*Y5*
Pre-Junior Certificate Examination, 2015
Triailscrúdú an Teastais Shóisearaigh, 2015
Mathematics
Paper 1
Ordinary Level
2 hours
300 marks
For examiner
Name:
Question
School:
Mark
1
2
Address:
3
Class:
4
Teacher:
6
5
7
8
9
10
11
12
13
14
15
Total
Grade
Running total
Page 1 of 24
Instructions
There are 15 questions on this examination paper.
Answer all questions.
Questions do not necessarily carry equal marks. To help you manage your time during this
examination, a maximum time for each question is suggested. If you remain within these times, you
should have about 10 minutes left to review your work.
Write your answers in the spaces provided in this booklet. You may lose marks if you do not do so.
There is space for extra work at the back of the booklet. You may also ask the superintendent for
more paper. Label any extra work clearly with the question number and part.
You will lose marks if all necessary work is not clearly shown.
Answers should include the appropriate units of measurement, where relevant.
Answers should be given in simplest form, where relevant.
Write the make and model of your calculator(s) here:
Page 2 of 24
Question 1
(a)
(Suggested maximum time: 5 minutes)
On the Venn diagram below, shade the region that represents A ∪ B .
A
(b)
B
On the Venn diagram below, shade the region that represents A ∩ B .
A
(c)
B
On the Venn diagram below, shade the region that represents A \ B .
A
(d)
B
If A represents the students in a school that do athletics as an after-school activity and B
represents the students in a school that do basketball as an after-school activity, explain in
your own words what the set A \ B represents.
Page 3 of 24
Question 2
(Suggested maximum time: 5 minutes)
P
Q
R
S
T
A
B
C
D
E
F
G
H
(a)
In the diagram above what fraction of row A is shaded?
(b)
In the same diagram, what fraction of column P is shaded?
(c)
Using the diagram, or otherwise, calculate the result when your fractions in part (a) and (b)
are multiplied.
×
=
Page 4 of 24
(d)
(i)
Shade in
2
on the diagram below.
3
(ii)
Shade in
3
on the diagram below.
4
(iii)
Calculate
3 2
− and explain your answer by using the diagram below or otherwise.
4 3
Page 5 of 24
Question 3
(a)
(Suggested maximum time: 4 minutes)
Three students were working on a science experiment. At different stages they used
a cylindrical test-tube and filled it with some water.
• John filled his test-tube so it was 0.7 full.
1
• Mary filled her test-tube so it was full.
5
• Sean filled his test-tube so it was 40% full.
On the cylindrical test-tube below, mark in the levels of water for each of the students.
(b)
4
of the pupils walk to school. The rest of the students go to
5
school by bus. How many students go to school by bus?
In a school of 800 pupils,
Page 6 of 24
Question 4
(Suggested maximum time: 5 minutes)
The table below shows the values of 3 when it is raised to certain values.
(a)
Complete the table.
Power of 3
Expanded
power of 3
Answer
31
3
3
32
3× 3
9
33
34
27
3× 3× 3× 3
35
243
729
Sean is to receive a bonus and is given two options:
• Option A: €2,000 cash today
or
• Option B: €3 today, €9 the next day, €27 the next day and trebling every day for
the 7 days in total.
(b)
Which option should Sean choose if he wants to get the largest bonus? Explain your answer.
Option:
Reason:
Page 7 of 24
Question 5
(Suggested maximum time: 8 minutes)
Emma went on a training cycle for 2 hours and 45 minutes and cycled at
an average speed of 18 kilometres per hour.
(a)
By rounding to the nearest whole number, estimate the total distance
covered by the cyclist in the training session.
(b)
Calculate the actual distance covered by the cyclist in the training session.
Allanah started her training cycling session at 9:40 and finished at 11:10.
(c)
Calculate the amount of time Allanah spent on her training session.
Answer:
Reason:
(d)
Allanah covered a total distance of 24 kilometres during her training cycle. Was Allanah
travelling at a greater average speed than Emma? Give a reason to support your answer.
Answer:
Reason:
Page 8 of 24
Question 6
(Suggested maximum time: 12 minutes)
(a)
Ali works in a supermarket and earns € 9 ⋅ 25 per hour. He works a 38 hour week. Find Ali’s
gross pay for the week.
(b)
Ali has to pay income tax at a rate of 20%. Find Ali’s gross tax.
(c)
He has a tax credit of €44 per week. Find Ali’s net tax
(d)
How much per week is he left with after tax?
(e)
Ali had €1,600 in the Credit Union at the beginning of a year. The Credit Union paid 3%
interest on his money. Find the interest earned in that year.
(f)
Ali withdrew €400 for spending money for his holiday in Manchester. Ali changed his
money into sterling (£) when the exchange rate was €1 = £ 0 ⋅ 79 sterling. How much
sterling did he receive?
Page 9 of 24
Question 7
(Suggested maximum time: 8 minutes)
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
P = {Odd numbers between 1 and 12}
Q = {Multiples of three between 1 and 12}
(a)
List the elements of the set P.
P ={
(b)
,
,
,
,
}
List the elements of the set Q.
Q = { 3,
(c)
,
,
}
,
Fill in the Venn diagram below placing all the elements of U in the correct regions.
P
(d)
List the elements of P ∩ Q .
P ∩Q = {
(e)
Q
,
}
Complete the sentence below:
If a number is outside both sets P and Q then it is not odd and it is
(f)
The number 18 is added to the universal set. If P and Q can include elements bigger than 12,
then place 12 in the correct region on the Venn diagram in part (c) and explain why you
placed it there.
Reason:
Page 10 of 24
Question 8
(Suggested maximum time: 8 minutes)
1
(a)
Calculate 25 2 .
(b)
Give two reasons why − 7 ⋅ 6 is not a natural number.
(c)
Write the number 2,345 ⋅ 287 :
(d)
(i)
Correct to the nearest whole number.
(ii)
Correct to one decimal place.
(iii)
Correct to one significant figure.
(iv)
Correct to the nearest 100.
When a house was sold the final sale price was rounded to the nearest thousand. If a house
is sold for €350,000, state the maximum and minimum price the house could have sold for.
Maximum price:
Minimum price:
Page 11 of 24
Question 9
(Suggested maximum time: 8 minutes)
Three students Padraic, Mary and Tariq were asked to solve the same equation.
Here are their solutions:
Padraic’s answer
Mary’s answer
3( x − 2) = 12
3x − 6 = 12
3x = 12 − 6
3( x − 2) = 12
3x − 6 = 12
3x = 6
x=2
(a)
3 x − 6 + 6 = 12 + 6
3 x = 18
3 x 18
=
3
3
x=6
Tariq’s answer
3( x − 2) = 12
3 x − 6 = 12
3 x − 6 + 6 = 12 + 6
3 x = 18
x = 18 − 3
x = 15
Identify the correct answer and verify the answer.
Answer:
Reason:
(b)
Circle a maths error on one of the other solutions and explain the error.
Page 12 of 24
Question 10
(a)
(Suggested maximum time: 10 minutes)
Find the value of the following expressions when a = 4 and b = 2 .
4a − 3b
(i)
4(
) − 3(
(ii)
3ab
3(
)(
(
(c)
)=
b 2 − 3b + 4
(iii)
(b)
)=
)2 − 3(
)+ 4 =
(i)
Solve the inequality 2 x − 5 ≤ 4, x ∈ ℕ.
(ii)
Show the solution set on the number line below.
(i)
Simplify 3 y 2 + 5 y + 7 + 5 y 2 + 3 y + 2 .
(3 y
2
(
) (
) (
)
)
+ 5y + 7 + 5y2 + 3y + 2 =
Simplify 5(2 x − 1) − 2(3 x − 2 ) .
(ii)
5(2 x − 1) − 2(3 x − 2 ) =
(iii)
Multiply 2m + 3 by m − 4 .
(2m + 3)(m − 4) =
Page 13 of 24
Question 11
(Suggested maximum time: 4 minutes)
On a Saturday Bill drove from Dublin to Galway. On his journey he stopped in Athlone to visit his
friend Monica. The graph below shows the different stages of his journey.
(a)
How far is it from Dublin to Athlone?
(b)
How long did Bill spend visiting his friend?
(c)
On one section of Bill’s journey he was driving faster than on any other section. Identify this
section and justify your answer by reference to the graph.
Page 14 of 24
(d)
How long did Bill take to drive from Athlone to Galway?
(e)
Bill’s car uses 1 litre of diesel for every 15 kilometres travelled. If diesel costs € 1⋅ 38 per
litre, calculate how much the journey from Dublin to Galway costs Bill.
Page 15 of 24
Question 12
(a)
(Suggested maximum time: 5 minutes)
Below is an algebra word problem broken down, in table form, into its different steps.
Complete the table.
Statement
I have a number
Algebra Statement
x
I add 5 to it
2( x + 5)
The result is a total of …
(b)
2(x + 5) = 16
Solve this equation and state what the original number was.
Original number:
(c)
Here is another table with just the algebra statements. Complete the table
Statement
Algebra Statement
y
y −5
3( y − 5)
3( y − 5) = 6
Page 16 of 24
Question 13
(a)
(Suggested maximum time: 10 minutes)
Five of the following terms have factors other than 1.
2 xy
4x 2
12 xy
5y
2x 2 − 6x
Write down the highest common factor of these terms.
(b)
Factorise fully each of the following:
5 p + 20q − 30t
(i)
= 5(
(ii)
)
5ab − bc + 5ad − dc
= b(
(iii)
)+ d(
)
x 2 − 5 x − 14
= ( x + 2 )(
)
Page 17 of 24
(c)
Divide x 2 − 9 x + 20 by x − 5 .
(d)
The cost for two adults and three children to go on a holiday is €2,200. John made the
equation to 2 x + 3 y = 2,200 to represent the cost of the holiday. The cost for one adult and
two children to go on the same holiday is €1,250.
(i)
Write down an equation to represent this cost.
(ii)
Use simultaneous equations to find the cost of a holiday for an adult and the cost of
a holiday for a child.
Cost for an adult:
Cost for a child:
Page 18 of 24
Question 14
(a)
(Suggested maximum time: 8 minutes)
When repairing a computer a company charges an initial fee of €20 and €30 per hour for
each hour they spend working on the repair. Complete the table and graph the resulting
couples.
Time in hours
Fee charged
0
1
2
3
4
5
€20
(b)
If a computer takes three and a half hours to repair, use your graph to estimate the cost.
(c)
Write down a formula that will calculate the cost of a repair. State clearly the meaning of
any letters you use in your formula.
Page 19 of 24
Question 15
(Suggested maximum time: 10 minutes)
The graphs of two functions f and g are shown below. The functions are:
f (x ) = x + 1
g (x ) = x 2 − 3x − 4
(a)
Match the graphs to the functions by writing f(x) or g(x) in the boxes beside the
corresponding graphs on the grid.
(b)
For both of the functions above , explain how you decided on your answer.
Function:
Explanation:
Function:
Explanation:
Page 20 of 24
(c)
Use the graph to find g ( x ) = 3 .
g (3) =
(d)
Verify your answer to (c) above by finishing the following calculation.
g (3) = (
(e)
)2 − 3(
)− 4
Use the graphs to find the values of x for which x 2 − 3 x − 4 = x + 1 .
x=
x=
Page 21 of 24
You may use this page for extra work.
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You may use this page for extra work.
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You may use this page for extra work.
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