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OCR Further Mechanics AS 2024 June Q7

11 marks Challenging +1.2

7 A particle $P$ of mass 3.5 kg is attached to one end of a rod of length 5.4 m . The other end of the rod is hinged at a fixed point $O$ and $P$ hangs in equilibrium directly below $O$. A horizontal impulse of magnitude 44.1 Ns is applied to $P$.
In an initial model of the subsequent motion of $P$ the rod is modelled as being light and inextensible and all resistance to the motion of $P$ is ignored. You are given that $P$ moves in a circular path in a vertical plane containing $O$. The angle that the rod makes with the downward vertical through $O$ is $\theta$ radians.

Determine the largest value of $\theta$ in the subsequent motion of $P$. In a revised model the rod is still modelled as being light and inextensible but the resistance to the motion of $P$ is not ignored. Instead, it is modelled as causing a loss of energy of 20 J for every metre that $P$ travels.
Show that according to the revised model, the maximum value of $\theta$ in the subsequent motion of $P$ satisfies the following equation. $$343 ( 1 + 2 \cos \theta ) = 400 \theta$$ You are given that $\theta = 1.306$ is the solution to the above equation, correct to $\mathbf { 4 }$ significant figures.
Determine the difference in the predicted maximum vertical heights attained by $P$ using the two models. Give your answer correct to $\mathbf { 3 }$ significant figures.
Suggest one further improvement that could be made to the model of the motion of $P$.

OCR Further Mechanics AS 2020 November Q1

5 marks Moderate -0.8

1 A car of mass 1200 kg is driven on a long straight horizontal road. There is a constant force of 250 N resisting the motion of the car. The engine of the car is working at a constant power of 10 kW .

The car can travel at constant speed $v \mathrm {~ms} ^ { - 1 }$ along the road. Find $v$.
Find the acceleration of the car at an instant when its speed is $30 \mathrm {~ms} ^ { - 1 }$.

OCR Further Mechanics AS 2020 November Q2

7 marks Moderate -0.8

2 A particle $P$ of mass 4.5 kg is moving in a straight line on a smooth horizontal surface at a speed of $2.4 \mathrm {~ms} ^ { - 1 }$ when it strikes a vertical wall directly. It rebounds at a speed of $1.6 \mathrm {~ms} ^ { - 1 }$.

Find the coefficient of restitution between $P$ and the wall.
Determine the impulse applied to $P$ by the wall, stating its direction.
Find the loss of kinetic energy of $P$ as a result of the collision.
State, with a reason, whether the collision is perfectly elastic.

OCR Further Mechanics AS 2020 November Q3

6 marks Standard +0.3

3 A particle $P$ of mass 5.6 kg is attached to one end of a light rod of length 2.1 m . The other end of the rod is freely hinged to a fixed point $O$. The particle is initially at rest directly below $O$. It is then projected horizontally with speed $5 \mathrm {~ms} ^ { - 1 }$. In the subsequent motion, the angle between the rod and the downward vertical at $O$ is denoted by $\theta$ radians, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-2_499_312_1905_244}

Find the speed of $P$ when $\theta = \frac { 1 } { 4 } \pi$.
Find the value of $\theta$ when $P$ first comes to instantaneous rest.

OCR Further Mechanics AS 2020 November Q4

12 marks Moderate -0.3

4 A particle $P$ of mass 2.4 kg is moving in a straight line $O A$ on a horizontal plane. $P$ is acted on by a force of magnitude 30 N in the direction of motion. The distance $O A$ is 10 m .

Find the work done by this force as $P$ moves from $O$ to $A$. The motion of $P$ is resisted by a constant force of magnitude $R \mathrm {~N}$. The velocity of $P$ increases from $12 \mathrm {~ms} ^ { - 1 }$ at $O$ to $18 \mathrm {~ms} ^ { - 1 }$ at $A$.
Find the value of $R$.
Find the average power used in overcoming the resistance force on $P$ as it moves from $O$ to $A$. When $P$ reaches $A$ it collides directly with a particle $Q$ of mass 1.6 kg which was at rest at $A$ before the collision. The impulse exerted on $Q$ by $P$ as a result of the collision is 17.28 Ns .
1. Find the speed of $Q$ after the collision.
2. Hence show that the collision is inelastic.

OCR Further Mechanics AS 2020 November Q5

9 marks Standard +0.3

5 A particle of mass $m$ moves in a straight line with constant acceleration $a$. Its initial and final velocities are $u$ and $v$ respectively and its final displacement from its starting position is $s$. In order to model the motion of the particle it is suggested that the velocity is given by the equation $\mathrm { v } ^ { 2 } = \mathrm { pu } ^ { \alpha } + \mathrm { qa } ^ { \beta } \mathrm { s } ^ { \gamma }$ where $p$ and $q$ are dimensionless constants.

Explain why $\alpha$ must equal 2 for the equation to be dimensionally consistent.
By using dimensional analysis, determine the values of $\beta$ and $\gamma$.
By considering the case where $s = 0$, determine the value of $p$.
By multiplying both sides of the equation by $\frac { 1 } { 2 } m$, and using the numerical values of $\alpha , \beta$ and $\gamma$, determine the value of $q$.

OCR Further Mechanics AS 2020 November Q6

12 marks Challenging +1.2

6 Three particles $A , B$ and $C$ are free to move in the same straight line on a large smooth horizontal surface. Their masses are $3.3 \mathrm {~kg} , 2.2 \mathrm {~kg}$ and 1 kg respectively. The coefficient of restitution in collisions between any two of them is $e$. Initially, $B$ and $C$ are at rest and $A$ is moving towards $B$ with speed $u \mathrm {~ms} ^ { - 1 }$ (see diagram). $A$ collides directly with $B$ and $B$ then goes on to collide directly with $C$. \includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-4_221_1342_552_246}

The velocities of $A$ and $B$ immediately after the first collision are denoted by $\mathrm { v } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }$ and $\mathrm { V } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }$ respectively.
After the collision between $B$ and $C$ there is a further collision between $A$ and $B$.
Determine the range of possible values of $e$.

OCR Further Mechanics AS 2020 November Q7

9 marks Standard +0.3

7 It is required to model the motion of a car of mass $m \mathrm {~kg}$ travelling at a constant speed $v \mathrm {~ms} ^ { - 1 }$ around a circular portion of banked track. The track is banked at $30 ^ { \circ }$ (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{0501e5a4-2137-4e7d-98ff-2ee81941cbf3-5_414_624_356_242} In a model, the following modelling assumptions are made.

The track is smooth.
The car is a particle.
The car follows a horizontal circular path with radius $r \mathrm {~m}$.
1. Show that, according to the model, $\sqrt { 3 } \mathrm { v } ^ { 2 } = \mathrm { gr }$.

For a particular portion of banked track, $r = 24$.

Find the value of $v$ as predicted by the model. A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.

Explain

OCR Further Mechanics AS 2021 November Q1

7 marks Easy -1.2

1 One end of a light inextensible string of length 2.8 m is attached to a fixed point $O$ on a smooth horizontal table. The other end of the string is attached to a particle $P$ which moves on the table, with the string taut, in a circular path around $O$. The speed of $P$ is constant and $P$ completes each circle in 0.84 seconds.

Find the magnitude of the angular velocity of $P$.
Find the speed of $P$.
Find the magnitude of the acceleration of $P$.
State the direction of the acceleration of $P$.

OCR Further Mechanics AS 2021 November Q2

9 marks Standard +0.3

2 A car has a mass of 800 kg . The engine of the car is working at a constant power of 15 kW . In an initial model of the motion of the car it is assumed that the car is subject to a constant resistive force of magnitude $R N$. The car is initially driven on a straight horizontal road. At the instant that its speed is $20 \mathrm {~ms} ^ { - 1 }$ its acceleration is $0.4 \mathrm {~ms} ^ { - 2 }$.

Show that $R = 430$.
Hence find the maximum constant speed at which the car can be driven along this road, according to the initial model. In a revised model the resistance to the motion of the car at any instant is assumed to be 60 v where $v$ is the speed of the car at that instant. The car is now driven up a straight road which is inclined at an angle $\alpha$ above the horizontal where $\sin \alpha = 0.2$.
Determine the speed of the car at the instant that its acceleration is $0.15 \mathrm {~ms} ^ { - 2 }$ up the slope, according to the revised model.

OCR Further Mechanics AS 2021 November Q3

13 marks Standard +0.3

3 A particle $A$ of mass 0.5 kg is moving with a speed of $3.15 \mathrm {~ms} ^ { - 1 }$ on a smooth horizontal surface when it collides directly with a particle $B$ of mass 0.8 kg which is at rest on the surface. The velocities of $A$ and $B$ immediately after the collision are denoted by $\mathrm { v } _ { \mathrm { A } } \mathrm { ms } ^ { - 1 }$ and $\mathrm { v } _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }$ respectively. You are given that $\mathrm { v } _ { \mathrm { B } } = 2 \mathrm { v } _ { \mathrm { A } }$.

Find the values of $\mathrm { V } _ { \mathrm { A } }$ and $\mathrm { V } _ { \mathrm { B } }$.
Find the coefficient of restitution between $A$ and $B$.
Explain why the coefficient of restitution is a dimensionless quantity.
Calculate the total loss of kinetic energy as a result of the collision.
State, giving a reason, whether or not the collision is perfectly elastic.
Calculate the impulse that $B$ exerts on $A$ in the collision.

OCR Further Mechanics AS 2021 November Q4

9 marks Standard +0.3

4 A small box $B$ of mass 4.2 kg is initially at rest at a point $O$ on rough horizontal ground. A horizontal force of magnitude 35 N is applied to $B$. $B$ moves in a straight line until it reaches the point $S$ which is 2.4 m from $O$. At the instant that $B$ reaches $S$ its speed is $4.5 \mathrm {~ms} ^ { - 1 }$.

1. Find the energy lost due to the resistive forces acting on $B$ as it moves from $O$ to $S$.
2. Deduce the magnitude of the average resistive force acting on $B$ as it moves from $O$ to $S$. When $B$ reaches $S$, the force is no longer applied. $B$ continues to move directly up a smooth slope which is inclined at $20 ^ { \circ }$ above the horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a65c4b75-b8b4-4a51-8abb-f857dc278271-3_275_1027_1866_244}
1. State an assumption required to model the motion of $B$ up the slope with only the information given.
2. Using the assumption made in part (b)(i), determine the distance travelled by $B$ up the slope until the instant when it comes to rest.

OCR Further Mechanics AS 2021 November Q5

10 marks Standard +0.3

5 The escape speed of an unpowered object is the minimum speed at which it must be projected to escape the gravitational influence of the Earth if it is projected vertically upwards from the Earth's surface. A formula for the escape speed $U$ of an unpowered object of mass $m$ is $U = \sqrt { \frac { 2 G m } { r } }$ where $r$ is the radius of the Earth and $G$ is a constant.

Show that the dimensions of $G$ are $\mathrm { M } ^ { - 1 } \mathrm {~L} ^ { 3 } \mathrm {~T} ^ { - 2 }$. A rocket is a powered object. A rocket is launched with a given launch speed and is then powered by engines which apply a constant force for a period of time after the launch. A student wishes to apply the formula given above to a rocket launch. They wish to model the minimum launch speed required for a rocket to escape the Earth's gravitational influence. They realise that the given formula is for unpowered objects and so they include an extra term in the formula to obtain $V = \sqrt { \frac { 2 G m } { r } } - \mathrm { kP } ^ { \alpha } \mathrm { W } ^ { \beta } \mathrm { t } ^ { \gamma }$. In their modified formula, $G$ and $r$ are the same as before. The other variables are defined as follows.

OCR Further Mechanics AS 2021 November Q6

12 marks Challenging +1.8

6 A smooth hemispherical shell of radius $r \mathrm {~m}$ is held with its circular rim horizontal and uppermost. The centre of the rim is at the point $O$ and the point on the inner surface directly below $O$ is $A$. A small object $P$ of mass $m \mathrm {~kg}$ is held at rest on the inner surface of the shell so that $\angle \mathrm { POA } = \frac { 1 } { 3 } \pi$ radians. At the instant that $P$ is released, an impulse is applied to $P$ in the direction of the tangent to the surface at $P$ in the vertical plane containing $O , A$ and $P$. The magnitude of the impulse is denoted by $I$ Ns. $P$ immediately starts to move along the surface towards $A$ (see diagram). $X$ is a point on the circular rim. $P$ leaves the shell at $X$. \includegraphics[max width=\textwidth, alt={}, center]{a65c4b75-b8b4-4a51-8abb-f857dc278271-5_512_860_829_242} In an initial model of the motion of $P$ it is assumed that $P$ experiences no resistance to its motion.

Find in terms of $r , g , m$ and $I$ an expression for the speed of $P$ at the instant that it leaves the shell at $X$.
Find in terms of $r , g , m$ and $I$ an expression for the maximum height attained by $P$ above $X$ after it has left the shell.
Find an expression for the maximum mass of $P$ for which $P$ still leaves the shell. In a revised model it is assumed that $P$ experiences a resistive force of constant magnitude $R$ while it is moving.
Show that, in order for $P$ to still leave the shell at $X$ under the revised model, $$I > \sqrt { m ^ { 2 } g r + \frac { 5 \pi m r R } { 3 } } .$$
Show that the inequality from part (d) is dimensionally consistent.

OCR Further Discrete AS 2018 June Q1

6 marks Moderate -0.3

1 Some jars need to be packed into small crates.
There are 17 small jars, 7 medium jars and 3 large jars to be packed.

A medium jar takes up the same space as four small jars.
A large jar takes up the same space as nine small jars.

Each crate can hold:

at most 12 small jars,
or at most 3 medium jars,
or at most 1 large jar (and 3 small jars),
or a mixture of jars of different sizes.
1. One strategy is to fill as many crates as possible with small jars first, then continue using the medium jars and finally the large jars.

Show that this method will use seven crates. The jars can be packed using fewer than seven crates.

The jars are to be packed in the minimum number of crates possible.

Some other numbers of the small, medium and large jars need to be packed into boxes.
The number of jars that a box can hold is the same as for a crate, except that

OCR Further Discrete AS 2018 June Q2

7 marks Standard +0.8

2 Mo eats exactly 6 doughnuts in 4 days.

What does the pigeonhole principle tell you about the number of doughnuts Mo eats in a day? Mo eats exactly 6 doughnuts in 4 days, eating at least 1 doughnut each day.
Show that there must be either two consecutive days or three consecutive days on which Mo eats a total of exactly 4 doughnuts. Mo eats exactly 3 identical jam doughnuts and exactly 3 identical iced doughnuts over the 4 days.
The number of jam doughnuts eaten on the four days is recorded as a list, for example $1,0,2,0$. The number of iced doughnuts eaten is not recorded.
Show that 20 different such lists are possible.

OCR Further Discrete AS 2018 June Q3

6 marks Standard +0.3

3 In the pay-off matrix below, the entry in each cell is of the form $( r , c )$, where $r$ is the pay-off for the player on rows and $c$ is the pay-off for the player on columns when they play that cell.

	P	Q	R
X	$( 1,4 )$	$( 5,3 )$	$( 2,6 )$
Y	$( 5,2 )$	$( 1,3 )$	$( 0,1 )$
Z	$( 4,3 )$	$( 3,1 )$	$( 2,1 )$

Show that the play-safe strategy for the player on columns is P .
Demonstrate that the game is not stable. The pay-off for the cell in row Y , column P is changed from $( 5,2 )$ to $( y , p )$, where $y$ and $p$ are real numbers.
What is the largest set of values $A$, so that if $y \in A$ then row Y is dominated by another row?
Explain why column P can never be redundant because of dominance.

OCR Further Discrete AS 2018 June Q4

8 marks Standard +0.3

4 The complete bipartite graph $K _ { 3,4 }$ connects the vertices $\{ 2,4,6 \}$ to the vertices $\{ 1,3,5,7 \}$.

How many arcs does the graph $K _ { 3,4 }$ have?
Deduce how many different paths are there that pass through each of the vertices once and once only. The direction of travel of the path does not matter. The arcs are weighted with the product of the numbers at the vertices that they join.
(a) Use an appropriate algorithm to find a minimum spanning tree for this network.
(b) Give the weight of the minimum spanning tree.

OCR Further Discrete AS 2018 June Q5

16 marks Standard +0.3

5 Greetings cards are sold in luxury, standard and economy packs.
The table shows the cost of each pack and number of cards of each kind in the pack.

Pack	Cost (£)	Handmade cards	Cards with flowers	Cards with animals	Other cards	Total number of cards
Luxury	6.50	10	5	5	0	20
Standard	5.00	5	10	5	10	30
Economy	4.00	0	10	10	20	40

Alice needs 25 cards, of which at least 8 must be handmade cards, at least 8 must be cards with flowers and at least 4 must be cards with animals.

Explain why Alice will need to buy at least two packs of cards. Alice does not want to spend more than $\pounds 12$ on the cards.
(a) List the combinations of packs that satisfy all Alice's requirements.
(b) Which of these is the cheapest? Ben offers to buy any cards that Alice buys but does not need. He will pay 12 pence for each handmade card and 5 pence for any other card. Alice does not want her net expenditure (the amount she spends minus the amount that Ben pays her) on the cards to be more than $\pounds 12$.
Show that Alice could now buy two luxury packs. Alice decides to buy exactly 2 packs, of which $x$ are luxury packs, $y$ are standard packs and the rest are economy packs.
Give an expression, in terms of $x$ and $y$ only, for the number of cards of each type that Alice buys. Alice wants to minimise her net expenditure.
Find, and simplify, an expression for Alice's minimum net expenditure in pence, in terms of $x$ and $y$. You may assume that Alice buys enough cards to satisfy her own requirements.
Find Alice's minimum net expenditure.

OCR Further Discrete AS 2018 June Q6

17 marks Standard +0.3

6 Sheona and Tim are making a short film. The activities involved, their durations and immediate predecessors are given in the table below.

	Activity	Duration (days)	Immediate predecessors	S	T
A	Planning	2	-	✓	✓
B	Write script	1	A	✓
C	Choose locations	1	A		✓
D	Casting	0.5	A	✓
E	Rehearsals	2	B, D	✓
F	Get permissions	1	C		✓
G	First day filming	1	E, F	✓
H	First day edits	1	G		✓
I	Second day filming	0.5	G	✓
J	Second day edits	2	H, I		✓
K	Finishing	1	J	✓	✓

By using an activity network, find:
- the minimum project completion time
- the critical activities
- the float on each non-critical activity.
- Give two reasons why the filming may take longer than the minimum project completion time.
Each activity will involve either Sheona or Tim or both.
- The activities that Sheona will do are ticked in the S column.
- The activities that Tim will do are ticked in the T column.
- They will do the planning and finishing together.
- Some of the activities involve other people as well.
An additional restriction is that Sheona and Tim can each only do one activity at a time.
Explain why the minimum project completion is longer than in part (i) when this additional restriction is taken into account.
The project must be completed in 14 days. Find:
1. the longest break that either Sheona or Tim can take,
2. the longest break that Sheona and Tim can take together,
3. the float on each activity.

OCR Further Discrete AS 2019 June Q1

5 marks Moderate -0.8

1 Alfie has a set of 15 cards numbered consecutively from 1 to 15.
He chooses two of the cards.

How many different sets of two cards are possible? Alfie places the two cards side by side to form a number with 2,3 or 4 digits.
Explain why there are fewer than ${ } ^ { 15 } \mathrm { P } _ { 2 } = 210$ possible numbers that can be made.
Explain why, with these cards, 1 is the lead digit more often than any other digit. Alfie makes the number 113, which is a 3-digit prime number. Alfie says that the problem of working out how many 3-digit prime numbers can be made using two of the cards is a construction problem, because he is trying to find all of them.
Explain why Alfie is wrong to say this is a construction problem.

OCR Further Discrete AS 2019 June Q2

10 marks Moderate -0.3

2 Two graphs are shown below. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8473d9aa-a4db-4001-ac71-e5fbbaee530c-2_396_353_1343_479} \captionsetup{labelformat=empty} \caption{Graph G1}

\end{figure} \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{8473d9aa-a4db-4001-ac71-e5fbbaee530c-2_399_328_1340_1233} \captionsetup{labelformat=empty} \caption{Graph G2}

\end{figure}

List the vertex degrees for each graph.
Prove that the graphs are non-isomorphic. The two graphs are joined together by adding an arc connecting J and T .
1. Explain how you know that the resulting graph is not Eulerian.
2. Describe how the graph can be made Eulerian by adding one more arc. The vertices of the graph $K _ { 3 }$ are connected to the vertices of the graph $K _ { 4 }$ to form the graph $K _ { 7 }$.
Explain why 12 arcs are needed connecting $K _ { 3 }$ to $K _ { 4 }$.

OCR Further Discrete AS 2019 June Q3

11 marks Moderate -0.5

Give an example of a standard sorting algorithm that can be used when some of the values are not known until after the sorting has been started. Becky needs to sort a list of numbers into increasing order.
She uses the following algorithm:
STEP 1: Let $L$ be the first value in the input list.
Write this as the first value in the output list and delete it from the input list.
STEP 2: If the input list is empty go to STEP 7.
Otherwise let $N$ be the new first value in the input list and delete this value from the input list. STEP 3: $\quad$ Compare $N$ with $L$.
STEP 4: If $N$ is less than or equal to $L$
- write the value of $N$ immediately before $L$ in the output list,
- replace $L$ with the first value in the new output list,
- then go to STEP 2.
STEP 5: If $N$ is greater than $L$
- if $L$ is the value of the last number in the output list, go to STEP 6;
- otherwise, replace $L$ with the next value in the output list and then go to STEP 3.
STEP 6: $\quad$ Write the value of $N$ immediately after $L$ in the output list. Let $L$ be the first value in the new output list and then go to STEP 2. STEP 7: Print the output list and STOP.
Trace through Becky's algorithm when the input list is $$\begin{array} { l l l l l l } 6 & 9 & 5 & 7 & 6 & 4 \end{array}$$ Complete the table in the Printed Answer Booklet, starting a new row each time that STEP 3 or STEP 7 is used.
You should not need all the lines in the Answer Booklet. Becky measures the efficiency of her sort by counting using the number of times that STEP 3 is used.
1. How many times did Becky use STEP 3 in sorting the list from part (b)?
2. What is the greatest number of times that STEP 3 could be used in sorting a list of 6 values? A computer takes 15 seconds to sort a list of 60 numbers using Becky's algorithm.
Approximately how long would you expect it to take the computer to sort a list of 300 numbers using the algorithm?

OCR Further Discrete AS 2019 June Q4

10 marks Standard +0.3

4 The table shows the activities involved in a project, their durations in hours and their immediate predecessors. The activities can be represented as an activity network.

Activity	A	B	C	D	E	F	G	H
Duration	2	4	5	4	3	3	2	4
Immediate predecessors	-	A	-	A, C	B, C	B, D	D, E	F, G

Use standard algorithms to find the activities that form
You must show working to demonstrate the use of the algorithms. Only one of the paths from part (a) has a practical interpretation.
What is the practical interpretation of the total weight of that path? The duration of activity E can be changed. No other durations change.
What is the smallest increase to the duration of E that will make activity E become part of a longest path through the network?

OCR Further Discrete AS 2019 June Q5

12 marks Moderate -0.3

5 Corey is training for a race that starts in 18 hours time. He splits his training between gym work, running and swimming.

At most 8 hours can be spent on gym work.
At least 4 hours must be spent running.
The total time spent on gym work and swimming must not exceed the time spent running.

Corey thinks that time spent on gym work is worth 3 times the same time spent running or 2 times the same time spent swimming. Corey wants to maximise the worth of the training using this model.

Formulate a linear programming problem to represent Corey's problem. Your formulation must include defining the variables that you are using. Suppose that Corey spends the maximum of 8 hours on gym work.
1. Use a graphical method to determine how long Corey should spend running and how long he should spend swimming.
2. Describe why this solution is not practical.
3. Describe how Corey could refine the LP model to make the solution more realistic.

	P	Q	R
X	\(( 1,4 )\)	\(( 5,3 )\)	\(( 2,6 )\)
Y	\(( 5,2 )\)	\(( 1,3 )\)	\(( 0,1 )\)
Z	\(( 4,3 )\)	\(( 3,1 )\)	\(( 2,1 )\)

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