Mathematics (Project Maths – Phase 2) · PDF fileBe sure to state the null hypothesis clearly, and to state the conclusion clearly. 2013 L.20 15/20 Page 15 of 19 Project Maths, Phase

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L.20

NAME

SCHOOL

TEACHER

Pre-Leaving Certificate Examination, 2013

Mathematics (Project Maths – Phase 2)

Paper 2

Higher Level

Time: 2 hours, 30 minutes

300 marks

For examiner

Question Mark

1

2

3

Centre stamp 4

5

6

7

8

Grade

Running total

Total

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Project Maths, Phase 2Paper 2 – 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 2 questions

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 8.

Write your answers in the spaces provided in this booklet. You will 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.

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:

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Section A Concepts and Skills 150 marks

Answer all six questions from this section.

Question 1 (25 marks) (a) The Venn diagram shows the probability of two events A and B occurring.

(i) Find the value of P(A B).

(ii) Verify that P(A B) P(A) P(B) P(A B)

and hence, state what this tells us about A and B.

(b) A fair coin is tossed four times giving either ‘heads’ or ‘tails’ each time.

(i) Complete the sample space below to show all the possible outcomes.

H H H H

H H H T

(ii) E is the probability that the first three results are ‘heads’ and F is the probability that the

fourth result is also ‘heads’. Find P(E) and P(F).

(iii) Show that P(E and F) P(E) P(F) and hence, state what this tells us about E and F.

A

S

B

0�4

0�1 0�3

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Question 2 (25 marks) One of the highlights of the Olympic Games is the marathon. It is a long-distance running event with an official distance of 42·195 kilometres that is usually run as a road race. It is one of the original events of the modern Olympic Games revived in 1896 although it did not feature in the original Games in ancient Greece. A sports researcher, wants to investigate the connection, if any, between athletes’ performances in the marathon and the athletes’ ages. He takes a random sample of 15 athletes who completed the race in the 2012 London Olympic Games and compares the race times of these athletes to each of their ages. The results are shown in the table below.

Race Time (minutes)

Age of Athlete (years)

129 37 131 35 132 33 133 34 134 32 135 32 135 33 136 32 137 30 138 28 139 27 140 27 141 25 143 26 145 24

(a) Explain the following terms in relation to the above information:

(i) Sample;

(ii) Population.

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(b) Create a suitable graphical representation to illustrate the data.

(c) What kind of relationship, if any, does the observed data suggest exists between athletes’

performances and the athletes’ ages in the marathon?

(d) Can you make the same hypothesis for all athletes

that compete in the marathon? Answer:

Explain your answer.

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Question 3 (25 marks) l is the line x 3y 11 0 and m is the line x 2y 1 0. (a) Write down the slope of l

and the slope of m.

Slope of l:

Slope of m:

(b) Find the measure of the acute angle, ,

between l and m.

(c) The line n also makes the same acute angle, , with l where it cuts the x-axis. Find the equation of n.

x

l m

x

y

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Question 4 (25 marks) c1 is a circle with centre (3, 2). The lines l1: 2x y 3 0 and l2: x 2y 6 0 are tangents to c1, as shown in the diagram. (a) Find the radius of c1 and hence,

write down the equation of c1.

(b) c2, c3 and c4 are images of c1 under different transformations. Describe fully the transformation in each case.

(c) Find the equations of the two circles that touch c1, c2, c3 and c4.

c2:

c3:

c4:

x

yl1

c1

c2

c3

c4

l2

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Question 5 (25 marks) (a) Show that sin 2x 2sin x cos x.

(b) Solve the equation sin 2x sin x 0 in the domain 0 ≤ x ≤ 2, x ℝ.

(c) Verify your solution from part (b) by sketching the graphs of the functions f : x sin 2x

and g : x sin x in the domain 0 ≤ x ≤ 2, x ℝ. Indicate clearly which is f and which is g.

0 � ��

x

y

2

�

2

3�

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Question 6 (25 marks)

Answer either 6A or 6B.

Question 6A

(a) Explain the difference between an axiom and a theorem.

(b) ABC is a triangle. Prove that, if a line l is parallel to BC and cuts [ AB ] in the ratio s : t, then it also cuts [ AC ]

in the same ratio.

Diagram:

Given:

To prove:

Proof:

Construction:

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OR

Question 6B

ACDE is a parallelogram.

The points A, D and E lie on the circle which cuts [ AC ] at B. (a) Prove that BCD is an isosceles triangle.

(b) Prove that | BDE | | AED |.

A

D

E

CB

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You may use this page for extra work.

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Section B Contexts and Applications 150 marks

Answer Question 7 and Question 8. Question 7 (75 marks) A confectionary manufacturer is launching a new range of chocolates. The disc-shaped chocolates, each of radius 1·5 cm and height 0·75 cm, are packed in two layers in a rectangular box. The configuration of chocolates in each layer is shown below.

(a) Calculate the internal volume of the box.

A production researcher suggests that if the configuration of the chocolates in each layer of the box was altered, the packaging costs of the chocolates could be reduced. The alternative design of each layer is shown below.

p

(b) Using the right-angled triangle on the alternative design, calculate p and hence,

find the internal volume of this box.

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(c) The original box costs €0·80 to produce. Calculate the ratio of the volumes of the two boxes and hence, find the potential savings of using the alternative design if the cost of producing each box is directly proportional to the volume of the box and the company projects sales of 150 000 boxes annually.

(d) There are three different types of chocolates in the box.

Half the chocolates are milk chocolates while the ratio of dark to white chocolates is 2 : 1.

(i) What is the probability that a chocolate chosen at random from the box will be a white chocolate?

(ii) If three chocolates are chosen at random from the box, find the probability that each

of the chocolates chosen is made from the same type of chocolate.

(iii) If three chocolates are chosen at random from the box, find the probability that each

of the chocolates chosen is made from a different type of chocolate.

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(e) The confectionary company claims that the minimum weight of chocolates in each box is 350 g. The company’s quality control department has determined that the weights of chocolates in each box are normally distributed with mean equal to the specified weight and standard deviation of 8 g. 50 boxes are selected at random and it is discovered that the mean weight of chocolates in these boxes is 348 g.

Use a hypothesis test at the 5% level of significance to decide whether there is sufficient evidence to validate the confectionary company’s claim.

Be sure to state the null hypothesis clearly, and to state the conclusion clearly.

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Question 8 (75 marks) Mary has a rectangular hallway measuring 6 m by 1·25 m. She plans on tiling the floor with regular pentagon-shaped tiles of side 6 cm using a mosaic pattern. She has decided to use a tiling contractor and he has given her a scaled drawing of the proposed work. (a) A tile on the scaled drawing is reproduced on the square grid as shown. Given the centre of enlargement, O, construct, using a compass and straight edge only,

a full-size tile.

O

2 cm

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(b) The tiling contractor has asked Mary to order the tiles she wants to use. She needs to find out the area that each tile covers in order to calculate the number of tiles needed to complete the work. However, the area of each tile is not shown on her drawing.

She knows that the internal angles of a regular pentagon are 108 and it can be divided up into three isosceles triangles as shown.

Find the area of each tile. Give your answer correct to one decimal place.

108�

6 cm

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(c) Mary asked her friend, Sean, to check her calculations. He uses an alternative method to find the area of each tile.

On the scaled drawing, Sean sketches a triangle ABC, where C is the centre of the pentagon. He calculates the area of the tile to be:

)54)(sin2)(71(

2

159

Explain what each of the following numbers represent:

(d) Mary’s hallway measures 6 m by 1·25 m. The tiling contractor has asked her to purchase

10% more tiles than are required to allow for cutting and wastage.

(i) Assuming the area of the triangular cut-offs required are negligible, how many boxes of tiles does Mary need to order if they are supplied in boxes of 60?

(ii) When Mary goes to order the tiles, she is informed that this particular size of tile has been

discontinued. However a similar regular pentagon-shaped tile of side 5 cm supplied in boxes of 72 is available. Calculate the number of boxes of tiles she now needs to order.

1·7:

5:

9:

54:

B

C

A

2 cm

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Pre-Leaving Certificate 2013 – Higher Level

Mathematics (Project Maths – Phase 2) – Paper 2 Time: 2 hours, 30 minutes

Documents

Mathematics (Project Maths – Phase 2) · PDF fileBe sure to state the null hypothesis clearly, and to state the conclusion clearly. 2013 L.20 15/20 Page 15 of 19 Project Maths, Phase