Mathematics (Project Maths – Phase 2) · PDF file(iii) The school has decided to add...

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J.20

NAME

SCHOOL

TEACHER

Pre-Junior Certificate Examination, 2014

Mathematics (Project Maths – Phase 2)

Paper 2

Higher Level

Time: 2 hours, 30 minutes

300 marks

For examiner

Question Mark Question Mark

1 11

2 12

School stamp 3 13

4 14

5 15

6 16

7

8

9

Grade

Running total

10 Total

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Project Maths, Phase 2Paper 2 – Higher Level

Instructions

There are 16 questions on this examination paper. Answer all questions.

Questions do not necessarily carry equal marks. To help you manage your time during this examination, a maximum time for each question is suggested. If you remain within these times, you should have about 10 minutes left to review your work.

Write your answers in the spaces provided in this booklet. You may lose marks if you do not do so. There is space for extra work at the back of the booklet. You may also ask the superintendent for more paper. Label any extra work clearly with the question number and part.

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

You will lose marks if all necessary work is not clearly shown.

Answers should include the appropriate units of measurement, where relevant.

Answers should be given in simplest form, where relevant.

Write the make and model of your calculator(s) here:

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Question 1 (suggested maximum time: 10 minutes)

The design of a company’s logo incorporates two identical semi-circles which fit exactly in a rectangle, as shown.

(i) From the diagram, measure the radius of the semi-circles.

(ii) Find the area enclosed by the two semi-circles, in terms of π.

(iii) Find the area of the shaded region. Give your answer correct to one decimal place.

(iv) Calculate the ratio of the area enclosed by the two semi-circles to the area of the rectangle, giving your answer in the form π : n, n ∈ ℕ.

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A vessel is in the shape of a cylinder on top of a cone, as shown. The diameter of the cylinder is 10 cm and the overall height of the vessel is 30 cm.

(i) Find the internal volume of the vessel. Give your answer in terms of π and h, the height of the cylinder.

(ii) Water is poured into the vessel. Under what conditions are the volumes of water in the cone and cylinder equal?

10 cm

30 cm

cmh

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Rob tested two different brands of batteries, A and B, to see which had the longer life-span. He placed batteries from each brand in a torch and recorded how long each lasted. The results are shown below in the back-to-back stem-and-leaf plot.

Brand A Brand B 65 1 66 3 0 67 8 9 4 4 68 2

9 4 4 3 69 4 9 9 0 70 4 0 71 7 8 9 7 1 72 0 0 3 6 9 73 1 2

Key: 65 1 means 651 minutes

(i) Find the median life-span of each brand of battery.

(ii) Find the interquartile range of each brand of battery.

(iii) Based on the above data, would you conclude that one brand is more reliable than the other? Give a reason for your answer.

Brand A: Brand B:

Brand A: Brand B:

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A group of 96 students from a school in Galway stopped at a food court in a shopping centre on the way home from visiting the Young Scientist Exhibition. When they got back on the bus, their teachers conducted a quick survey to determine which food outlet they had chosen to eat in at the food court. The results are recorded in the table below.

Burger Joint Sandwich Bar Wok Station

Boys 24 16 6

Girls 11 x 32

(i) Assuming that all students ate at one of the food outlets, find the value of x.

(ii) If one student is chosen at random, what is the probability that the student chosen is a girl who ate at the Wok Station?

(iii) If one student is chosen at random, what is the probability that the student chosen is a boy?

(iv) If the student chosen at random is a boy, what is the probability that he did not eat at the Sandwich Bar?

(v) Place each of your answers to parts (ii), (iii) and (iv) at its correct position on the probability scale below.

0 1

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Students in a school have been asked to select the subjects that they would like to study for their Leaving Certificate next year. Each student can only select one subject from each of the following options lines.

Option Subjects

A Biology Business Geography Accounting

B French Art

C German History Chemistry

(i) How many different subject selections are possible?

(ii) Sophie does not want to study any of the languages offered. From how many different subject selections can she pick?

(iii) The school has decided to add Music to one of the option lines. On which option line should Music be placed, in order to maximise the possible subject

selections from which students can pick? Explain your answer.

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Third-year students were asked to choose their favourite sport offered in their school. The results are recorded in the table below.

Sport Soccer Tennis Rugby Basketball

Number of Students 40 10 25 15

(i) Display the data on a pie-chart, showing clearly how the size of each angle is calculated.

(ii) Why is this an appropriate method to display the above data?

(iii) What percentage of students chose Basketball as their favourite sport in school? Give your answer correct to the nearest whole number.

(iv) It was later discovered that the surveys from one class of 30 students had been mislaid. When these surveys were accounted for, the number of students who chose Rugby increased by 9. Calculate the measure of the angle that would now represent the students who chose Rugby.

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PQRS is a parallelogram. X is the image of R under axial symmetry in the line QS. Y is the point of intersection of QX and PS.

(i) Prove that triangle QSP and triangle QSX are congruent.

(ii) Show that | PY | = | YX |.

Q

P

S

X

R

Y

Y

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Prove that in a right-angled triangle, the square of the hypotenuse is the sum of the squares of the other two sides.

Diagram:

Given:

To Prove:

Proof:

Construction:

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Lisa has a set of eight coloured plastic strips as shown below.

9 cm

4 cm

4 cm

20 cm

21 cm

29 cm

6 cm 6 cm

(i) Is it possible for Lisa to make an equilateral triangle using any three of the strips shown above? Give a reason for your answer.

(ii) What strips could Lisa use to make a parallelogram?

(iii) How many different isosceles triangles can Lisa form using the strips shown above? Explain your answer.

(iv) If Lisa wanted to form a right-angled triangle, which three strips could she use? Explain your answer.

Answer:

Reason:

Answer:

Explanation:

Answer:

Explanation:

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The table below gives the equations of five lines.

Line Equation Slope

a y = 2x + 3

b 2x + y = 3

c 3x + y = t

d 2y = −x + 2

e y = 5x + 6

(i) Write down the slope of each line in the table above.

(ii) Which line has the greatest slope? Give a reason for your answer.

(iii) Which lines are perpendicular? Give a reason for your answer.

(iv) The line c contains the point (1, −1). Find the value of t.

(v) Which lines make equal intercepts of the y-axis? Give a reason for your answer.

Answer:

Reason:

Answer:

Reason:

Answer:

Reason:

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Using only a compass and a straight-edge, divide the line segment below into four equal parts. Show all construction work.


(i) Construct a right-angled triangle containing an angle A such that cos A = 7

2.

(ii) Find tan A, giving your answer in surd form.

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(i) In the table below, show the values of cos and sin of the angles listed. Give your answers correct to three significant figures.

cos sin

30°

40°

50°

60°

(ii) What can you conclude from the above results? Give a reason for your answer.

(iii) Brid notices during an examination that her calculator is faulty. The sin function is giving her an error message. Assuming all other functions are working correctly, explain how she might use her calculator to calculate the value of sin 42°.

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The Costa Concordia disaster was the partial sinking of the Italian cruise ship when it ran aground at Isola del Giglio during the night of 13 January 2012.

When planning to raise the ship, engineers calculated various angles and distances.

Using the information shown in the diagram, find the height of the top of the smoke stack, marked A, above the level of the water.

A

34� 2 5m·

76�

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A map of Ireland has been placed over a co-ordinate plane.

(i) By identifying the co-ordinates of Dublin (D) and Fermoy (F), estimate the distance between Dublin and Fermoy, given that each unit on the co-ordinate plane represents 60 km.

(ii) If it takes Sean 2·5 hours to drive from Dublin to Fermoy, estimate how long it will take him to drive from Dublin to Sligo (S), assuming that he drives at the same speed.

2

1

2

3

4

5

6

7

8

9

3 4 5 6 7 8

x

y

1

D

F

S

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l is the line x + y + 4 = 0.

(i) Draw and label the line l on the co-ordinate plane.

(ii) Draw and label a line k, through the point of intersection of the line l and the y-axis, which is perpendicular to the line l.

(iii) Find the equation of the line k.

(iv) Calculate the area of the triangle formed by the line l, the line k and the x-axis.

2

�1

1

�2

2

�3

3

�4

4

5

�5

3 4 5 6

x

y

1�1�2�3�4�5�6

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You may use this page for extra work.

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You may use this page for extra work.

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Pre-Junior Certificate, 2014 – Higher Level

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

Mathematics (Project Maths – Phase 2) · PDF file(iii) The school has decided to add...

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