate how many minutes it takes to travel the route $$( 0 ; 0 ) - ( 1 ; 1 ) - ( 2 ; 2 ) - ( 3 ; 3 ) - ( 4 ; 4 ) - ( 5 ; 0 ) .$$ The friends then realise that if they try to find the quickest route using dynamic programming with this (stage; state) formulation, they will get the route $( 0 ; 0 ) - ( 1 ; 1 ) - ( 2 ; 2 ) - ( 3 ; 3 ) - ( 4 ; 1 ) - ( 5 ; 0 )$, or this in reverse, taking 27 minutes.
Explain why the route $( 0 ; 0 ) - ( 1 ; 1 ) - ( 2 ; 2 ) - ( 3 ; 3 ) - ( 4 ; 1 ) - ( 5 ; 0 )$ is not a solution to the friends' problem. Instead, the friends set up a dynamic programming tabulation with stages and states as described above, except that now the states also show, in brackets, any birds that have already been visited. So, for example, state $1 ( 234 )$ means that they are currently visiting the kite and have already visited the other three birds in some order. The partially completed dynamic programming tabulation is shown opposite.
For the last completed row, i.e. stage 2, state 1(3), action 4(13), explain where the value 18 and the value 6 in the working column come from.

Complete the table in the insert and hence find the order in which the birds should be visited to give a quickest route and find the corresponding minimum journey time.

Stage

State

Action

Working

Suboptimal minimum

\multirow{4}{*}{4}

1(234)

0

10

2(134)

0

14

3(124)

0

12

4(123)

0

17

\multirow{12}{*}{3}

1(23)

4(123)

$17 + 6 = 23$

23

1(24)

3(124)

$12 + 2 = 14$

14

1(34)

2(134)

$14 + 3 = 17$

17

2(13)

4(123)

$17 + 4 = 21$

21

2(14)

3(124)

$12 + 2 = 14$

14

2(34)

1(234)

$10 + 3 = 13$

13

3(12)

4(123)

$17 + 3 = 20$

20

3(14)

2(134)

$14 + 2 = 16$

16

3(24)

1(234)

$10 + 2 = 12$

12

4(12)

3(124)

$12 + 3 = 15$

15

4(13)

2(134)

$14 + 4 = 18$

18

4(23)

1(234)

$10 + 6 = 16$

16

\multirow{12}{*}{2}

1(2)

3(12) 4(12)

$20 + 2 = 22$

21

1(3)

2(13) 4(13)

$21 + 3 = 24 18 + 6 = 24$

24

1(4)

2(1)

2(3)

2(4)

3(1)

3(2)

3(4)

4(1)

4(2)

4(3)

\multirow{4}{*}{1}

1

2

3

4

0

1

2

3

4

OCR H240/01 2019 June Q1

4 marks Moderate -0.8

1 In this question you must show detailed reasoning. Solve the inequality $10 x ^ { 2 } + x - 2 > 0$.

OCR H240/01 2019 June Q7

8 marks Standard +0.8

7 In this question you must show detailed reasoning. A sequence $u _ { 1 } , u _ { 2 } , u _ { 3 } \ldots$ is defined by $u _ { n } = 25 \times 0.6 ^ { n }$.
Use an algebraic method to find the smallest value of $N$ such that $\sum _ { n = 1 } ^ { \infty } u _ { n } - \sum _ { n = 1 } ^ { N } u _ { n } < 10 ^ { - 4 }$.

OCR H240/02 2018 June Q13

12 marks Standard +0.8

13 In this question you must show detailed reasoning. The probability that Paul's train to work is late on any day is 0.15 , independently of other days.

The number of days on which Paul's train to work is late during a 450-day period is denoted by the random variable $Y$. Find a value of $a$ such that $\mathrm { P } ( Y > a ) \approx \frac { 1 } { 6 }$. In the expansion of $( 0.15 + 0.85 ) ^ { 50 }$, the terms involving $0.15 ^ { r }$ and $0.15 ^ { r + 1 }$ are denoted by $T _ { r }$ and $T _ { r + 1 }$ respectively.
Show that $\frac { T _ { r } } { T _ { r + 1 } } = \frac { 17 ( r + 1 ) } { 3 ( 50 - r ) }$.
The number of days on which Paul's train to work is late during a 50-day period is modelled by the random variable $X$.
1. Find the values of $r$ for which $\mathrm { P } ( X = r ) \leqslant \mathrm { P } ( X = r + 1 )$.
2. Hence find the most likely number of days on which the train will be late during a 50-day period.

OCR H240/02 2021 November Q5

8 marks Standard +0.3

5 In this question you must show detailed reasoning. Points $A , B$ and $C$ have coordinates $( 0,6 ) , ( 7,5 )$ and $( 6 , - 2 )$ respectively.

Find an equation of the perpendicular bisector of $A B$.
Hence, or otherwise, find an equation of the circle that passes through points $A , B$ and $C$.

OCR H240/03 2018 June Q3

6 marks Moderate -0.3

3 In this question you must show detailed reasoning. A gardener is planning the design for a rectangular flower bed. The requirements are:

the length of the flower bed is to be 3 m longer than the width,
the length of the flower bed must be at least 14.5 m ,
the area of the flower bed must be less than $180 \mathrm {~m} ^ { 2 }$.

The width of the flower bed is $x \mathrm {~m}$.
By writing down and solving appropriate inequalities in $x$, determine the set of possible values for the width of the flower bed.

OCR PURE Q11

4 marks Challenging +1.2

11 In this question you must show detailed reasoning. A biased four-sided spinner has edges numbered $1,2,3,4$. When the spinner is spun, the probability that it will land on the edge numbered $X$ is given by $P ( X = x ) = \begin{cases} \frac { 1 } { 2 } - \frac { 1 } { 10 } x & x = 1,2,3,4 , \\ 0 & \text { otherwise } . \end{cases}$

Draw a table showing the probability distribution of $X$. The spinner is spun three times and the value of $X$ is noted each time.
Find the probability that the third value of $X$ is greater than the sum of the first two values of $X$.

OCR PURE Q8

9 marks Standard +0.3

8 In this question you must show detailed reasoning. The diagram shows part of the graph of $y = 2 x ^ { \frac { 1 } { 3 } } - \frac { 7 } { x ^ { \frac { 1 } { 3 } } }$. The shaded region is enclosed by the curve, the $x$-axis and the lines $x = 8$ and $x = a$, where $a > 8$. \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-5_577_1164_477_438} Given that the area of the shaded region is 45 square units, find the value of $a$.

OCR PURE Q1

3 marks Easy -1.2

1 In this question you must show detailed reasoning. Solve the equation $x ( 3 - \sqrt { 5 } ) = 24$, giving your answer in the form $a + b \sqrt { 5 }$, where $a$ and $b$ are positive integers.

OCR PURE Q3

7 marks Moderate -0.8

3 In this question you must show detailed reasoning. Find the equation of the normal to the curve $y = 4 \sqrt { x } - 3 x + 1$ at the point on the curve where $x = 4$. Give your answer in the form $a x + b y + c = 0$, where $a , b$ and $c$ are integers.

OCR PURE Q4

9 marks Standard +0.3

4 In this question you must show detailed reasoning. The cubic polynomial $6 x ^ { 3 } + k x ^ { 2 } + 57 x - 20$ is denoted by $\mathrm { f } ( x )$. It is given that $( 2 x - 1 )$ is a factor of $\mathrm { f } ( x )$.

Use the factor theorem to show that $k = - 37$.
Using this value of $k$, factorise $\mathrm { f } ( x )$ completely.
1. Hence find the three values of $t$ satisfying the equation $6 \mathrm { e } ^ { - 3 t } - 37 \mathrm { e } ^ { - 2 t } + 57 \mathrm { e } ^ { - t } - 20 = 0$.
2. Express the sum of the three values found in part (c)(i) as a single logarithm.

OCR PURE Q6

6 marks Moderate -0.3

6 In this question you must show detailed reasoning.

\includegraphics[max width=\textwidth, alt={}]{7fc02f90-8f8b-4153-bba1-dc0807124e96-4_650_661_1765_242}

The diagram shows the line $3 y + x = 7$ which is a tangent to a circle with centre $( 3 , - 2 )$.
Find an equation for the circle.

OCR PURE Q2

4 marks Standard +0.3

2 In this question you must show detailed reasoning. Solve the equation $3 x + 1 = 4 \sqrt { x }$.

OCR PURE Q2

4 marks Moderate -0.3

2 In this question you must show detailed reasoning. Solve the equation $x \sqrt { 5 } + 32 = x \sqrt { 45 } + 2 x$. Give your answer in the form $a \sqrt { 5 } + b$, where $a$ and $b$ are integers to be determined.

OCR PURE Q8

7 marks Moderate -0.3

8 In this question you must show detailed reasoning. Given that $\int _ { 4 } ^ { a } \left( \frac { 4 } { \sqrt { x } } + 3 \right) \mathrm { d } x = 7$, find the value of $a$.

OCR Further Statistics AS 2018 June Q6

5 marks Standard +0.8

6 In this question you must show detailed reasoning. The random variable $T$ has a binomial distribution. It is known that $\mathrm { E } ( T ) = 5.625$ and the standard deviation of $T$ is 1.875 . Find the values of the parameters of the distribution.

Activity	Duration
\(A\)	5
\(B\)	3
\(C\)	4
\(D\)	2
\(E\)	1
\(F\)	3
\(G\)	5
\(H\)	2
\(I\)	4
\(J\)	3

Activity	\(A\)	\(B\)	\(C\)	\(D\)	\(E\)	\(F\)	\(G\)	\(H\)	\(I\)	\(J\)
Number of workers	4	1	2	2	3	2	3	3	1	2

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OCR D1 2007 June Q5

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