*P35*
PRE-LEAVING CERTIFICATE EXAMINATION, 2023
TRIAILSCRÚDÚ NA HARDTEISTIMÉIREACHTA, 2023
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APPLIED MATHEMATICS — HIGHER LEVEL
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TIME: 2½ hours
400 marks
Eight questions to be answered. All questions carry equal marks.
A Formulae and Tables booklet may be obtained from the Superintendent.
Take the value of g to be 9.8 m s–2.
Marks may be lost if necessary work is not clearly shown.
Marks may be lost for omission of correct units with numerical answers.
_____________________
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Instructions
There are eight questions on this paper. Answer all questions.
Write your details in the box on the front cover.
Write your answers in blue or black pen. You may use pencil in graphs and diagrams only.
All of your work should be presented in the answer areas, or on the given graphs,
networks or other diagrams. Anything that you write outside of these areas may not be seen by
the examiner.
Write all answers into this booklet. There is space for extra work at the back of the booklet. If you
need to use it, label any extra work clearly with the question number and part.
The superintendent will give you a copy of the Formulae and Tables booklet. You must return it at
the end of the examination. You are not allowed to bring your own copy into the examination.
You may lose marks if your solutions do not include relevant supporting work.
You may lose marks if the appropriate units of measurement are not included, where relevant.
You may lose marks if your answers are not given in their simplest form, where relevant.
Diagrams are generally not drawn to scale.
Unless otherwise indicated, take the value of 𝑔, the acceleration due to gravity, to be 9.8 m s–2.
Unless otherwise indicated, 𝚤⃗ and 𝚥⃗ are unit perpendicular vectors in the horizontal and vertical
directions, respectively, or eastwards and northwards, respectively, as appropriate to the
question.
Write the make and model of your calculator(s) here:
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(a)
A particle of mass 5 kg hangs at the end of an elastic string
of natural length 1 metre and elastic constant k N/m.
When the string is extended the string is 2.4 m in length.
(i)
Find the value of k.
(ii)
Find the length of the string if a particle of mass 8 kg was used instead of a 5 kg
mass.
(b)
P
Q
R
S
P
1
1
1
1
Q
1
0
2
0
R
0
1
1
2
S
1
0
0
0
The adjoining matrix A above is for a graph G.
(i)
A = table
How can you tell from the matrix that G is a directed graph?
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(ii)
Draw the graph of G.
(iii)
Find 𝐴2 and 𝐴4 .
(iv)
How many walks of length 4 are there from P to R?
(v)
How many walks of length 2 are there from R to S?
(vi)
Mary says “There are at least 10 walks of length 2 or 4 from any vertex to any other
vertex”. Is she right? Justify your answer.
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2.
(a)
Given that the length of the minimum
spanning tree for the network is of
length 88, find the value of x.
(b)
The matrix represents a network of roads between six villages: A, B, C, D, E and F.
The values in the matrix are the distances (in km) along these roads.
(i)
A
B
C
D
E
F
A
---
7
3
---
9
11
B
7
---
4
2
---
7
C
3
4
---
5
8
---
D
---
2
5
---
6
3
E
9
---
8
6
---
---
F
11
7
---
3
---
---
Show this information in this diagram:
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(ii)
Use Kruskal’s algorithm to determine the minimum spanning tree for the network
and its total length.
(iii)
Draw the minimum spanning tree.
(iv)
Starting at A, find the minimum spanning tree using Prim’s algorithm with
the matrix.
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3.
𝑑𝑦
(a)
Given the differential equation 𝑥 𝑑𝑥 = √9 – 𝑦²
(i)
Find the general solution given that 𝑦 = 0 when 𝑥 = 1.
(ii)
Find the value of 𝑦 when 𝑥 = √𝑒 /2 .
(b)
A particle of mass m is projected vertically upwards from ground level with a speed
1
of 84 ms-1. In addition to the weight, there is air-resistance of magnitude 20 mv ²,
where its speed is v, both forces being measured in newtons.
(i)
Find the maximum height reached by the particle.
(ii)
Find the approximate time taken to reach maximum height.
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4.
(a)
The network above shows the times (in minutes) of a short project.
(i)
Fill in any missing early or late times.
(ii)
What is the minimum time for the completion of the project?
(iii)
Name the critical activities.
(iv)
Write down the critical paths.
(v)
What is the total float for D?
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(b)
A new car costs €28,000. Maintenance costs are as follows:
€500 in the first year, €800 in the second year and €1,200 in the third year.
The resale value of a second-hand car is €20,000 after 1 year, €18,000 after 2 years
and €16,000 after 3 years. Sheila buys a new car and wants to use dynamic
programming to decide what is the best strategy over the next six years; how often
should she replace her car over these six years and in which years? Assume that
there is no resale value after three years (she must sell any new car after 1, 2 or
3 years) and that she will sell whatever car she has at the end of year 6.
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5.
(a)
A ball is projected horizontally from
a point P above a smooth horizontal
plane with speed 3 m𝑠 –1.
The ball first hits the plane at a point
whose horizontal displacement from
P is 0.6 m.
The ball next strikes the plane at a horizontal displacement of 1.5 m from P.
The coefficient of restitution between the ball and the plane is e. Find the value of e.
(b)
Two smooth spheres, A and B, have masses 2m and m respectively, and velocity
vectors 3u 𝚤⃗ + 4u 𝚥⃗ and –4u 𝚤⃗ + 3u 𝚥⃗ respectively, when they collide, with their line
of centres parallel to the unit vector 𝚤⃗, where 𝚤⃗ is along their line of centres at
impact.
The impact causes a loss of energy equal to the original kinetic energy of B.
𝑎
Find the coefficient of restitution between the spheres in the form √𝑏 , a,b  N.
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6.
(a)
(i)
If the weights in this multi-stage network represent distances in km, find the
shortest path from start (S) to destination (D).
(ii)
If, instead, the weights represent profit (in €100s) in a shop over 4 days, find
the optimal path and the maximum profit.
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(b)
A particle of mass 3 m is connected by a light inextensible string
passing over a smooth, fixed, pulley to a smooth pulley P of mass M.
Over this pulley passes a similar string carrying masses m and 2m
at its ends.
The system is released from rest.
(i)
Show that the acceleration of P is:
𝑔(3𝑀−𝑚)
(3𝑀+17𝑚)
If P is replaced by a pulley of negligible mass, find:
(ii)
the new acceleration of P.
(iii)
the tension in the strings.
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7.
(a)
A particle falls from rest at a point A under gravity. After it has fallen a distance a,
another particle is given a downward speed √8𝑎𝑔 from the same starting point A.
(i)
Show that the two particles collide.
(ii)
Find the time and distance from A at which they collide.
(b)
A car X, moving with constant acceleration, is travelling at 30 kmℎ–1 as it passes
a fixed point P. After travelling a further quarter of a kilometre, the car reaches
a speed of v kmℎ–1 and is thereafter driven at this speed. Six seconds after X
passes P, another car, Y, travelling at 45 kmℎ–1 in the same direction on the same
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road and accelerating at 33 m𝑠 –2, also passes P. On reaching a speed of 75 kmℎ–1,
car Y is thereafter driven at this speed. Y passes X having travelled 1 km beyond P.
(i)
Find the time taken by Y to cover the first kilometre beyond P.
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(ii)
Calculate the value of v.
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8.
(a)
The population of an island off the west coast of Scotland was given by 𝑃0 = 3,200
last year. This year it had dwindled to 𝑃1 = 2,400. Census experts estimate 𝑃𝑛 ,
the island’s population after 𝑛 years, obeys the difference equation
𝑃𝑛 =
1
{16 𝑃𝑛−1 − 5 𝑃𝑛−2 } for 𝑛 > 1
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(i)
Solve this difference equation.
(ii)
Estimate the population in 5 years’ time.
(iii)
Estimate the population in 10 years’ time.
(iv)
Show that the population will dwindle to nothing as time goes by.
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(b)
Caroline takes out a loan of €150,000 to build an extension to her house.
The bank agrees to a 15-year loan at a monthly percentage rate (MPR) of 0.5%.
(i)
If 𝐷𝑛 is the amount of debt owing after 𝑛 months and 𝐴 the amount she pays back
each month, write down a difference equation in 𝐷𝑛 .
(ii)
Write down the value of 𝐷0 .
(iii)
Solve the difference equation.
(iv)
Find (to the nearest cent) the amount she will have to pay back every month.
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(v)
If after 60 months the bank increases its MPR to 0.8%, how much will she have to
pay back every month (to the nearest euro) in order to clear the loan in the agreed
time of 15 years?
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