pm61 - jc-maths-higher-paper 2 · number of items 0 – 4 4 – 10 10 – 20 20 – 30 30 – 50...

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Pre-Junior Certificate Examination, 2013

Triailscrúdú an Teastais Shóisearaigh, 2013

_______________

Project Maths

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

Total

Name:

School:

Address:

Class:

Teacher:

*PM61*

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Instructions There are fourteen 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 and 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: 20 minutes) (a) The volume of a hemisphere of is 144π cm3. Find the radius of the hemisphere.

(b) A square of side length 10 cm is inscribed in a circle,

as shown in the diagram.

Calculate:

(i) The radius of the circle in simplest surd form.

(ii) The area of the shaded region correct to one decimal place.

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(c) A solid cone of wax has a radius of 4 cm and has a slant height of 52 cm.

(i) Find the volume of wax in the cone in terms of π . (ii) The cone is melted and all the wax is used to make a cylindrical candle of

height 4 cm. Calculate the radius of this candle and write your answer in the form a b where ,a b∈ .

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Question 2 (Suggested maximum time: 10 minutes) Below are two fair circular spinners, X and Y. Spinner X has 4 equal sectors labelled A, B, C, D. Spinner Y has 5 equal sectors labelled 1, 2, 3, 4, 5. Spinner X Spinner Y Both spinners are spun. (a) What is the probability of Spinner X landing on the letter A? (b) What is the probability that the letter B on Spinner X and the number 5 on Spinner Y are

spun?

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(c) What is the probability that the letter D on Spinner X and an even number on Spinner Y is spun? (d) The number of possible outcomes when both these spinners are spun is 20. By adding extra letters to Spinner X and adding extra numbers to Spinner Y, outline how the number of outcomes can be increased to 42.

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Question 3 (Suggested maximum time: 10 minutes) On a sunny day a man of height 1.65 metres casts a shadow 4.29 metres in length. At the same time a building casts a shadow 6.51 metres longer than the man’s shadow. (a) Explain how this information can be used to find the height of the building. (b) Calculate the height of the building correct to one decimal place. (c) Calculate the angle of elevation of the sun, at the time the measurements were taken, to

the nearest degree.

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Question 4 (Suggested maximum time: 5 minutes)

(a) Construct a right-angled triangle containing an angle A such that 4cos7

A = .

(b) Calculate, from your triangle, sin A in surd form.

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Question 5 (Suggested maximum time: 5 minutes) (a) Write down the equation of the line l. (b) On the same graph draw the line k through ( )0, 4 with a slope of 3− .

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Question 6 (Suggested maximum time: 10 minutes) A P.E. teacher asked the members of his basketball squad to do a fitness test, called a Beep Test. Higher scores in the Beep Test indicate higher fitness levels. The results were as follows:

Test 1: 6 8, 8 3, 7 3, 6 3, 6 9, 6 6, 7 8, 7 2, 8 2, 7 5, 6 9, 7 5⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ The teacher put the squad on an eight week fitness programme and then at the end of the eight week re-tested the squad members. The results for the second Beep Test were as follows:

Test 2 : 7 4, 9 5, 9 2, 7 5, 8 2, 8 3, 8 9, 9 1, 7 8, 8 7, 8 5, 7 7⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ (a) Construct a back-to-back stem-and-leaf plot of the above data. Test 1 Test 2 (b) How many members were in the squad? (c) What is the range for the squad? Test 1: Test 2: (d) Based on the data and the diagram, do you think that the 8-week fitness programme

Improved the squad member’s results in the Beep Test? Justify your answer.

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Question 7 (Suggested maximum time: 5 minutes) In a town in America, Mario’s Pizza chain has two pizza restaurants located at A and B. The company runs a delivery service and when orders are received they are sent from the nearest restaurant. By using an appropriate construction, show how to partition the town so that orders are sent from the nearest available restaurant.

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Question 8 (Suggested maximum time: 10 minutes) A survey was done of shoppers, in a supermarket, concerning the number of items they bought while doing their shopping. The results are shown in the table below.

Number of Items 0 – 4 4 – 10 10 – 20 20 – 30 30 – 50

Shoppers 15 10 x 10 9

(Note: 10 − 20 means 10 items or more but less than 20 items.) Taking mid-interval values, it was found that the mean number of items bought by each shopper was 16 items. Find the value of x.

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Question 9 (Suggested maximum time: 10 minutes) Over a three hour period the age and sex of 200 visitors to an Aqua Park were recorded as follows.

Category Frequency Relative Frequency Daily Frequency

for Saturday (Part (d) below)

Male aged under 10 32

Male aged 10 to 25 35

Male aged over 25

Female aged under 10 45

Female aged 10 to 25 28

Female aged over 25 34

(a) Calculate the number of males, aged over 25, that attended the park and write it into

the table. (b) Calculate the relative frequency of each category and write these into the table.

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(c) Based on the information in the table above what is the probability that the next visitor to the centre is female and over 10.

(d) On a busy Saturday in summer the centre had 1800 visitors. Using the information above fill

in the expected daily frequency for Saturday in the final column of the table on the previous page.

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Question 10 (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.

Diagram:

Given: To Prove: Construction: Proof:

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(b) In the diagram below, AC and BD are diameters of the circle with centre O. Diameters AC and BD are perpendicular to each other. Using the theorem above, or otherwise, prove that 2 2 24AC BD AB+ = .

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Question 11 (Suggested maximum time: 5 minutes) Find the measures of the angles labelled P and Q in the diagram below given that k n .

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Question 12 (Suggested maximum time: 10 minutes) The point P is shown on the diagram. (a) Write down the co-ordinates of P. (b) On the same diagram plot

the following points:

Q R S

( )1, 1− ( )4, 4− ( )1, 6−

(c) Find the slope of the line QR. (d) Find PR and QS and leave your answer in surd form.

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(e) A student was asked to check if the shape PQRS is a parallelogram. The student concluded that the shape was not a parallelogram because the two distances in question (d) were different. Can you say why the student’s reasoning is incorrect? (f) PQRS is in fact a parallelogram. Using co-ordinate geometry or otherwise, show clearly how you would justify that it is a parallelogram.

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Question 13 (Suggested maximum time: 10 minutes) The Wellington monument is right in the middle of the Phoenix Park in Dublin County. This type of monument is called an obelisk and the Wellington monument is Europe’s tallest obelisk. The base of the steps around the monument form a square with side of length 34m. Bernie, who is 1.55 metres tall, walks away from the bottom of the steps of the monument on the path in the diagram. Bernie stops when the perpendical distance between herself and the middle of the square base of the monument is 30m. At this point Bernie uses a clinometer and finds that the angle of elevation of the top of the monument is 52o. (a) Explain how Bernie can use her

measurements and the information about the base of the monument to find the full height of the monument from the ground to it’s top.

(b) Draw a suitable diagram and calculate the height of the monument to the nearest metre.

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Question 14 (Suggested maximum time: 10 minutes) X, Y, Z, W are points on a circle as shown. O is the centre of the circle.

150XOZ∠ = ° and XW XY= Find: (a) XWY∠ . (b) XYZ∠ . (c) By joining W to Z, or otherwise, find OWZ∠ .

pm61 - jc-maths-higher-paper 2 · number of items 0 – 4 4 – 10 10 – 20 20 – 30 30 – 50...

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