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Pre-Junior Certificate Examination, 2012

Triailscrúdú an Teastais Shóisearaigh, 2012

_______________

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

*PM61*

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Instructions There are fifteen 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. 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 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:

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Question 1 (Suggested maximum time: 12 minutes) (a) The tyre of a bicycle has a radius of 49 cm.

(i) What distance will the tyre cover in 20 complete turns? Give your answer correct to the nearest metre.

(ii) How many complete revolutions will the tyre make in a 10 km road race?

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(b) An industrial oil tank is in the shape of a cylinder with dimensions as shown.

The maximum capacity of the tank is 24,000 litres of oil.

(i) Calculate x, the length of the tank, correct to the nearest whole number.

(ii) During a 3 month period 65% of the oil is used. If oil costs €0.84 per litre, calculate the cost of refilling the tank.

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Question 2 (Suggested maximum time: 10 minutes) A class of second year students carried out a survey of cars that collected students from school on a certain day. They noted the colour of each car and recorded their results using a tally count.

Colour Tally Frequency

Red

Silver

Black

Other

(a) How many cars collected students from school that day?

(b) What is the probability that a student was collected in a black car?

(c) The students repeated the survey over a number of weeks. If they recorded a total of 1450 cars how many should they expect to be red?

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Question 3 (Suggested maximum time: 10 minutes) (a) Explain the difference between a survey and a census.

(b) List one advantage and one disadvantage of carrying out a postal survey.

Advantage:

Disadvantage:

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(c) Susan is designing a survey. She designs the following questions to gather information from the public. Examine each question and explain why they may not be suitable for a survey.

(i) Are you:

YOUNG MIDDLEAGED OLD

(ii) What is your yearly salary?

LOW MEDIUM HIGH

(iii) From what you have learned about designing surveys, suggest a way that each question above can be asked in a different manner that is more suitable to a survey.

First Question:

Second Question:

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Question 4 (Suggested maximum time: 5 minutes) A car dealership offers three models of cars in four different colours. (a) If the dealership must display one of each model and colour, how many cars will be in the showroom?

(b) If a customer can choose airconditioning or cruise control or alloy wheels as an additional extra, how many cars would the dealer need to display in the showroom?

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Question 5 (Suggested maximum time: 13 minutes) A student recorded the number of computer games a group of students from two classes bought in a certain month. The results were recorded in the following table:

Number of Games 0 1 2 3 4 5 6 7

Class A 2 3 4 6 7 4 3 1

Class B 1 3 4 5 6 7 2 2

(a) Calculate the mean number of games bought in Class A.

(b) Calculate the mean number of games bought in Class B.

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(c) Choose a suitable graphical method to display both sets of data.

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(d) Compare both sets of data in terms of the shape of the distribution, the range of values and the median of each set.

Shape: Range: Median:

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Question 6 (Suggested maximum time: 13 minutes) The length of time taken by a group of students to complete an obstacle course during a school sports day was recorded as follows:

BOYS GIRLS

180 155 165 168 187 181 171 165 183 143

150 177 151 177 142 151 158 141 175 165

164 176 181 142 177 142 164 183 158 158

169 168 150 179 150 156 181 152 165 183

168 174 184 168 156 143 164 172 159 176

173 155 178 141 184 187 153 177 187 166

(a) Draw a back-to-back stem and leaf plot to display both sets of data. Indicate the key you have used clearly. Boys: Girls: 14 15 16 17 18 Key:

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(b) Comment on the shape of both distributions.

(c) Calculate the inter-quartile range for both sets of data.

Boys: Girls:

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(d) On a second run of the course it was found that the average time taken for girls to complete the course dropped by 5 seconds while the time taken by the boys remained unchanged. Compare the mean value for both sets of data on the second run of the course.

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Question 7 (Suggested maximum time: 8 minutes) A gardener recorded the height in centimetres of a number of pea plants in his greenhouse after two weeks of growing. He recorded the results in the following table:

Height (cm) 0-10 10-20 20-30 30-40 40-50

No. of Plants 3 7 14 21 15

(a) How many pea plants were growing?

(b) Taking mid-interval values, calculate the mean height of the plants correct to one decimal place. (c) What is the greatest number of plants that could be taller than 25 cm?

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Question 8 (Suggested maximum time: 10 minutes) The map of Dublin below has points for a road race marked. The race will start at the Red Cow Roundabout (Point A) and travel via Harold’s Cross (Point B), Ringsend (Point C), Dublin City Centre (Point D), Phoenix Park (Point E) and will finish in Castleknock (Point F). (a) Write down the co-ordinates of each point on the map.

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(b) The route was designed to have two sections of equal length. Verify that Harold’s Cross is half way between the Red Cow Roundabout and Ringsend.

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Question 9 (Suggested maximum time: 8 minutes) The lines l, m and p are shown on the coordinate diagram opposite.

(a) If the slopes of the three lines shown are 2 3 5, and3 2 4

− assign each line its correct slope.

LINE SLOPE

l

m

p

(b) Prove that two of the lines are perpendicular.

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Question 10 (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 three equal segments showing all construction lines clearly.

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Question 11 (Suggested maximum time: 8 minutes) Two tourist buses depart Galway city. One bus travels to the Cliffs of Moher while the other visits Bunratty Castle in Limerick. Both buses finish their journey at the same hotel near the Lakes of Killarney Prove that both buses travel the same distance.

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Question 12 (Suggested maximum time: 10 minutes) Prove that the angle at the centre of circle standing on a given arc is twice the angle at any point of the circle standing on the same arc.

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Question 13 (Suggested maximum time: 10 minutes) A set of goals has dimensions 3.6 m 1.7 m 0.5 m× × . The back support of the goal forms a triangle as shown where 90CAB∠ = ° and 71ABC∠ = ° . Calculate the BC correct to one decimal place.

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Question 14 (Suggested maximum time: 10 minutes) A goalie kicks out a ball from point A as shown below. The ball reaches its maximum height at the point B and hits the ground at the point D. (a) Find the height of the ball above the ground at the point B, correct to the nearest metre.

6 m

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(b) Find the total horizontal distance travelled by the ball, correct to the nearest metre.

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Question 15 (Suggested maximum time: 10 minutes) A tourist is taking photographs of the Leaning Tower of Pisa. He stands at the point E, a distance of 65.17 m from the point C. He then moves nearer the tower and takes a picture at the point F a distance of 56.44 m from the point D.

(a) Calculate the height of the Leaning Tower of Pisa on the lower side [ ]BC , correct to two decimal places.

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(b) Calculate the difference in height above ground between the low side [ ]BC and the high

side [ ]AD of the tower, correct to two decimal places.

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