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Pre-Junior Certificate Examination, 2014

Triailscrúdú an Teastais Shóisearaigh, 2014

_______________

Mathematics (Project Maths – Phase 2)

Paper 2

Higher Level

2½ hours

300 marks

For examiner Question Mark

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15

Total

*W8*

Name:

School:

Address:

Class:

Teacher:

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Instructions There are 15 questions on this examination paper. Answer all questions. Questions do not necessarily carry equal marks. To help you manage your time during this examination, a maximum time for each question is suggested. If you remain within these times, you should have about 10 minutes left to review your work. Write your answer in the spaces provided in this booklet. There is space for extra work at the back of the booklet. Label any extra work clearly with the question number and part. 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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Question 1 (Suggested maximum time: 10 minutes) The diagram shows two spinners. The first wheel is divided into four equal segments numbered 1, 2, 3 and 4. The second wheel is divided into three equal segments labelled A, B and C. A game consists of spinning the two spinners and noting the result from each spinner.

For example, the outcome shown above is (3, A). (a) List all the possible outcomes of spinning the wheels. (b) What is the probability that the outcome is (2, C)? (c) What is the probability that the outcome contains an odd number with the letter A? (d) What is the probability that the outcome includes the letter C?

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Question 2 (Suggested maximum time: 10 minutes) The nationality of 240 visitors at the Cliffs of Moher was noted over a two-hour period. (a) Calculate the number of European visitors in the two-hour period. (b) Calculate the relative frequency of each category and write these into the table above. (c) Based on the information in the table above what is the probability that the next visitor to the centre is not Irish? (d) On a busy Saturday in summer the centre had 2,400 visitors. Using the information above estimate how many of those visitors were American.

Category Irish European American Other

Frequency 48 82 33

Relative frequency

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Question 3 (Suggested maximum time: 10 minutes) The volume of a cylindrical tin of beans of height 10 cm is 490π cm3. Find the radius of the cylindrical tin.

Question 4 (Suggested maximum time: 10 minutes) In the diagram [ ]AC is parallel to [ ]BE ,

80BCA∠ = ° and 55CAB∠ = ° . (a) Calculate the value of X. (b) Calculate the value of Y.

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Question 5 (Suggested maximum time: 10 minutes)

( )2,1P − and ( )4, 5Q are two points. (a) Plot the points P and Q on a co-ordinate diagram.

(b) Find the slope of [ ]PQ . K is the line 3 2 9 0x y+ − = . (c) Show that K passes through the midpoint of [ ]PQ .

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Question 6 (Suggested maximum time: 10 minutes) The graphs below represent the equations of six lines A, B, C, D, E and F.

Fill in the table, matching each equation with its particular graph.

Equation Graph

3 6y x= +

4y x= +

2 1y x= +

8 8 0x y+ + =

2 6 0x y+ − =

2 0x y+ =

A B C

D E F

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Question 7 (Suggested maximum time: 10 minutes) (a) Explain what is meant by the term “axiom”. (b) Give an example of an axiom you have studied. (c) By using construction methods show how to cut the line segment below into three equal sections.

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Question 8 (Suggested maximum time: 10 minutes) The Burj Al Arab in Dubai is the world’s fourth largest hotel. The hotel stands on an artificial island which is 268 m from the beach. A tourist standing on the beach, who is 1 65⋅ m tall, found using a clinometer that the angle of elevation to the top of the tower is 50° . (a) Draw a suitable diagram and calculate the height of the hotel to the nearest metre. (b) Research shows that using a clinometer can lead to error of at most plus or minus

5 percentage when measuring the angle of elevation. Based on this information and the information above, what is the maximum possible estimated height of the Burj Al Arab to the nearest metre?

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Question 9 (Suggested maximum time: 10 minutes) Find the value of the angles A, B, C and D in the following diagram. Give a reason for each value.

Angle A =

Reason:

Angle B =

Reason:

Angle C =

Reason:

Angle D =

Reason:

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Question 10 (Suggested maximum time: 5 minutes) Prove that the sum of the angles in a triangle is 180° .

Given:

To prove:

Construction:

Proof:

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Question 11 (Suggested maximum time: 10 minutes) A solid cone of radius 9 cm has a slant height of 15 cm. (a) Find h, the perpendicular height of the cone. (b) Show that the volume of the cone is 324π cm3. (c) One-eighteenth of the volume of the cone drips into a hemisphere

and fills it. Calculate the radius of the hemisphere.

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Question 12 (Suggested maximum time: 5 minutes) The picture below shows two container types: cylindrical and cone shaped. Their circular bases are congruent to each other and their heights are also equal. A cylindrical container is filled with water and we will transfer the water into the cone-shaped containers.

One of the pictures A through E shows how the cone-shaped containers will be filled up after all the water from the cylindrical container is transferred. Select the correct picture and justify your answer. A B C D E

Selection:

Reason:

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Question 13 (Suggested maximum time: 10 minutes) The triangle ABC is an equilateral triangle with side length 2 units.

(a) Construct on the diagram above the bisector of the angle ABC. (b) Using your diagram only, to justify your answer, write down the value sin 60° in surd form. (c) Using the same diagram, a student noticed correctly that the cos30° was the same as sin 60° . Give a reason to support the student’s claim. (d) On further investigation the student came up with the idea that if any two angles added up to 90° , i.e. 90A B+ = ° , then the sin cosA B= . Give a reason to support or contradict the student’s claim.

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Question 14 (Suggested maximum time: 10 minutes) 20 Irish teenagers were asked how long they spent online gaming in a month. Their times, in full hours, are listed below: 6 22 24 29 36

19 22 22 23 19 24 24 48 45 38 9 17 49 38 35

(a) Use the data to complete the grouped frequency table below. (b) Calculate the mean number of hours spent on gaming by these Irish teenagers. Round your answer to the nearest full hour. (c) A student decided to calculate the median for the data above and found that it was smaller than the mean. The students concluded that, for this data set, the median was a better measure of central tendency than the mean. Say whether you agree or disagree with

the student’s statement and give a reason to back up your answer.

Number of hours spent gaming 0 – 10 10 – 20 20 – 30 30 – 40 40 – 50

Students

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(iv) 20 teenagers were asked how long they spent online gaming in a month. Their results are below, along with the Irish data from the previous page. American data: 6 7 24 29 36

9 8 8 23 19 19 24 18 14 38 9 17 17 32 35

Irish data: 6 22 24 29 36

19 22 22 23 19 24 24 48 45 38 9 17 49 38 35

Choose a suitable graphical representation to compare the two sets of data. (v) A student looked at both data sets and concluded that ‘Irish 15-year-olds spend more time

online gaming than American 15-year-olds’. By analysing your graphs from question (d) above or by using an appropriate statistic can you come up with evidence either to support or contradict the student’s claim?

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Question 15 (Suggested maximum time: 10 minutes) In the parallelogram PQRS the points T and W are on the diagonal [ ]PR such that

PQT WSR∠ = ∠ .

(a) Prove that PT WR= . (b) Hence, or otherwise, show that the triangles PWS and QTR are congruent.

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