Questions - OCR - A-Level Maths

This diagram shows a network. The insert has a copy of this network together with a list of the arcs, sorted into increasing order of weight. Use Kruskal's algorithm on the insert to find a minimum spanning tree for this network. Draw your tree and give its total weight.
Use your answer to part (i) to find the weight of a minimum spanning tree for the network with vertex $E$, and all the arcs joined to $E$, removed. Hence find a lower bound for the travelling salesperson problem on the original network.
Show that the nearest neighbour method, starting from vertex $A$, fails on the original network.
Apply the nearest neighbour method, starting from vertex $B$, to find an upper bound for the travelling salesperson problem on the original network.
Apply Dijkstra's algorithm to the copy of the network in the insert to find the least weight path from $A$ to $G$. State the weight of the path and give its route.
The sum of the weights of all the arcs is 300 . 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. The weights of least weight paths from vertex $A$ should be found using your answer to part (v); the weights of other such paths should be determined by inspection.

OCR D1 2009 January Q4

12 marks Easy -1.2

4 Answer this question on the insert provided. The list of numbers below is to be sorted into decreasing order using shuttle sort. $$\begin{array} { l l l l l l l l l } 21 & 76 & 65 & 13 & 88 & 62 & 67 & 28 & 34 \end{array}$$

How many passes through shuttle sort will be required to sort the list? After the first pass the list is as follows. $$\begin{array} { l l l l l l l l l } 76 & 21 & 65 & 13 & 88 & 62 & 67 & 28 & 34 \end{array}$$
State the number of comparisons and the number of swaps that were made in the first pass.
Write down the list after the second pass. State the number of comparisons and the number of swaps that were used in making the second pass.
Complete the table in the insert to show the results of the remaining passes, recording the number of comparisons and the number of swaps made in each pass. You may not need all the rows of boxes printed. When the original list is sorted into decreasing order using bubble sort there are 30 comparisons and 17 swaps.
Use your results from part (iv) to compare the efficiency of these two methods in this case. Katie makes and sells cookies. Each batch of plain cookies takes 8 minutes to prepare and then 12 minutes to bake. Each batch of chocolate chip cookies takes 12 minutes to prepare and then 12 minutes to bake. Each batch of fruit cookies takes 10 minutes to prepare and then 12 minutes to bake. Katie can only bake one batch at a time. She has the use of the kitchen, including the oven, for at most 1 hour.
[0pt]
Each batch of cookies must be prepared before it is baked. By considering the maximum time available for baking the cookies, explain why Katie can make at most 4 batches of cookies. [2] Katie models the constraints as $$\begin{gathered} x + y + z \leqslant 4 \\ 4 x + 6 y + 5 z \leqslant 24 \\ x \geqslant 0 , y \geqslant 0 , z \geqslant 0 \end{gathered}$$ where $x$ is the number of batches of plain cookies, $y$ is the number of batches of chocolate chip cookies and $z$ is the number of batches of fruit cookies that Katie makes.
Each batch of cookies that Katie prepares must be baked within the hour available. By considering the maximum time available for preparing the cookies, show how the constraint $4 x + 6 y + 5 z \leqslant 24$ was formed.
In addition to the constraints, what other restriction is there on the values of $x , y$ and $z$ ? Katie will make $\pounds 5$ profit on each batch of plain cookies, $\pounds 4$ on each batch of chocolate chip cookies and $\pounds 3$ on each batch of fruit cookies that she sells. Katie wants to maximise her profit.
Write down an expression for the objective function to be maximised. State any assumption that you have made.
Represent Katie's problem as an initial Simplex tableau. Perform one iteration of the Simplex algorithm, choosing to pivot on an element from the $x$-column. Show how each row was obtained. Write down the number of batches of cookies of each type and the profit at this stage. After carrying out market research, Katie decides that she will not make fruit cookies. She also decides that she will make at least twice as many batches of chocolate chip cookies as plain cookies.
Represent the constraints for Katie's new problem graphically and calculate the coordinates of the vertices of the feasible region. By testing suitable integer-valued coordinates, find how many batches of plain cookies and how many batches of chocolate chip cookies Katie should make to maximise her profit. Show your working.

OCR D1 2010 January Q1

11 marks Standard +0.3

1 Answer this question on the insert provided. \includegraphics[max width=\textwidth, alt={}, center]{e1495f6b-c09f-46a1-a6f8-02354e28887a-02_533_1353_342_395}

Apply Dijkstra's algorithm to the copy of this network in the insert to find the least weight path from $A$ to $F$. State the route of the path and give its weight.
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. Write down a closed route that has this least weight. An extra arc is added, joining $B$ to $E$, with weight 2 .
Write down the new least weight path from $A$ to $F$. Explain why the new least weight closed route, that uses every arc at least once, has no repeated arcs.

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.
\href{http://physicsandmathstutor.com}{physicsandmathstutor.com}

OCR D1 2009 June Q4

25 marks Standard +0.8

4 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[max width=\textwidth, alt={}, center]{fe06fa02-9f5d-4082-8e96-feea705d3fa2-4_812_1198_443_475}

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. 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? 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.) 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? 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 $A$.
Apply the nearest neighbour method starting from $C$ to find an upper bound for the distance that Amber must travel.
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.

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.

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

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 $X$.
(a) Find the values of $r$ for which $\mathrm { P } ( X = r ) \leqslant \mathrm { P } ( X = r + 1 )$.
(b) 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 H240/03 2020 November Q6

11 marks Challenging +1.2

6 In this question you must show detailed reasoning.

\includegraphics[max width=\textwidth, alt={}]{373fa8e4-9c10-4fcf-9e00-e497161b4c6d-06_495_800_312_244}

The diagram shows the curve with equation $4 x y = 2 \left( x ^ { 2 } + 4 y ^ { 2 } \right) - 9 x$.

Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 4 x - 4 y - 9 } { 4 x - 16 y }$. At the point $P$ on the curve the tangent to the curve is parallel to the $y$-axis and at the point $Q$ on the curve the tangent to the curve is parallel to the $x$-axis.
Show that the distance $P Q$ is $k \sqrt { 5 }$, where $k$ is a rational number to be determined.

OCR H240/03 2022 June Q6

8 marks Standard +0.3

6 In this question you must show detailed reasoning.

\includegraphics[max width=\textwidth, alt={}]{e69f8d73-764e-4f13-a126-faec02c4ad08-06_611_1344_306_242}

The diagram shows the curves $y = \sqrt { 2 x + 9 }$ and $y = 4 \mathrm { e } ^ { - 2 x } - 1$ which intersect on the $y$-axis. The shaded region is bounded by the curves and the $x$-axis. Determine the area of the shaded region, giving your answer in the form $p + q \ln 2$ where $p$ and $q$ are constants to be determined.

OCR H240/03 2022 June Q7

8 marks Standard +0.8

7 In this question you must show detailed reasoning.

Show that the equation $m \sec \theta + 3 \cos \theta = 4 \sin \theta$ can be expressed in the form $$m \tan ^ { 2 } \theta - 4 \tan \theta + ( m + 3 ) = 0 .$$
It is given that there is only one value of $\theta$, for $0 < \theta < \pi$, satisfying the equation $m \sec \theta + 3 \cos \theta = 4 \sin \theta$. Given also that $m$ is a negative integer, find this value of $\theta$, correct to $\mathbf { 3 }$ significant figures.

OCR H240/03 2023 June Q12

13 marks Standard +0.3

12 In this question you should take the acceleration due to gravity to be $10 \mathrm {~ms ^ { - 2 }$.}

\includegraphics[max width=\textwidth, alt={}]{977ffad6-2440-46bf-9f17-0f30817d2ddf-09_410_1344_324_244}

A small ball $P$ is projected from a point $A$ with speed $39 \mathrm {~ms} ^ { - 1 }$ at an angle of elevation $\theta$, where $\sin \theta = \frac { 5 } { 13 }$ and $\cos \theta = \frac { 12 } { 13 }$. Point $A$ is 20 m vertically above a point $B$ on horizontal ground. The ball first lands at a point $C$ on the horizontal ground (see diagram). The ball $P$ is modelled as a particle moving freely under gravity.

Find the maximum height of $P$ above the ground during its motion. The time taken for $P$ to travel from $A$ to $C$ is $T$ seconds.
Determine the value of $T$.
State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). At the instant that $P$ is projected, a second small ball $Q$ is released from rest at $B$ and moves towards $C$ along the horizontal ground. At time $t$ seconds, where $t \geqslant 0$, the velocity $v \mathrm {~ms} ^ { - 1 }$ of $Q$ is given by
$v = k t ^ { 3 } + 6 t ^ { 2 } + \frac { 3 } { 2 } t$,
where $k$ is a positive constant.
Given that $P$ and $Q$ collide at $C$, determine the acceleration of $Q$ immediately before this collision.
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-10_607_803_303_246} The diagram shows a small block $B$, of mass 2 kg , and a particle $P$, of mass 4 kg , which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of $60 ^ { \circ }$ with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion $P$ moves down the plane and $B$ does not reach the pulley. It is given that the coefficient of friction between $P$ and the inclined plane is twice the coefficient of friction between $B$ and the horizontal surface.
Determine, in terms of $g$, the tension in the string. When $P$ is moving at $2 \mathrm {~ms} ^ { - 1 }$ the string breaks. In the 0.5 seconds after the string breaks $P$ moves 1.9 m down the plane.
Determine the deceleration of $B$ after the string breaks. Give your answer correct to 3 significant figures.

OCR PURE 2018 May Q8

6 marks Standard +0.8

8 In this question you must show detailed reasoning. The lines $y = \frac { 1 } { 2 } x$ and $y = - \frac { 1 } { 2 } x$ are tangents to a circle at $( 2,1 )$ and $( - 2,1 )$ respectively. Find the equation of the circle in the form $x ^ { 2 } + y ^ { 2 } + a x + b y + c = 0$, where $a , b$ and $c$ are constants.

OCR PURE 2020 October 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 $X$.

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

Questions — OCR (4619 questions)

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