Mathematics (Project Maths – Phase 3) - Folens · Answers should be given in simplest form, where...

2012. M329 S

Coimisiún na Scrúduithe Stáit

State Examinations Commission

Leaving Certificate Examination, 2012 Sample Paper

Mathematics (Project Maths – Phase 3)

Paper 1

Higher Level

Time: 2 hours, 30 minutes

300 marks

Examination number For examiner

Question Mark

1

2

3

4

5

6

7

8

9

Total

Centre stamp

Running total Grade

PriscillaOC

Text Box

Leaving Certificate 2012 – Sample Paper of 19 Project Maths, Phase 3 Paper1 – Higher Level

Instructions

There are two sections in this examination paper:

Section A Concepts and Skills 150 marks 6 questions

Section B Contexts and Applications 150 marks 3 questions

Answer all nine questions.

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 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.

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:


Section A Concepts and Skills 150 marks Answer all six questions from this section. Question 1 (25 marks)

(a) 1 3w i= − + is a complex number, where 2 1i = − . (i) Write w in polar form.

(ii) Use De Moivre’s theorem to solve the equation 2 1 3z i= − + , giving your answer(s) in rectangular form.

(b) Four complex numbers 1z , 2z , 3z and 4z are shown on the

Argand diagram. They satisfy the following conditions: 12 izz =

3 1z kz= , where k ∈

324 zzz += .

The same scale is used on both axes. (i) Identify which number is which, by

labelling the points on the diagram. (ii) Write down the approximate value of k. Answer:

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Im(z)

Re(z)

w Im(z)

Re(z)


Question 2 (25 marks)

(a) (i) Prove by induction that, for any n, the sum of the first n natural numbers is ( 1)

2

n n +.

(ii) Find the sum of all the natural numbers from 51 to 100, inclusive.

(b) Given that logcp x= , express log log ( )c cx cx+ in terms of p.


Question 3 (25 marks) A cubic function f is defined for x∈ as

3 2: (1 )f x x k x k+ − + , where k is a constant. (a) Show that k− is a root of f. (b) Find, in terms of k, the other two roots of f. (c) Find the set of values of k for which f has exactly one real root.

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Question 4 (25 marks) (a) Solve the simultaneous equations,

2 8 3 1

2 3 2 2

2 5.

x y z

x y z

x y z

+ − = −− + =+ + =

(b) The graphs of the functions : 3f x x − and : 2g x are shown in the diagram.

(i) Find the co-ordinates of the points A, B, C and D. (ii) Hence, or otherwise, solve the inequality 3 2x − < .

( )y f x=

( )y g x=

x

y

A B

C

D

A = ( , ) B = ( , )

C = ( , ) D = ( , )



A is the closed interval [ ]0,5 . That is, { }0 5,A x x x= ≤ ≤ ∈ .

The function f is defined on A by: 3 2: : 5 3 5f A x x x x→ − + + .

(a) Find the maximum and minimum values of f. (b) State whether f is injective. Give a reason for your answer.

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Question 6 (25 marks) (a) (i) Write down three distinct anti-derivatives of the function

3 2: 3 3g x x x− + , x∈ . 1. 2. 3. (ii) Explain what is meant by the indefinite integral of a function f. (iii) Write down the indefinite integral of g, the function in part (i). Answer: (b) (i) Let ( ) lnh x x x= , for , 0x x∈ > . Find ( )h x′ .

(ii) Hence, find ln x dx .


Section B Contexts and Applications 150 marks Answer all three questions from this section. Question 7 (50 marks) A company has to design a rectangular box for a new range of jellybeans. The box is to be assembled from a single piece of cardboard, cut from a rectangular sheet measuring 31 cm by 22 cm. The box is to have a capacity (volume) of 500 cm3. The net for the box is shown below. The company is going to use the full length and width of the rectangular piece of cardboard. The shaded areas are flaps of width 1 cm which are needed for assembly. The height of the box is h cm, as shown on the diagram. (a) Write the dimensions of the box, in centimetres, in terms of h.

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Bottom Top Side Side

Side

Side

22 cm

31 cm

h

h

height = h cm length = cm width = cm


(b) Write an expression for the capacity of the box in cubic centimetres, in terms of h. (c) Show that the value of h that gives a box with a square bottom will give the correct capacity. (d) Find, correct to one decimal place, the other value of h that gives a box of the correct

capacity.


(e) The client is planning a special “10% extra free” promotion and needs to increase the capacity of the box by 10%. The company is checking whether they can make this new box from a piece of cardboard the same size as the original one (31 cm × 22 cm). They draw the graph below to represent the box’s capacity as a function of h. Use the graph to explain why it is not possible to make the larger box from such a piece of cardboard.

Explanation:

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2010 15

-200

200

400

600

800

5

h

Cap

acit

y in

cm

3


Question 8 (50 marks) Pádraig is 25 years old and is planning for his pension. He intends to retire in forty years’ time, when he is 65. First, he calculates how much he wants to have in his pension fund when he retires. Then, he calculates how much he needs to invest in order to achieve this. He assumes that, in the long run, money can be invested at an inflation-adjusted annual rate of 3%. Your answers throughout this question should therefore be based on a 3% annual growth rate. (a) Write down the present value of a future payment of €20,000 in one years’ time. (b) Write down, in terms of t, the present value of a future payment of €20,000 in t years’ time. (c) Pádraig wants to have a fund that could, from the date of his retirement, give him a payment

of €20,000 at the start of each year for 25 years. Show how to use the sum of a geometric series to calculate the value on the date of retirement of the fund required.


(d) Pádraig plans to invest a fixed amount of money every month in order to generate the fund calculated in part (c). His retirement is 40 × 12 = 480 months away.

(i) Find, correct to four significant figures, the rate of interest per month that would, if paid

and compounded monthly, be equivalent to an effective annual rate of 3%. (ii) Write down, in terms of n and P, the value on the retirement date of a payment of €P

made n months before the retirement date. (iii) If Pádraig makes 480 equal monthly payments of €P from now until his retirement,

what value of P will give the fund he requires? (e) If Pádraig waits for ten years before starting his pension investments, how much will he then

have to pay each month in order to generate the same pension fund?

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(a) Let 2( ) 0·5 5 0·98f x x x= − + − , where x∈ . (i) Find the value of (0·2)f (ii) Show that f has a local maximum point at (5, 11·52). (b) A sprinter’s velocity over the course of a particular

100 metre race is approximated by the following model, where v is the velocity in metres per second, and t is the time in seconds from the starting signal:

2

0, for 0 0·2

( ) 0·5 5 0·98, for 0·2 5

11·52, for 5

t

v t t t t

t

≤ <= − + − ≤ < ≥

Note that the function in part (a) is relevant to v(t) above. (i) Sketch the graph of v as a function of t for the first 7 seconds of the race.

1 2 3 4 5 6 7

5

10

time (seconds)

velo

city

(m

/s)

Photo: William Warby. Wikimedia Commons. CC BY 2.0


(ii) Find the distance travelled by the sprinter in the first 5 seconds of the race. (iii) Find the sprinter’s finishing time for the race. Give your answer correct to two decimal

places.

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(c) A spherical snowball is melting at a rate proportional to its surface area. That is, the rate at which its volume is decreasing at any instant is proportional to its surface area at that instant.

(i) Prove that the radius of the snowball is decreasing at a constant rate. (ii) If the snowball loses half of its volume in an hour, how long more will it take for it to

melt completely?

Give your answer correct to the nearest minute.


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 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.

Leaving Certificate 2012 – Higher Level

Mathematics (Project Maths – Phase 3) – Paper 1 Sample Paper Time: 2 hours 30 minutes

2011. M229 S

Coimisiún na Scrúduithe Stáit State Examinations Commission

Leaving Certificate Examination 2011 Sample Paper


Paper 1

Higher Level


300 marks


Question Mark

1

2

3

4

5

6

7

8

9

Total

Centre stamp

Running total Grade

Leaving Certificate 2011 – Sample Paper Page 2 of 19 Project Maths, Phase 2 Paper1 – Higher Level

Instructions

There are three sections in this examination paper:



Section C Functions and Calculus (old syllabus) 100 marks 3 questions

Answer questions as follows:

In Section A, answer all four questions

In Section B, answer both Question 5 and Question 6

In Section C, answer any two of the three questions.







Section A Concepts and Skills 100 marks Answer all four questions from this section. Question 1 (25 marks)

(a) 1 3w i= − + is a complex number, where 2 1i = − . (i) Write w in polar form. (ii) Use De Moivre’s theorem to solve the equation 2 1 3z i= − + ,

giving your answer(s) in rectangular form. (b) Four complex numbers 1z , 2z , 3z and 4z are shown on the

Argand diagram. They satisfy the following conditions: 12 izz = 3 1z kz= , where k∈ 324 zzz += . The same scale is used on both axes. (i) Identify which number is which, by

labelling the points on the diagram. (ii) Write down the approximate value of k. Answer:

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Im(z)

Re(z)

w Im(z)

Re(z)



(a) (i) Prove by induction that, for any n, the sum of the first n natural numbers is ( 1)2

n n + .

(ii) Find the sum of all the natural numbers from 51 to 100, inclusive. (b) Given that logcp x= , express log log ( )c cx cx+ in terms of p.



A cubic function f is defined for x∈ as 3 2: (1 )f x x k x k+ − + , where k is a constant.

(a) Show that k− is a root of f. (b) Find, in terms of k, the other two roots of f. (c) Find the set of values of k for which f has exactly one real root.

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Question 4 (25 marks) (a) Solve the simultaneous equations,

2 8 3 12 3 2 22 5.

x y zx y zx y z

+ − = −− + =+ + =

(b) The graphs of the functions : 3f x x − and : 2g x are shown in the diagram. (i) Find the co-ordinates of the points A, B, C and D. (ii) Hence, or otherwise, solve the inequality 3 2x − < .

( )y f x=

( )y g x=

x

y

A B

C

D

A = ( , ) B = ( , )

C = ( , ) D = ( , )


Section B Contexts and Applications 100 marks Answer both Question 5 and Question 6. Question 5 (50 marks) A company has to design a rectangular box for a new range of jellybeans. The box is to be assembled from a single piece of cardboard, cut from a rectangular sheet measuring 31 cm by 22 cm. The box is to have a capacity (volume) of 500 cm3. The net for the box is shown below. The company is going to use the full length and width of the rectangular piece of cardboard. The shaded areas are flaps of width 1 cm which are needed for assembly. The height of the box is h cm, as shown on the diagram. (a) Write the dimensions of the box, in centimetres, in terms of h.

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Bottom Top Side Side

Side

Side

22 cm

31 cm

h

h

height = h cm length = cm width = cm


(b) Write an expression for the capacity of the box in cubic centimetres, in terms of h. (c) Show that the value of h that gives a box with a square bottom will give the correct capacity. (d) Find, correct to one decimal place, the other value of h that gives a box of the correct

capacity.


(e) The client is planning a special “10% extra free” promotion and needs to increase the capacity of the box by 10%. The company is checking whether they can make this new box from a piece of cardboard the same size as the original one (31 cm × 22 cm). A graph of the box’s capacity as a function of h is shown below. Use the graph to explain why it is not possible to make the larger box from such a piece of cardboard.

Explanation:

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20 10 15

-200

200

400

600

800

5

h

capacity


Question 6 (50 marks) Pádraig is 25 years old and is planning for his pension. He intends to retire in forty years’ time, when he is 65. First, he calculates how much he wants to have in his pension fund when he retires. Then, he calculates how much he needs to invest in order to achieve this. He assumes that, in the long run, money can be invested at an inflation-adjusted annual rate of 3%. Your answers throughout this question should therefore be based on a 3% annual growth rate. (a) Write down the present value of a future payment of €20,000 in one years’ time. (b) Write down, in terms of t, the present value of a future payment of €20,000 in t years’ time. (c) Pádraig wants to have a fund that could, from the date of his retirement, give him a payment

of €20,000 at the start of each year for 25 years. Show how to use the sum of a geometric series to calculate the value on the date of retirement of the fund required.


(d) Pádraig plans to invest a fixed amount of money every month in order to generate the fund calculated in part (c). His retirement is 40 × 12 = 480 months away.

(i) Find, correct to four significant figures, the rate of interest per month that would, if paid

and compounded monthly, be equivalent to an effective annual rate of 3%. (ii) Write down, in terms of n and P, the value on the retirement date of a payment of €P

made n months before the retirement date. (iii) If Pádraig makes 480 equal monthly payments of €P from now until his retirement,

what value of P will give the fund he requires? (e) If Pádraig waits for ten years before starting his pension investments, how much will he then

have to pay each month in order to generate the same pension fund?

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Section C Functions and Calculus (old syllabus) 100 marks Answer any two of the three questions from this section. Question 7 (50 marks)

(a) The equation 3 2 4 0x x+ − = has only one real root. Taking 13

2x = as the first approximation

to the root, use the Newton-Raphson method to find 2x , the second approximation. (b) Parametric equations of a curve are:

212

+−

=ttx ,

2+=

tty , where \{ 2}t∈ − .

(i) Find .dxdy

(ii) What does your answer to part (i) tell you about the shape of the graph?


(c) A curve is defined by the equation .122432 =++ yxyx

(i) Find dxdy in terms of x and y.

(ii) Show that the tangent to the curve at the point ( )6 ,0 is also the tangent to it at

the point ( ).0 ,3

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(a) Differentiate 2x with respect to x from first principles.

(b) Let cos sin cos sin

x xyx x+

=−

.

(i) Find dydx

.

(ii) Show that 2 1 dy ydx

= + .


(c) The function ( ) ( ) ( )1 log 1ef x x x= + + is defined for x > − 1.

(i) Show that the curve ( )y f x= has a turning point at 1 1, ee e− −

.

(ii) Determine whether the turning point is a local maximum or a local minimum.

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(a) Find ( )4sin2 xx e dx+∫ .

(b) The curve 3 212 48 36y x x x= − + crosses the

x-axis at 0, 1 and 3x x x= = = , as shown. Calculate the total area of the shaded regions

enclosed by the curve and the x-axis.

1 2 3 x

y


(c) (i) Find, in terms of a and b, cos

1 sin

b

a

xI dxx

=+∫

(ii) Find in terms of a and b,

sin 1 cos

b

a

xJ dxx

=+∫ .

(iii) Show that if 2

a b π+ = , then I J= .

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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, questions 7, 8, and 9 in Section C on paper 1 will be the same questions as those that appear as 6, 7, and 8 on the examination for candidates who are not in the initial schools. On this sample paper, the corresponding questions from the 2010 examination have been inserted to illustrate.



2011. M230S

Coimisiún na Scrúduithe Stáit State Examinations Commission

Leaving Certificate Examination, 2011 Sample Paper


Paper 2

Higher Level


300 marks

Examination number For examiner Question Mark

1 2 3 4 5 6 7 8

Total

Centre stamp

Running total Grade

Leaving Certificate 2011 – Sample Paper Page 2 of 19 Project Maths, Phase 2 Paper 2 – Higher Level

Instructions

There are two sections in this examination paper.



Answer all eight questions, as follows:

In Section A, answer:

Questions 1 to 5 and

either Question 6A or Question 6B.

In Section B, answer Question 7 and Question 8.







Section A Concepts and Skills 150 marks Answer all six questions from this section. Question 1 (25 marks) The random variable X has a discrete distribution. The probability that it takes a value other than 13, 14, 15 or 16 is negligible. (a) Complete the probability distribution table below and hence calculate E(X), the expected

value of X.

x 13 14 15 16

P(X = x) 0·383 0·575 0·004 (b) If X is the age, in complete years, on 1 January 2010 of a student selected at random from

among all second-year students in Irish schools, explain what E(X) represents. (c) If ten students are selected at random from this population, find the probability that exactly

six of them were 14 years old on 1 January 2010. Give your answer correct to three significant figures.

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Question 2 (25 marks) (a) Explain what is meant by stratified sampling and cluster sampling. Your explanation should

include: a clear indication of the difference between the two methods one reason why each method might be chosen instead of simple random sampling.

(b) A survey is being conducted of voters’ opinions on several different issues. (i) What is the overall margin of error of the survey, at 95% confidence, if it is based on a

simple random sample of 1111 voters? (ii) A political party had claimed that it has the support of 23% of the electorate. Of the

voters in the sample above, 234 stated that they support the party. Is this sufficient evidence to reject the party’s claim, at the 5% level of significance?



(a) Show that, for all k∈ , the point ( )4 2, 3 1P k k− + lies on the line 1 : 3 4 10 0l x y− + = .

(b) The line 2l passes through P and is perpendicular to 1l . Find the equation of 2l , in terms of k. (c) Find the value of k for which 2l passes through the point Q(3, 11). (d) Hence, or otherwise, find the co-ordinates of the foot of the perpendicular from Q to 1l .

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x

y Q

P

1l



The centre of a circle lies on the line 2 6 0x y+ − = . The x-axis and the y-axis are tangents to the circle. There are two circles that satisfy these conditions. Find their equations.



The diagram below shows the graph of the function : sin 2f x x . The line 2 1y = is also shown. (a) On the same diagram above, sketch the graphs of : sing x x and : 3sin 2h x x . Indicate clearly which is g and which is h. (b) Find the co-ordinates of the point P in the diagram.

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( )y f x= 2 1y =

-3

-2

-1

0

1

2

3

π 2π

P


Question 6 (25 marks) Answer either 6A or 6B.

Question 6A Explain, with the aid of an example, what is meant by proof by contradiction.

Note: you do not need to provide the full proof involved in your example. Give sufficient outline to illustrate how contradiction is used. Explanation: Example:


OR Question 6B ABC is a triangle.

D is the point on BC such that AD BC⊥ . E is the point on AC such that BE AC⊥ .

AD and BE intersect at O. Prove that DOC DEC∠ = ∠ .

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A

B C

D

E

O


Section B Contexts and Applications 150 marks Answer Question 7 and Question 8. Question 7 (75 marks) The King of the Hill triathlon race in Kinsale consists of a 750 metre swim, followed by a 20 kilometre cycle, followed by a 5 kilometre run. The questions below are based on data from 224 athletes who completed this triathlon in 2010. Máire is analysing data from the race, using statistical software. She has a data file with each competitor’s time for each part of the race, along with various other details of the competitors. Máire gets the software to produce some summary statistics and it produces the following table. Three of the entries in the table have been removed and replaced with question marks (?).

Swim Cycle Run

Mean 18.329 41.927 ? Median 17.900 41.306 ? Mode #N/A #N/A #N/A Standard Deviation ? 4.553 3.409 Sample Variance 10.017 20.729 11.622 Skewness 1.094 0.717 0.463 Range 19.226 27.282 20.870 Minimum 11.350 31.566 16.466 Maximum 30.576 58.847 37.336 Count 224 224 224

Máire produces histograms of the times for the three events. Here are the three histograms, without their titles.

0 10 20 30 40 50 60 70

16 20 24 28 32 36

Com

petit

ors

Time (minutes)

0 5

10 15 20 25 30 35 40 45

30 34 38 42 46 50 54 58

Com

petit

ors

Time (minutes)

0 10 20 30 40 50 60 70 80

10 14 18 22 26 30

Com

petit

ors

Time (minutes)

Lizzie Lee, winner of the women’s event


(a) (i) Use the summary statistics in the table to decide which histogram corresponds to each event. Write the answers above the histograms.

(ii) The mean and the median time for the run are approximately equal. Estimate this value

from the corresponding histogram. mean ≈ median ≈ (iii) Estimate from the relevant histogram the standard deviation of the times for the swim. standard deviation ≈ (iv) When calculating the summary statistics, the software failed to find a mode for the data

sets. Why do you think this is? Máire is interested in the relationships between the athletes’ performance in the three different events. She produces the following three scatter diagrams (b) Give a brief summary of the relationship between performance in the different events, based

on the scatter diagrams.

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30354045505560

10 15 20 25 30 35

cycl

e tim

e

swim time

Cycle vs. Swim

15

20

25

30

35

40

10 15 20 25 30 35

run

time

swim time

Run vs. Swim

15

20

25

30

35

40

30 35 40 45 50 55 60

run

time

cycle time

Run vs. Cycle


(c) The best-fit line for run-time based on swim-time is 0·53 15·2y x= + . The best-fit line for run-time based on cycle-time is 0·58 0·71y x= + . Brian did the swim in 17·6 minutes and the cycle in 35·7 minutes. Give your best estimate of Brian’s time for the run, and justify your answer.

The mean finishing time for the overall event was 88·1 minutes and the standard deviation was 10·3 minutes. (d) Based on an assumption that the distribution of overall finishing times is approximately

normal, use the empirical rule to complete the following sentence: “95% of the athletes took between and minutes to complete the race.” (e) Using normal distribution tables, estimate the number of athletes who completed the race in

less than 100 minutes. (f) After the event, a reporter wants to interview two people who took more than 100 minutes to

complete the race. She approaches athletes at random and asks them their finishing time. She keeps asking until she finds someone who took more than 100 minutes, interviews that person, and continues until she finds a second such person. Assuming the athletes are cooperative and truthful, what is the probability that the second person she interviews will be the sixth person she approaches?


Question 8 (75 marks) (a) A stand is being used to prop up a portable solar panel. It consists of a

support that is hinged to the panel near the top, and an adjustable strap joining the panel to the support near the bottom.

By adjusting the length of the strap, the angle between the panel and

the ground can be changed. The dimensions are as follows: 30 cmAB =

5 cmAD CB= =

22 cmCF =

4 cmEF = . (i) Find the length of the strap [DE] such that the angle α between the panel and the ground

is 60°.

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A

B

C (hinge)

panel

support

strap D E

F α


(ii) Find the maximum possible value of α, correct to the nearest degree. (b) A regular tetrahedron has four faces, each of

which is an equilateral triangle. A wooden puzzle consists of several pieces

that can be assembled to make a regular tetrahedron. The manufacturer wants to package the assembled tetrahedron in a clear cylindrical container, with one face flat against the bottom.

If the length of one edge of the tetrahedron is

2a, show that the volume of the smallest

possible cylindrical container is 38 69

aπ

.


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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.



2010. M130 S

Coimisiún na Scrúduithe Stáit

State Examinations Commission

Leaving Certificate Examination

Sample Paper

Mathematics (Project Maths)

Paper 2

Higher Level


300 marks


Question Mark

1

2

3

4

5

6

7

8

9

Total

Centre stamp

Running total

Grade

Leaving Certificate 2010 – Sample Paper Page 2 of 19 Project Maths, Paper 2 – Higher Level

Instructions

There are two sections in this examination paper.



Answer all nine questions, as follows:

In Section A, answer all six questions

In Section B, answer:

Question 7

Question 8

either Question 9A or Question 9B.

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 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.





Section A Concepts and Skills 150 marks

Answer all six questions from this section.


The events A and B are such that ( ) 0·7P A = , ( ) 0·5P B = and ( ) 0·3P A B∩ = .

(a) Find ( )P A B∪

(b) Find ( | )P A B

(c) State whether A and B are independent events, and justify your answer.

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The shapes of the histograms of four different sets of data are shown below.

(a) Complete the table below, indicating whether the statement is correct (�) or incorrect (�)

with respect to each data set.

A B C D

The data are skewed to the left

The data are skewed to the right

The mean is equal to the median

The mean is greater than the median

There is a single mode

(b) Assume that the four histograms are drawn on the same scale.

State which of them has the largest standard deviation, and justify your answer.

Answer:

Justification:


The co-ordinates of three points A, B, and C are: A(2, 2), B(6, –6), C(–2, –3).

(See diagram on facing page.)

(a) Find the equation of AB.

A B C D


(b) The line AB intersects the y-axis at D.

Find the coordinates of D.

(iii) Find the perpendicular distance from C to AB.

(iv) Hence, find the area of the triangle ADC.

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x

y

A

B

C

D



(a) Write down the equation of the circle with centre (–3, 2) and radius 4.

(b) A circle has equation 2 2 2 4 15 0x y x y+ − + − = .

Find the values of m for which the line 2 7 0mx y+ − = is a tangent to this circle.



The function ( ) 3sin(2 )f x x= is defined for x∈� .

(i) Complete the table below

x 0 π

4

π

2

3π

4 π

2x

sin(2 )x

3sin(2 )x

(ii) Draw the graph of ( )y f x= in the domain 0 π, .x x≤ ≤ ∈�

(iii) Write down the range and the period of f.

Range = Period =

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π

4

π

2

3π

4

π

x

y

0



ABCD is a parallelogram in which [ ]AC is a diagonal.

2a i j= −� ��

, 5 3b i j= +

� � �, and c i j= − −

� ��.

(i) Express d� in terms of i

� and j

�.

(ii) Find AC��

and AB��

.

(iii) Hence, find CAB∠ , correct to the nearest degree.



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Section B Contexts and Applications 150 marks

Answer Question 7, Question 8, and either Question 9A or Question 9B.

Question 7 Probability and Statistics (50 marks)

An economics student is interested in finding out whether the length of time people spend in

education affects the income they earn. The student carries out a small study. Twelve adults are

asked to state their annual income and the number of years they spent in full-time education. The

data are given in the table below, and a partially completed scatter plot is given.

Years of

education

Income

/€1,000

11 28

12 30

13 35

13 43

14 55

15 38

16 45

16 38

17 55

17 60

17 30

19 58

(i) The last three rows of data have not been included on the scatter plot. Insert them now.

(ii) Calculate the correlation coefficient.

Answer:

(iii) What can you conclude from the scatter plot and the correlation coefficient?

(iv) Add the line of best fit to the completed scatter plot above.

(v) Use the line of best fit to estimate the annual income

of somebody who has spent 14 years in education.

Answer:

20

30

40

50

60

70

10 12 14 16 18 20

Years of education

Annual income /€1000


(vi) By taking suitable readings from your diagram, or otherwise, calculate the slope of the line of

best fit.

(vii) Explain how to interpret this slope in this context?

(viii) The student collected the data using a telephone survey. Numbers were randomly chosen

from the Dublin area telephone directory. The calls were made in the evenings, between 7

and 9 pm. If there was no answer, or if the person who answered did not agree to participate,

then another number was chosen at random.

List three possible problems regarding the sample and how it was collected that might make

the results of the investigation unreliable. In each case, state clearly why the issue you

mention could cause a problem.

Problem 1:

Problem 2:

Problem 3:

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Question 8 Geometry and Trigonometry (50 marks)

Two surveyors want to find the height of an electricity pylon.

There is a fence around the pylon that they cannot cross for

safety reasons. The ground is inclined at an angle. They have a

clinometer (for measuring angles of elevation) and a 100 metre

tape measure. They have already used the clinometer to

determine that the ground is inclined at 10° to the horizontal.

(a) Explain how they could find the height of the pylon.

Your answer should be illustrated on the diagram below.

Show the points where you think they should take

measurements, write down clearly what measurements

they should take, and outline briefly how these can be

used to find the height of the pylon.

Diagram:

Measurements to be taken:

Procedure used to find the height:

10°


(b) Write down possible values for the measurements taken, and use them to show how to find

the height of the pylon. (That is, find the height of the pylon using your measurements, and

showing your work.)

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Question 9A Probability and Statistics (50 marks)

A car rental company has been using Evertread tyres on their fleet of economy cars. All cars in this

fleet are identical. The company manages the tyres on each car in such a way that the four tyres all

wear out at the same time. The company keeps a record of the lifespan of each set of tyres. The

records show that the lifespan of these sets of tyres is normally distributed with mean 45 000 km

and standard deviation 8000 km.

(i) A car from the economy fleet is chosen at random. Find the probability that the tyres on this

car will last for at least 40 000 km.

(ii) Twenty cars from the economy fleet are chosen at random. Find the probability that the tyres

on at least eighteen of these cars will last for more than 40 000 km.


(iii) The company is considering switching brands from Evertread tyres to SafeRun tyres, because

they are cheaper. The distributors of SafeRun tyres claim that these tyres have the same mean

lifespan as Evertread tyres. The car rental company wants to check this claim before they

switch brands. They have enough data on Evertread tyres to regard these as a known

population. They want to test a sample of SafeRun tyres against it.

The company selects 25 economy cars at random from the fleet and fits them with the new

tyres. For these cars, it is found that the mean life span of the tyres is 43 850 km.

Test, at the 5% level of significance, the hypothesis that the mean lifespan of SafeRun tyres is

the same as the known mean of Evertread tyres. State clearly what the company can conclude

about the tyres.

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Question 9B Geometry and Trigonometry (50 marks)

(a) Prove that, if two triangles ∆ABC and ∆A'B'C' are similar, then their sides are proportional, in

order:

AB BC CA

A B B C C A= =

′ ′ ′ ′ ′ ′.

Diagram:

Given:

To prove:

Construction:

Proof:


(b) Anne is having a new front gate made and has decided on the design below.

The gate is 2 metres wide and 1·5 metres high. The horizontal bars are 0·5 metres apart.

(i) Calculate the common length of the bars [AF] and [DE], in metres, correct to three

decimal places.

(ii) In order to secure the bar [AF] to [DE], the manufacturer needs to know:

- the measure of the angle EGF, and

- the common distance AG = DG .

Find these measures. Give the angle correct to the nearest degree and the length correct

to three decimal places.

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G

A

D

B

E

F

C



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Note to readers of this document:

This sample paper is intended to help teachers and candidates prepare for the June 2010

examination in the Project Maths initial schools. The content and structure do not necessarily

reflect the 2011 or subsequent examinations in the initial schools or in all other schools.

Leaving Certificate – Higher Level

Mathematics (Project Maths) – Paper 2

Sample Paper

Time: 2 hours 30 minutes

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