pre-junior certificate examination, 2015 triailscrúdú an...

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Pre-Junior Certificate Examination, 2015

Triailscrúdú an Teastais Shóisearaigh, 2015

Mathematics

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

Total

Grade

*W8*

Name:

School:

Address:

Class:

Teacher:

Running total

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Instructions There are 13 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. Label any extra work clearly with the question number and part. 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: 8 minutes) A grocer has 100 boxes of strawberries. (i) He weighs 10 of the boxes. Which three words describe the data he collects? Put a tick () in the correct box below. Continuous Discrete Sample Primary Secondary (ii) Outline a suitable sampling method to obtain a simple random sample of 10 boxes from the 100 boxes.

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Question 2 (Suggested maximum time: 12 minutes) (a) Rhianna plays a game. The probability that she will lose the game is 0.37. The probability that she will draw the game is 0.05. Rhianna is going to play the game 200 times. Work out an estimate for the number of times she will win the game. (b) Another game involves throwing a die and spinning

the spinner as shown here.

(i) List the elements of the sample space that show all possible outcomes of the game.

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John plays the game. What is the probability that:

(ii) the same number occurs on the die and the spinner?

(iii) the number on the spinner is greater than the number on the die?

(iv) the total is 7 when the number on the die and the number on the spinner are added?

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Question 3 (Suggested maximum time: 5 minutes) Find the measures of the angles labelled P and Q in the diagram below given that nk .

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Question 4 (Suggested maximum time: 14 minutes) (a) The diagram shows the plan of a playground. Bill is going to cover the playground with

tarmac. It costs €9.35 to cover each square metre with tarmac. Work out the total cost of the tarmac Bill needs.

(b) Over a four hour period the age and sex of 150 visitors to the playground were recorded

as follows:

Category Frequency Relative

Frequency

Daily Frequency for Saturday

(part (iv) below)

Male aged under 5 25

Male aged 5 to 12

Male aged over 12 18

Female aged under 5 18

Female aged 5 to 12 31

Female aged over 12 24

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(i) Calculate the number of males, aged 5 to 12, that attended the park and write it into the table on the previous page. (ii) Calculate the relative frequency of each category and write these into the table on the

previous page. (iii) Based on the information in the table on the previous page, what is the probability that the next visitor to the playground is female and over 5? (iv) On a busy Saturday in summer the playground had 900 visitors. Using the information from the table on the previous page, fill in the expected daily frequency for Saturday in the final column of the table.

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Question 5 (Suggested maximum time: 8 minutes) (a) What is meant by the term axiom? (b) Give an example of an axiom you have studied. (c) Divide the line segment [AB] into four equal segments showing all construction lines clearly.

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Question 6 (Suggested maximum time: 12 minutes) A spherical golf ball has a diameter of 4 cm. (a) Find the volume of the golf ball in terms of π . A cylindrical hole in a golf course is 10 cm in diameter and 12 cm deep. The hole is half full of rainwater. (b) Calculate the volume of water in the hole, in terms of π . The golf ball is dropped into the hole. (c) Find the rise in the level of water, correct to two decimal places.

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Question 7 (Suggested maximum time: 8 minutes) ABCD is a parallelogram. Angle °= 38ADB . Angle °= 41BEC . Angle °=120DAB . Calculate the size of angle x. Give reasons for your answer.

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Question 8 (Suggested maximum time: 15 minutes) (a) Prove that in a right angled-triangle the square on the hypotenuse is equal to the sum of the squares on the other two sides.

Given:

To prove:

Construction:

Proof:

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(b) ABCD is a square with a side length of 4x. M is the midpoint of DC. N is the point on AD where ND = x. BMN is a right-angled triangle. (i) Find an expression in x for BM .

(ii) Find an expression, in terms of x, for the area of triangle BMN. Give your expression in its simplest form.

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Question 9 (Suggested maximum time: 10 minutes) During a trigonometry lesson a group of students wrote down some statements about trigonometry: (1) The value from any of these trigonometric functions will never be greater than 1. (2) If the size of the angle is doubled then the value from the trigonometric functions will also double. (3) The value from all of the trigonometric functions will increase if the size of the angle is increased. (4) I do not need to use a calculator to find Sin 45°. I can do it by drawing a right angled isosceles triangle. The answer will be in surd form. (a) Do you think that (1) is correct? Give a reason to justify your answer.

Answer:

Reason:

(b) Do you think that (2) is correct? Give a reason to justify your answer.

Answer:

Reason:

(c) Do you think that (3) is correct? Give a reason to justify your answer.

Answer:

Reason:

(d) Show how a right angled isosceles triangle in which the equal sides are 1 unit each, can be used to find Sin 45° in surd form.

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Question 10 (Suggested maximum time: 5 minutes)

(a) Through the point A draw a line 1l with a slope of 4

3.

(b) Through the point A draw another line 2l with a slope of 3

4− .

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Question 11 (Suggested maximum time: 8 minutes) The diagram shows a ladder leaning against a vertical wall. The ladder stands on horizontal ground. The length of the ladder is 225 cm. The ladder is safe to use when the angle marked y is 75°. (a) Is the ladder safe to use when it is 70 cm from the base of the house? Justify your answer. (b) What is the height the ladder can reach up the wall when it is in a safe position?

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Question 12 (Suggested maximum time: 20 minutes) A magazine did some testing to see how long three different types of camera batteries lasted before they needed recharging. (Several cameras and test conditions were used). Below is a stem and leaf plot of the results for Battery A.

BATTERY A’S RESULTS Key : 5 | 6 = 56 hours STEM LEAF

0 4 5 5 7 1 0 4 5 5 7 2 0 2 3 6 7 3 2 4 6 4 5 6 6 7 5 (a) Calculate the range for battery life results for Battery A. (b) Calculate the median battery life results for Battery A. (c) Calculate the mean battery life results for Battery A, to the nearest full hour.

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The results, measured in number of hours, for Battery B, were as follows:

14, 19, 23, 24, 25,

15, 17, 29, 31, 32,

35, 38, 39, 29, 36,

29, 35, 36, 36. (d) Draw the stem and leaf plot for this data. (e) Martin claims that Battery B is more reliable. Find two reasons to support his claim. (f) Below are the battery life results of Battery C.

Number of hours before recharging 10-20 20-30 30-40 40-50

Number of batteries 4 5 7 3

The magazine, by using one measure of central tendency, recommended one of the three batteries. Identify which one they recommended and justify their choice.

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Question 13 (Suggested maximum time: 15 minutes)

( )1,2 −−P , ( )3,2Q and ( )5,4−R are three points. (a) Plot these points on the grid below.

(b) Find the midpoint of [ ]PQ .

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(c) On the diagram above construct the perpendicular bisector of [ ]PQ . (d) Verify, by the use of an appropriate co-ordinate geometry formula or otherwise, that the triangular PQR is an isosceles triangle.

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You may use this page for extra work.

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