Mathematics (Project Maths – Phase 3) - Folens · Mathematics (Project Maths – Phase 3) Paper 1 ... Answers should be given in simplest form, ... Junior Certificate 2013 –sample

2013. S332 S

Coimisiún na Scrúduithe Stáit

State Examinations Commission

Junior Certificate Examination, 2013 Sample Paper

Mathematics (Project Maths – Phase 3)

Paper 1

Ordinary Level

Time: 2 hours

300 marks

Examination number

Centre stamp

Running total

For examiner

Question Mark Question Mark

1 11

2 12

3 13

4 14

5

6

7

8

9

10 Total

Grade

Junior Certificate 2013 –sample paper Page 2 of 19 Project Maths, Phase 3 Paper 1 – Ordinary Level

Instructions

There are 14 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. 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.

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.

Marks will be lost 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:


Question 1 (Suggested maximum time: 5 minutes)

(a) On the Venn diagram below, shade the region that represents .A B∪

(b) On the Venn diagram below, shade the region that represents \ .A B

(c) Using your answers to (a) and (b) above or otherwise, shade in the region ( ) \ ( \ ).A B A B∪

(d) If A represents the students in a class who like apples and B represents the students in the same class who like bananas, write down what the set \A B represents.

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A B

A B

A B



In the game of Scrabble, players score points by forming words from individual lettered tiles and placing them on a board. The points for each letter are written on the tile. In a game, Maura selects these seven tiles.

She then arranges them to form the word below.

J8 U1 K5 E1 B3 O1 X8 (a) Find the total number of points that Maura would score for the above word.

Certain squares on the board can be used to gain extra points for letters or words. The scores for the letters are calculated first. Part of one line of the board is shown below. Maura places her word on this line with one letter in each adjacent box.

(b) Place Maura’s word on the board below in a way that gives the maximum possible score. Double

letter score

Double word score

Double letter score

(c) Maura also gets a bonus of 50 points for using all her letters. Calculate the total number of

points that Maura scores for this word.

K5 X8

U1 O1

B3

E1 J8



(a) Write as a decimal.

(b) Show the approximate height of water in the glass if the glass is

full.

(c) Represent the numbers and 0·4 on the number line below.

(d) How could the number line in (c) above help you decide which is the bigger of the two numbers?

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0 1



(a) In the diagram below what fraction of row A is shaded?

(b) In the same diagram what fraction of column R is shaded?

(c) Using the diagram, or otherwise, calculate the result when your fractions in part (a) and (b) are multiplied.

(d) Tim claims that the two fractions shown by the shading of the strips A and B below are the same. Is Tim correct? Give a reason for your answer.

R S T U

A

B

C

D

E

F

G

× =

A

B

Answer:

Reason:



Dermot has €5000 and would like to invest it for two years. A special savings account is offering an annual interest rate of 4% if the money remains in the account for the two years. (a) Calculate the interest he would earn for year one.

(b) Tax of 27% will be deducted each year from the interest earned. Calculate the tax (27%) on

the interest from part (a).

(c) Find the amount of the investment at the start of year two.

(d) Find the total amount of Dermot’s investment at the end of year 2, after the tax is deducted

from the interest. Give your answer correct to the nearest euro.

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€5000 + Interest – Tax



(a) A rectangular solid is 24·9 cm long, 20·3 cm wide and 19·6 cm

high. Select which of the values A, B, C, or D is the best estimate for the volume of the solid.

A B C D

1 000 cm3 100 cm3 10 000 cm3 65 cm3

(b) Using a calculator, or otherwise, calculate the exact volume of the solid in cm3.


(a) Write 2 2 2 2 2 2× × × × × in the form 2x where .x ∈ (b) If 3 8 ,pa a a× = write down the value of p. p = _________

(c) Simplify 5 6

4 3

2 2.

2 2

××

Estimation:

Best estimate

2

5 6

4 3

2 2

2 2

××

=



Olive cycled to the shop to get some milk for her tea. She cycled along a particular route and returned by the same route. The graph below shows the different stages of her journey.

(a) How long did Olive spend cycling? _____________ (b) How far from her home is the shop? _____________ ____ (c) Compare the speed of her trip to the shop with her speed on the way home. (d) Write a paragraph to describe her journey.

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Dis

tan

ce f

rom

hom

e al

ong

rou

te (

km

)

Time (minutes) 10 20 30 40 50

1

2

3

4

5



Tina is standing beside a race-track. A red car and a blue car are travelling in the same direction at steady speeds on the track. At a particular time the red car has gone 70 m beyond Tina and its speed is 20 m/s. At the same instant the blue car has gone 20 m beyond Tina and its speed is 30 m/s. (a) Complete the table below to show the distance between the red car and Tina and the Blue car

and Tina during the next 9 seconds.

(b) After how many seconds will both cars be the same distance from Tina? __________

(c) After 8 seconds which car is furthest away from Tina and how far ahead of the other car is it?

Furthest from Tina =

Distance between cars =

(d) On the diagram on the next page draw graphs of the distance between the red car and Tina and the distance between the blue car and Tina over the 9 seconds.

Time Red Car

Distance (m)Blue Car

Distance (m)

0 70 20

1 90 50

2

3

4

5

6

7

8

9


(e) Write down a formula to represent the distance between the red car and Tina for any given time. State clearly the meaning of any letters used in your formula.

(f) Write down a formula to represent the distance between the blue car and Tina for any given time. _____________________________

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Dis

tan

ce (

m)

Time (seconds) 0 1 2 3 4 5 6 7 8 9 10

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

Red car


(g) Use your formulas from (e) and (f) to verify the answer that you gave to part (b) above.


(a) John is three times as old as Mary. Mary’s age in years is represented by x. Select one expression from A, B, C, D and E below which represents John’s age in years.

A B C D E

x+ 3 3x 3x +3 3x + 9 x –3

Expression: ______________

(b) Select one expression from A, B, C, D and E above which represents John’s age in three years time.

Expression: ______________

(c) Select one expression from A, B, C, D and E above which represents Mary’s age in three years time.

Expression: ______________

(d) In three years time, John’s age added to Mary’s age will give a total of 26 years. Write down an equation in x to represent this statement. (e) Solve your equation to find Mary’s age now.

Equation

Mary’s age =



(a) 1000 people attended a concert. Of these, x were adults and y were children. Use this information to write an equation in x and y.

(b) Each adult ticket cost €10 and a child's ticket cost €5. The total amount collected through

ticket sales was €8750. Use this information to write another equation in x and y.

(c) Solve your two equations to find the number of adults and the number of children who attended the concert.

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Equation 2

Equation 1



(a) Find the value of the expression 2 1 3 5

3 2

x x+ −+ when x = 7.

(b) Express 2 1 3 5

3 2

x x+ −+ as a single fraction. Give your answer in its simplest form.

(c) Suggest a method to check that your answer to (b) above is correct. Perform this check.

(d) Solve the equation 2 1 3 5 13

.3 2 2

x x+ −+ =

( ) ( )2 1 3 5

3 2

+ −+ =

2 1 3 5

3 2

x x+ −+ =

Method:

Check:



Some students are asked to write down linear and quadratic expressions that have (x + 2) as a factor.

(a) The expressions 3x + 5, x + 1 and 2x – 10 are examples of linear expressions in x. Write down a linear expression in x, other than x + 2, that has x + 2 as a factor. (b) Anton writes down a quadratic expression of the form x2 – k, where k is a number. For what value of k will Anton’s expression have x +2 as a factor?

(c) To get her quadratic expression, Denise multiplies x + 2 by 2x + 3. Find Denise’s expression.

( 2)(2 3)x x+ +

(d) Fiona’s expression is 23 11 10.x x+ + She uses division to check whether x + 2 is a factor. Explain how division will allow her to check this.

(e) Divide 23 11 10x x+ + by x + 2.

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(f) Write down a quadratic expression, other than those already given or calculated above, that has x + 2 as a factor.



The graphs of two functions f and g are shown on the grid below. The functions are:

(a) Match the graphs to the functions by putting f(x) or g(x) in the boxes beside the corresponding graphs on the grid. (b) For one of the functions above, explain how you decided on your answer. (c) Use the graph to find f(2). (d) Verify your answer to (c) above by finishing the following calculation. (e) Use the graph to find the value of x for which g(x) = 3. x = . (f) Sketch the function h(x) = x+2 on the grid above. (g) Use the graph to find the values of x for which 2 2 3 1.x x x− − = +

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2( ) 2 3

( ) 1

f x x x

g x x

= − −= +

Function: Explanation:

f(2) =

f(2) = (2)2 – 2(2) – 3 =

-3 -2 -1 1 2 3 4 5

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9


You may use this page for extra work.



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Note to readers of this document:

This sample paper is intended to help teachers and candidates prepare for the June 2013 Mathematics examination in the Project Maths initial schools. The content and structure do not necessarily reflect the 2014 or subsequent examinations in the initial schools or in all other schools. The number of questions on the examination paper may vary somewhat from year to year.

Junior Certificate – Ordinary Level

Mathematics (Project Maths – Phase 3) – Paper 1 Sample Paper, 2013 Time: 2 hours

2013. S333 S



Junior Certificate Examination, 2013 Sample Paper


Paper 2

Ordinary Level

Time: 2 hours

300 marks

Examination number

Centre stamp

Running total

For examiner


1 11

2 12

3 13

4 14

5 15

6 16

7

8

9

10 Total

Grade

Junior Certificate 2013 – sample paper Page 2 of 19 Project Maths, Phase 3 Paper 2 – Ordinary Level

Instructions



Questions do not necessarily carry equal marks.


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.







The shape below, on the right, consists of 6 squares. Each side is 2 cm long. It can be folded to form a cube. Find the surface area of the cube.


A food production company has to decide between a closed cylindrical tin A or a rectangular carton B to hold a product they are marketing for the first time. Both containers have the same volume.

(a) Tin A has a radius of 3 cm and a height of 10 cm. Find the volume of tin A.

(b) Carton B has a square base of length 5 cm. Use the volume you got in (a) above to find the height of carton B. Give your answer correct to one decimal place.

(c) Which one of the above containers do you think the company might choose? Give a reason

for your answer.

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Container: Reason:

A

B



Mary is planning to fly to London. The table shows the flights leaving Dublin Airport (DUB) and arriving in London Heathrow Airport (LHR) on a particular day.

(a) Mary needs to arrive in Heathrow by 10:30 am. What is the departure time of the latest flight that she can take from Dublin Airport? _________________ (b) Find the time of her flight in hours and minutes.

(c) Mary would like to arrive in Dublin Airport 75 minutes before that flight leaves. At what time should she arrive at the airport?

(d) Mary checks in one bag. The cost of checking in a bag is €25. The fare summary for her

journey is given in the table below. Find how much the taxes and charges amount to.

Fare Summary (€)

From Dublin to London/Heathrow

Fare 74·99

Taxes and Charges

Baggage 25·00

Total 133·88


(e) The distance from Dublin to London is 464 km. Find the average speed of the airplane during the flight, in km/h.


A survey was conducted among third year students. The answers to survey questions can be classified as

(i) Categorical data where the categories are not ordered (ii) Ordered categorical data (iii) Discrete numerical data (iv) Continuous numerical data

In each row in the table below, write a short question that you could include in a survey and that will give the type of data stated. Question:

Type of data:

Q1.

Categorical data where the categories are not ordered

Q2.

Ordered categorical data

Q3.

Discrete numerical data

Q4.

Continuous numerical data



The following question was asked on the phase 9 CensusAtSchool questionnaire: “Approximately how many hours per week do you spend on social networking sites?” The data below are from two samples of students chosen at random from the UK and Ireland.

Number of hours UK

Number of students Ireland

Number of students 1 2 1 1 3 2 3 4 1 2 5 2 2 6 7 2 7 3 8 9 1 5 10 2 11 3 12 3 13 4 4 14 1 2 15 5 16 5 5 17 2 1 18 4 2 19 5 4 20 3 2 21 2 22 3 23 1 24 25 1 4

(a) How many students are in each sample? UK ______ Ireland ____________

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(b) Display the data in a way that allows you to compare the two samples. (Use a separate display for each sample.)

(c) Based on your answer to part (b), write down one similarity and one difference between the two samples?

(d) Is it true to say that there are differences between Irish and UK people regarding the time they spend on social networking sites? Explain your answer.

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Answer: Reason:



A bag contains red disks, blue disks and white disks. In an experiment, each student in a class of 24 takes out a disk, records the colour and replaces it. This is repeated ten times. The results from the class are recorded in the table below.

Colour Red Blue White Total

Frequency 123 78 39

Relative frequency

% of total (Relative frequency ×100)

(a) In your opinion, why is the number recorded for red greater than for blue or white? (b) Complete the table above. (c) Use the results from the table above to estimate the probability of getting each colour when a disk is taken from the bag.

Colour Red Blue White

Probability

(d) Anne says that she thinks there are ten discs in the bag. Is this a reasonable suggestion? Explain your answer.

(e) Based on the information in the table, how many disks of each colour do you think are in the bag? Give a reason for your answer.

Frequency Total

Answer: Reason:

Answer: Reason:



(a) Estimate the probability for each of the events A, B, C, D and E listed below, and write your answers into the table.

Probability

A name is picked at random from a list of 50 girls and 50 boys. A = A girl’s name is picked.

A fair coin is tossed once. B = A head is the outcome.

One card is drawn at random from a pack of playing cards. C = The card is a diamond.

A day is chosen at random from a list of the days of the week. D = The name of the day contains the letter a.

One number is picked at random from the set {1, 2, 3, 4, 5, 7, 11, 13}. E = The number chosen is a prime number.

(b) Place the letter for each of the events at the most appropriate position on the probability scale below.

(c) Write down another event that you think has a probability similar to that of C in the scale above.

(d) Write down another event that you think has a probability similar to that of D in the scale above.

(e) In a multiple choice quiz, three possible answers are given to a question. James does not know the answer and guesses which one is correct. Put an X on the scale above to show the probability that he has chosen the correct answer.

0 0·5 1



(a) The mean of a list of five numbers is 8. Write down two different lists of numbers for which the above statement is true. (b) The mode of a list of six numbers is 7. Write down two different lists of numbers for which the above statement is true.


(a) Use a protractor to measure ABC∠ and .DEF∠ | ABC∠ | = __________________ | DEF∠ | = __________________ (b) The four angles ∠ M, ∠ N, ∠ O and ∠ P are shown in the diagrams below.

Starting with the smallest, arrange the four angles in order of magnitude.

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List 1

List 2

List 1

List 2

A

B C

E F

D

N M

OP



(a) From the diagram opposite write down three angles which together add up to °180 .

(b) From the diagram opposite write down two angles which together add up to °180 .

(c) What can you conclude from your two statements about the relationship between | ∠ D| and (| ∠ A| +| ∠ B|)

(d) Find α in the diagram.

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°α

112˚

47˚

A

D

B

C

+ + = °180

+ = °180


Question 11 (Suggested maximum time: 15 minutes) The triangle ABC is isosceles.

.36°=∠BAC

(a) Calculate ACB∠ .

(b) On the diagram construct the bisector of ABC∠ . Show all construction lines clearly. (c) Mark in the point D where your bisector meets the line AC. (d) Calculate all the angles in the triangle BDC and write them into the diagram. (e) Can you conclude that the triangle BDC is also isosceles? Give a reason for your answer. (f) Measure AC and BC .

AC = _________cm BC = ___________cm

(g) Calculate the ratio BC

AC correct to three places of decimals.

BC

AC = ___________

C B

A

°36



During a trigonometry lesson a group of students made some predictions about what they expected to find for the values of the trigonometric functions of some angles. They then found the sine, cosine and tangent of °25 and °50 . (a) In the table given, show, correct to three decimal places, the values they found. (b) (i) Maria had said “The value from any of these trigonometric functions will always be less than 1”. Was Maria correct? Give a reason for your answer.

Answer: Reason:

(ii) Sharon had said “If the size of the angle is doubled then the value from any of these

trigonometric functions will also double.” Was Sharon correct? Give a reason for your answer.

Answer: Reason:

(iii) James had said “The value from all of these trigonometric functions will increase if the size of the angle is increased.” Was James correct? Give a reason for your answer. Answer: Reason:

sin °25 = cos °25 = tan °25 =

sin °50 = cos °50 = tan °50 =



Anne wanted to create a question which would use sin 22° in its solution. She drew the diagram and wrote the question in the box below. (a) Anne has not enough information to answer the question. Put in an appropriate measurement on the diagram to complete it for her. (b) Using your measurement, find the height h in the diagram.


The following diagram shows a square.

Draw in all its axes of symmetry.

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°22

Find the height h of the kite above the ground.

h String



(a) Write down the coordinates of the point A and the point B on the diagram.

(b) Use the distance formula to find |AB|. (c) Write down the distance from O to A and the distance from O to B.

(d) Use the theorem of Pythagoras to find the length of the hypotenuse of the triangle OAB.

|OA| = |OB| =

A

B

O -2 -1 1 2 3 4 5

-2

-1

1

2

3

4

5

( , )

( , )



A computer game shows the location of four flowers A(1, 7), B(1, 2), C(6, 2),and D(5, 6) on a grid. The object of the game is to collect all the nectar from the flowers in the shortest time.

(a) A bee found a hidden flower half way between flower B and flower D. Find the coordinates of this hidden flower.

(b) Another flower E can be located by completing the square ABCE. Write down the coordinates of the point E. E = ______________ (c) Bee 1 and Bee 2 are on flower A. Bee 1 flies directly from flower A to B and then on to C.

Bee 2 flies from flower A directly to D and then on to C. Write down which bee has travelled the shortest total distance. Give a reason for your answer.

Answer: ________________________ Reason:

A(1, 7)

B(1, 2) C(6, 2)

D(5, 6)

-1 1 2 3 4 5 6 7 8-1

1

2

3

4

5

6

7

8



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This sample paper is intended to help teachers and candidates prepare for the June 2013 Mathematics examination in the Project Maths initial schools. The content and structure do not necessarily reflect the 2014 or subsequent examinations in the initial schools or in all other schools. The number of questions on the examination paper may vary somewhat from year to year.

Junior Certificate – Ordinary Level

Mathematics (Project Maths – Phase 3) – Paper 2 Sample Paper, 2013 Time: 2 hours

2012. S232S



Junior Certificate Examination 2012 Sample Paper


Paper 1

Ordinary Level

Time: 2 hours

300 marks

Examination number

Centre stamp

Running total

For examiner


1 11

2 12

3 13

4 14

5

6

7

8

9

10 Total

Grade


Instructions



Question 14 carries a total of 50 marks.


The superintendent will give you a copy of the booklet of Formulae and Tables. You must return it at the end of the examination. You are not allowed to bring your own copy into the examination.







(a) On the Venn diagram below, shade the region that represents .A B∪

(b) On the Venn diagram below, shade the region that represents \ .A B

(c) Using your answers to (a) and (b) above or otherwise, shade in the region ( ) \ ( \ ).A B A B∪

(d) If A represents the students in a class who like apples and B represents the students in the same class who like bananas, write down what the set \A B represents.

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A B

A B

A B



In the game of Scrabble, players score points by forming words from individual lettered tiles and placing them on a board. The points for each letter are written on the tile. In a game, Maura selects these seven tiles.

She then arranges them to form the word below.

J8 U1 K5 E1 B3 O1 X8 (a) Find the total number of points that Maura would score for the above word.

Certain squares on the board can be used to gain extra points for letters or words. The scores for the letters are calculated first. Part of one line of the board is shown below. Maura places her word on this line with one letter in each adjacent box.

(b) Place Maura’s word on the board below in a way that gives the maximum possible score. Double

letter score

Double word score

Double letter score

(c) Maura also gets a bonus of 50 points for using all her letters. Calculate the total number of

points that Maura scores for this word.

K5 X8

U1 O1

B3

E1 J8



(a) Write as a decimal.

(b) Show the approximate height of water in the glass if the glass is

full.

(c) Represent the numbers and 0·4 on the number line below.

(d) How could the number line in (c) above help you decide which is the bigger of the two numbers?

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0 1



(a) In the diagram below what fraction of row A is shaded?

(b) In the same diagram what fraction of column R is shaded?

(c) Using the diagram, or otherwise, calculate the result when your fractions in part (a) and (b) are multiplied.

(d) Tim claims that the two fractions shown by the shading of the strips A and B below are the same. Is Tim correct? Give a reason for your answer.

R S T U

A

B

C

D

E

F

G

× =

A

B

Answer:

Reason:



Dermot has €5000 and would like to invest it for two years. A special savings account is offering an annual interest rate of 4% if the money remains in the account for the two years. (a) Calculate the interest he would earn for year one.

(b) Tax of 27% will be deducted each year from the interest earned. Calculate the tax (27%) on

the interest from part (a).

(c) Find the amount of the investment at the start of year two.

(d) Find the total amount of Dermot’s investment at the end of year 2, after the tax is deducted

from the interest. Give your answer correct to the nearest euro.

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€5000 + Interest – Tax



(a) A rectangular solid is 24·9 cm long, 20·3 cm wide and 19·6 cm

high. Select which of the values A, B, C, or D is the best estimate for the volume of the solid.

A B C D

1 000 cm3 100 cm3 10 000 cm3 65 cm3

(b) Using a calculator, or otherwise, calculate the exact volume of the solid in cm3.


(a) Write 2 2 2 2 2 2× × × × × in the form 2x where .x∈ (b) If 3 8 ,pa a a× = write down the value of p. p = _________

(c) Simplify 5 6

4 3

2 2.

2 2

××

Estimation:

Best estimate

2

5 6

4 3

2 2

2 2

××

=



Olive cycled to the shop to get some milk for her tea. She cycled along a particular route and returned by the same route. The graph below shows the different stages of her journey.

(a) How long did Olive stay in the shop? _____________ (b) How far from her home is the shop? _____________ ____ (c) Compare the speed of her trip to the shop with her speed on the way home. (d) Write a paragraph to describe her journey.

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Dis

tan

ce f

rom

hom

e al

ong

rou

te (

km

)

Time (minutes) 10 20 30 40 50

1

2

3

4

5



Tina is standing beside a race-track. A red car and a blue car are travelling in the same direction at steady speeds on the track. At a particular time the red car has gone 70 m beyond Tina and its speed is 20 m/s. At the same instant the blue car has gone 20 m beyond Tina and its speed is 30 m/s. (a) Complete the table below to show the distance between the red car and Tina and the Blue car

and Tina during the next 9 seconds.

(b) After how many seconds will both cars be the same distance from Tina? __________

(c) After 8 seconds which car is furthest away from Tina and how far ahead of the other car is it?

Furthest from Tina =

Distance between cars =

(d) On the diagram on the next page draw graphs of the distance between the red car and Tina and the distance between the blue car and Tina over the 9 seconds.

Time Red Car

Distance (m)Blue Car

Distance (m)

0 70 20

1 90 50

2

3

4

5

6

7

8

9


(e) Write down a formula to represent the distance between the red car and Tina for any given time. State clearly the meaning of any letters used in your formula.

(f) Write down a formula to represent the distance between the blue car and Tina for any given time. _____________________________

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Dis

tan

ce (

m)

Time (seconds) 0 1 2 3 4 5 6 7 8 9 10

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

Red car


(g) Use your formulas from (e) and (f) to verify the answer that you gave to part (b) above.


(a) John is three times as old as Mary. Mary’s age in years is represented by x. Select one expression from A, B, C, D and E below which represents John’s age in years.

A B C D E

x+ 3 3x 3x +3 3x + 9 x –3

Expression: ______________

(b) Select one expression from A, B, C, D and E above which represents John’s age in three years time.

Expression: ______________

(c) Select one expression from A, B, C, D and E above which represents Marys’ age in three years time.

Expression: ______________

(d) In three years time, John’s age added to Marys’ age will give a total of 26 years. Write down an equation in x to represent this statement.. (e) Solve your equation to find Mary’s age.

Equation

Mary’s age =



(a) 1000 people attended a concert. Of these, x were adults and y were children. Use this information to write an equation in x and y.

(b) Each adult ticket cost €10 and a child's ticket cost €5. The total amount collected through

ticket sales was €8750. Use this information to write another equation in x and y.

(c) Solve your two equations to find the number of adults and the number of children who attended the concert.

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Equation 2

Equation 1



(a) Find the value of the expression 2 1 3 5

3 2

x x+ −+ when x = 7.

(b) Express 2 1 3 5

3 2

x x+ −+ as a single fraction. Give your answer in its simplest form.

(c) Suggest a method to check that your answer to (b) above is correct. Perform this check.

(d) Solve the equation 2 1 3 5 13

.3 2 2

x x+ −+ =

( ) ( )2 1 3 5

3 2

+ −+ =

2 1 3 5

3 2

x x+ −+ =

Method:

Check:



Some students are asked to write down linear and quadratic expressions that have (x + 2) as a factor.

(a) The expressions 3x + 5, x + 1 and 2x – 10 are examples of linear expressions in x. Write down a linear expression in x other than x + 2, that has x + 2 as a factor. (b) Anton writes down a quadratic expression of the form x2 – k, where k is a number. For what value of k will Anton’s expression have x +2 as a factor?

(c) To get her quadratic expression, Denise multiplies x + 2 by 2x + 3. Find Denise’s expression.

( 2)(2 3)x x+ +

(d) Fiona’s expression is 23 11 10.x x+ + She uses division to check whether x + 2 is a factor. Explain how division will allow her to check this.

(e) Divide 23 11 10x x+ + by x + 2.

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(f) Write down one quadratic expression, other than those already given above, that has x + 2 as a factor.

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(a) f (x) = 2x – 7. Find:

(b) Draw the graph of the function g : x 2x² – 4x + 1 in the domain −1 ≤ x ≤ 3, where x ∈ ℝ.

(i) f(4)

(ii) f(–3)

-2 -1 1 2 3 4

-2

-1

1

2

3

4

5

6

7

8

9


(c) (i) Given that y = x – 1, complete the table below.

(ii) On the grid below the graph of the line y = 3 – x is drawn. Using your answers from (i), draw the graph of y = x – 1 on the same grid.

(iii) Use the graphs drawn in (c) (ii) to write down the co-ordinates of the point of intersection of the two lines y = 3 – x and y = x – 1.

x 1 2 3 4 y

Answer to be written here.

-1 1 2 3 4 5

-1

1

2

3

4

y = 3 – x



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This sample paper is intended to help teachers and candidates prepare for the June 2012 examination in the Project Maths initial schools. The content and structure do not necessarily reflect the 2013 or subsequent examinations in the initial schools or in all other schools. In the 2012 examination, one question will be the same as a question on the examination for candidates who are not in the initial schools. On this sample paper, the corresponding question from the 2011 examination has been inserted, as question 14, to illustrate.

Junior Certificate 2012 – Ordinary Level

Mathematics (Project Maths, Phase 2) – Paper 1 Sample Paper Time: 2 hours

2012. S233S



Junior Certificate Examination Sample Paper


Paper 2

Ordinary Level

Time: 2 hours

300 marks

Examination number

Centre stamp

Running total

For examiner


1 11

2 12

3 13

4 14

5 15

6 16

7

8

9

10 Total

Grade


Instructions


Questions do not necessarily carry equal marks.









Some of the nets below can be folded to form a cube. Draw a circle around any net that can form a cube in this way.


A food production company has to decide between two closed cylindrical tins A or B to hold a fruit they are marketing for the first time. Both tins have the same volume.

(a) Tin A has a radius of 4 cm and a height of 8 cm. Find the volume of tin A.

(b) The radius of tin B is 3·5 cm. Use the volume you got in (a) above to find the height of tin B. Give your answer correct to one place of decimals.

(c) Which one of the above tins do you think the company might choose? Give a reason for your

answer.

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Tin: Reason:

A B

A

B



Mary is planning to fly to London. The table shows the flights leaving Dublin Airport (DUB) and arriving in London Heathrow Airport (LHR) on a particular day.

(a) Mary needs to arrive in Heathrow by 10:30 am. What is the departure time of the latest flight

that she can take from Dublin Airport? _________________

(b) Find the time of her flight in hours and minutes.

(c) Mary would like to arrive in Dublin Airport 75 minutes before the flight leaves. At what time should she arrive at the airport?

(d) Mary checks in one bag. The cost of checking in a bag is €25. The fare summary for her

journey is given in the table below. Find how much the taxes and charges amount to.

Fare Summary (€)

From Dublin to London/Heathrow

Fare 74·99

Taxes and Charges

Baggage 25·00

Total 133·88

Departing Arriving


(e) The distance from Dublin to London is 464 km. Find the average speed of the airplane during the flight, in km/h.


A survey was conducted among third year students. The answers to survey questions can be classified as


In each row in the table below, write a short question that you could include in a survey and that will give the type of data stated. Question:

Type of data:

Q1.


Q2.


Q3.


Q4.


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The following question was asked on the phase 9 CensusAtSchool questionnaire: “Approximately how many hours per week do you spend on social networking sites?” The data below are from two samples of students chosen at random from the UK and Ireland.

Number of hours UK

Number of students Ireland

Number of students 1 2 1 1 3 2 3 4 1 2 5 2 2 6 7 2 7 3 8 9 1 5 10 2 11 3 12 3 13 4 4 14 1 2 15 5 16 5 5 17 2 1 18 4 2 19 5 4 20 3 2 21 2 22 3 23 1 24 25 1 4

(a) How many students are in each sample? UK ______ Ireland ____________



(c) Based on your answer to part (b), what similarities and differences are there between the two samples?

(d) Is it safe to say that there are differences between Irish and UK people regarding the time they spend on social networking sites? Explain your answer.

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A bag contains red disks, blue disks and white disks. In an experiment, each student in a class of 24 takes out a disk, records the colour and replaces it. This is repeated ten times. The results from the class are recorded in the table below.


Frequency 123 78 39

Relative frequency

% of total (Relative frequency ×100)

(a) In your opinion, why is the number for red greater than for blue or white? (b) Complete the table above. (c) Use the results from the table above to estimate the probability of getting each colour when a disk is taken from the bag.

Colour Red Blue White

Probability

(d) Anne says that she thinks there are ten discs in the bag. Is this a reasonable suggestion? Explain your answer.

(e) Based on the information in the table, how many disks of each colour do you think are in the bag?

Frequency Total



(a) For each of the events A, B, C, D and E below, estimate its probability and place the letter at the most appropriate position on the probability scale below.

Probability


A fair coin is tossed twice. B = A head is the outcome on each toss.




(b) Write down another event that you think has a probability similar to that of C in the scale above.

(c) Write down another event that you think has a probability similar to that of D in the scale above.

(d) In a multiple choice quiz, three possible answers are given to a question. James does not know the answer and guesses which one is correct. Put an X on the scale above to show the probability that he has chosen the correct answer.

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0 0.5 1



(a) The mean of a list of five numbers is 8. Write down two different lists of numbers for which the above statement is true. (b) The mode of a list of six numbers is 7. Write down two different lists of numbers for which the above statement is true.


(a) Use a protractor to measure the angles ABC∠ and .DEF∠ | ABC∠ | = __________________ | DEF∠ | = __________________ (b) The four angles ∠ M, ∠ N, ∠ O and ∠ P are shown in the diagrams below.

Starting with the smallest, put the four angles in order.

List 1

List 2

List 1

List 2

A

B C

E F

D

N M

OP



(a) From the diagram opposite write down three angles which together add up to °180 .

(b) From the diagram opposite write down two angles which together add up to °180 .

(c) What can you conclude from your two statements about the relationship between | ∠ D| and (| ∠ A| +| ∠ B|)

(d) Find α in the diagram.

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°α

112˚

47˚

A

D

B

C

+ + = °180

+ = °180


Question 11 (Suggested maximum time: 15 minutes) The triangle ABC is isosceles.

.36°=∠BAC

(a) Calculate ACB∠ .

(b) On the diagram construct the bisector of ABC∠ . Show all construction lines clearly. (c) Mark in the point D where your bisector meets the line AC. (d) Calculate all the angles in the triangle BDC and write them into the diagram. (e) Can you conclude that the triangle BDC is also isosceles? Give a reason for your answer. (e) Measure AC and BC .

AC = _________cm BC = ___________cm

(f) Calculate the ratio BC

AC correct to three places of decimals.

BC

AC = ___________

C B

A

°36



During a trigonometry lesson a group of students made some predictions about what they expected to find for the values of the trigonometric functions of some angles. They then found the Sin, Cos and Tan of °25 and °50 . (a) In the table given, show, correct to three decimal places, the values they found. (b) (i) Maria had said “The value from any of these trigonometric functions will always be less than 1”. Was Maria correct? Give a reason for your answer.

Answer: Reason:

(ii) Sharon had said “If the size of the angle is doubled then the value from any of these


Answer: Reason:

(iii) James had said “The value from all of these trigonometric functions will increase if the size of the angle is increased.” Was James correct? Give a reason for your answer. Answer: Reason:

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Sin °25 = Cos °25 = Tan °25 =

Sin °50 = Cos °50 = Tan °50 =



Anne wanted to create a question which would use Sin °22 in its solution. She drew the diagram and wrote the question in the box below. (a) Anne has not enough information to answer the question. Put in an appropriate measurement on the diagram to complete it for her. (b) Using your measurement, find the height h in the diagram.


The following diagram shows a square.

Draw in all its axes of symmetry.

°22

Find the height h of the kite above the ground.

h String



(a) Write down the coordinates of the point A and the point B on the diagram.

(b) Use the distance formula to find |AB| (c) Write down the distance from O to A and the distance from O to B.

(d) Use the theorem of Pythagoras to find the length of the hypotenuse of the triangle OAB.

|OA| = |OB| =

A

B

O -2 -1 1 2 3 4 5

-2

-1

1

2

3

4

5

( , )

( , )



A computer game shows the location of four flowers A(1, 7), B(1, 2), C(6, 2),and D(5, 6) on a grid. The object of the game is to collect all the nectar from the flowers in the shortest time.

(a) A bee found a hidden flower half way between flower B and flower D. Find the coordinates of this hidden flower.

(b) Another flower E can be located by completing the square ABCE. Write down the coordinates of the point E. E = ______________ (c) Bee 1 and Bee 2 are on flower A. Bee 1 flies directly from flower A to B and then on to C.

Bee 2 flies from flower A directly to D and then on to C. Write down which bee has travelled the shortest distance giving a reason for your answer.

Answer: ________________________ Reason:

A(1, 7)

B(1, 2) C(6, 2)

D(5, 6)

-1 1 2 3 4 5 6 7 8-1

1

2

3

4

5

6

7

8



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This sample paper is intended to help teachers and candidates prepare for the June 2012 examination in the Project Maths initial schools. The content and structure do not necessarily reflect the 2013 or subsequent examinations in the initial schools or in all other schools. The number of questions on the examination paper may vary somewhat from year to year.


Mathematics (Project Maths – Phase 2) – Paper 2 Sample Paper Time: 2 hours

2011. S133S

Coimisiún na Scrúduithe Stáit State Examinations Commission

Junior Certificate Examination Sample Paper


Paper 2

Ordinary Level

Time: 2 hours

300 marks

Examination number

Centre stamp

Running total

For examiner


1 11

2 12

3 13

4 14

5 15

6 16

7

8

9

10 Total

Grade

Junior Certificate 2011 – sample paper Page 2 of 19 Project Maths, Phase1 Paper 2 – Ordinary Level

Instructions

There are sixteen 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.







Question 1 (suggested maximum time: 2 minutes) A parallelogram has dimensions as shown in the diagram. Find, in cm2, the area of the parallelogram. Question 2 (suggested maximum time: 5 minutes) (a) Dara left Lucan by car at 09:25 and arrived in Sligo at 11:55. How long did it take Dara to travel from Lucan to Sligo? Give your answer in hours and minutes. (b) The distance from Lucan to Sligo is 195 km. Calculate Dara's average speed, in km/h.

(c) On the return journey from Sligo to Lucan, Dara's average speed was 60 km/h. How long, in hours and minutes, did the return journey take?

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5 cm

14 cm


Question 3 (suggested maximum time: 10 minutes) A park is in the shape of a rectangle with a semicircular end. The rectangle is 150 m long and 28 m wide. The diameter of the semicircular end is also 28 m. There is a path around the park which is used for walking and jogging. (a) Taking π as 3×142, calculate the length of the semicircular end. Give your answer to the nearest metre. (b) Calculate the total length of the path around the park. (c) Barbara wishes to jog 2×5 km. How many laps of the path must she complete to ensure that she jogs this distance?

Path

150 m

28 m


Question 4 (suggested maximum time: 5 minutes) The answers to survey questions can be classified as


In each row in the table below, write a short question that you could include in a survey and that will give the type of data stated.

Question: Type of data:

Q1.


Q2.


Q3.


Q4.


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Question 5 (suggested maximum time: 10 minutes) The following question was asked on the phase 9 CensusAtSchool questionnaire: “Approximately how many hours per week do you spend on social networking sites?” The data below are from two samples of students chosen at random from the UK and Ireland.

Number of hours UK Number of students

Ireland Number of students

1 2 1 1 3 2 3 4 1 2 5 2 2 6 7 2 7 3 8 9 1 5 10 2 11 3 12 3 13 4 4 14 1 2 15 5 16 5 5 17 2 1 18 4 2 19 5 4 20 3 2 21 2 22 3 23 1 24 25 1 4

(a) How many students are in each sample? UK ________ Ireland ________



(c) Based on your answer to part (b), what similarities and differences are there between the two samples? (d) Is it safe to say that there are differences between Irish and UK people regarding the time they spend on social networking sites? Explain your answer.

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Question 6 (suggested maximum time: 10 minutes) A bag contains red disks, blue disks and white disks. In an experiment, each student in a class of 24 takes out a disk, records the colour and replaces it. This is repeated ten times. The results from the class are recorded in the table below.


Frequency 123 78 39

Relative frequency:

frequencytotal

% of total (Relative frequency × 100)

(a) In your opinion, why is the number for red greater than for blue or white? (b) Complete the table above. (c) Use the results from the table above to estimate the probability of getting each colour when a

disk is taken from the bag.

Colour Red Blue White Probability

(d) Anne says that she thinks there are ten disks in the bag. Is this a reasonable suggestion? Explain your answer. (e) Based on the information in the table, how many disks of each colour do you think are in the bag?


Question 7 (suggested maximum time: 10 minutes) (a) For each of the events A, B, C, D and E below, estimate its probability and place the letter at the most appropriate position on the probability scale below.

Probability


A fair coin is tossed twice. B = A head is the outcome on each toss.




(b) Write down another event that you think has a probability similar to that of C in the scale above. (c) Write down another event that you think has a probability similar to that of D in the scale above.

(d) In a multiple choice quiz, three possible answers are given to a question. James does not know the answer and guesses which one is correct. Put an X on the scale above to show the probability that he has chosen the correct answer.

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0 0·5 1


Question 8 (suggested maximum time: 5 minutes) (a) The mean of a list of five numbers is 8. Write down two different lists of numbers for which the above statement is true. (b) The mode of a list of six numbers is 7. Write down two different lists of numbers for which the above statement is true. Question 9 (suggested maximum time: 5 minutes) (a) Use a protractor to measure the angles ABC∠ and .DEF∠ | ABC∠ | = __________________ | DEF∠ | = __________________ (b) The four angles ∠M, ∠N, ∠O and ∠ P are shown in the diagrams below. Starting with the smallest, put the four angles in order.

List 1:

List 2:

List 1:

List 2:

A

B C

E F

D

M N O

P


Question 10 (suggested maximum time: 10 minutes) (a) From the diagram opposite write

down three angles which together add up to °180 .

(b) From the diagram opposite write

down two angles which together add up to °180 .

(c) What can you conclude from your two statements about the relationship between |∠D| and (|∠A| +| ∠B|) (d) Find α in the diagram.

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+ + = °180

+ = °180 A D

B

C

°α

112˚

47˚


Question 11 (suggested maximum time: 15 minutes) The triangle ABC is isosceles.

.36°=∠BAC

(a) Calculate ACB∠ . (b) On the diagram construct the bisector of

ABC∠ . Show all construction lines clearly.

(c) Mark in the point D where your bisector

meets the line AC. (d) Calculate all the angles in the triangle BDC and write them into the diagram. (e) Can you conclude that the triangle BDC is also isosceles? Give a reason for your answer. (e) Measure AC and BC . AC = _________cm BC = ___________cm

(f) Calculate the ratio BCAC

correct to three places of decimals. BCAC

= ___________

°36

C B

A


Question 12 (suggested maximum time: 5 minutes) During a trigonometry lesson a group of students made some predictions about what they expected to find for the values of the trigonometric functions of some angles. They then found the sine, cosine and tangent of °25 and °50 . (a) In the table given, show, correct to three decimal places, the values they found.

sin 25°= cos 25°= tan 25°=

sin 50°= cos50°= tan 50°=

(b) (i) Maria had said “The value from any of these trigonometric functions will always be less

than 1”. Was Maria correct? Give a reason for your answer. Answer: Reason: (ii) Sharon had said “If the size of the angle is doubled then value from any of these


Answer: Reason: (iii) James had said “The value from all of these trigonometric functions will increase if the size of the angle is increased.” Was James correct? Give a reason for your answer. Answer: Reason:

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Question 13 (suggested maximum time: 5 minutes) Anne wanted to create a question which would use sin °22 in its solution. She drew the diagram and wrote the question in the box below. (a) Anne has not given enough information to answer the question. Put in an appropriate

measurement on the diagram to complete it for her. (b) Using your measurement, find the height h in the diagram. Question 14 (suggested maximum time: 2 minutes) The following diagram shows a square. Draw in all its axes of symmetry.

°22Find the height h of the kite above the ground.

h

String


Question 15 (suggested maximum time: 5 minutes) (a) Write down the coordinates of the point A and

the point B on the diagram.

(b) Use the distance formula to find |AB| (c) Write down the distance from O to A and the distance from O to B. (d) Use the theorem of Pythagoras to find the length of the hypotenuse of the triangle OAB.

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A

B

O -2 -1 1 2 3 4 5

-2

-1

1

2

3

4

5

( , )

( , )

|OA| = |OB| =


Question 16 (suggested maximum time: 5 minutes) A computer game shows the location of some flowers on a grid. The object of the game is to collect all the nectar from the flowers in the shortest time.

(a) A bee found a hidden flower half way between flower B and flower D. Find the coordinates of this hidden flower. (b) Another flower E can be located by completing the square ABCE. Write down the coordinates of the point E. E = ______________ (c) Bee 1 and Bee 2 are on flower A. Bee1 flies directly from flower A to B and then on to C.

Bee 2 flies from flower A directly to D and then on to C. Write down which bee has travelled the shortest distance, giving a reason for your answer.

Answer: ________________________ Reason:

A

B C

D

-1 -1 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

8


You may use this page for extra work

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Note to readers of this document: This sample paper is intended to help teachers and candidates prepare for the June 2011 examination in the Project Maths initial schools. The content and structure do not necessarily reflect the 2012 or subsequent examinations in the initial schools or in all other schools. In the 2011 examination, the material in some questions will be compiled from questions 1 and 2 on the examination for candidates who are not in the initial schools. On this sample paper, portions of questions from the 2010 examination have been inserted to illustrate. The number of questions on the examination paper may vary somewhat from year to year.


Mathematics (Project Maths – Phase 1) – Paper 2 Sample Paper Time: 2 hours

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