Mathematics (Project Maths – Phase...

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Pre-Leaving Certificate Examination, 2014

Triailscrúdú na hArdteistiméireachta, 2014

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Mathematics (Project Maths – Phase 3)

Paper 2

Ordinary Level

2½ hours

300 marks

*B7*

For examiner Question Mark

1 2 3 4 5 6 7 8

Total

Name:

School:

Address:

Class:

Teacher:

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Instructions There are two sections in this examination paper: Section A Concepts and Skills 150 marks 6 questions Section B Contexts and Applications 150 marks 2 questions Answer all eight questions, as follows: In Section A, answer:

Questions 1 to 5 and

either Question 6A or Question 6B.

In Section B, answer Question 7 and Question 8. Write your answers in the spaces provided in this booklet. You may lose marks if you do not do so. 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. 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 down the make and model of your calculator(s) here:

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Section A Concepts and Skills 150 marks

Answer all six questions from this section. Question 1 (25 marks) 80 patients took part in a medical trial. 54 were given treatment A, 47 were given treatment B and 24 were given both treatment A and treatment B. (a) Draw a Venn diagram to represent this data. (b) How many patients received neither treatment? (c) If a patient is chosen at random what is the probability they are receiving treatment B only?

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Question 2 (25 marks) The scores of 20 students in a maths exam were recorded as follows.

79 10 89 82 83 90 6 89 91 85 12 90 100 95 3 99 89 89 84 93

(a) Calculate the mean mark for the class. (b) Write down the modal mark in the class. (c) Calculate the median mark in the class. (d) Identify the outliers in the class. Explain your answer. (e) Which of the measures of central tendency are most affected by the outliers? Explain your answer fully.

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Question 3 (25 marks) (a) If the point ( )4, t lies on the line 3 2 6x y+ = , find the value of t. (b) Show that the points ( )2,7− , ( )4,3 and ( )3,8 form an isosceles triangle.

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(c) The line l is shown below.

(i) Find the slope of the line l.

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(ii) The line n is perpendicular to the line l and passes through the point B. Find the equation of the line n in the form 0ax by c+ + = .

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Question 4 (25 marks) (a) Match each circle with its correct equation.

Equation Circle

( ) ( )2 23 4 4x y+ + − =

( ) ( )2 25 4 16x y− + − =

( ) ( )2 25 2 1x y− + + =

2 2 1x y+ =

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(b) Write down the equation of a circle which has centre ( )3, 2− and radius 5. (c) Find the equation of the tangent to the circle shown at the point C in the form

0ax by c+ + = .

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Question 5 (25 marks) A cylinder of height 60 cm and radius 10 cm is full with water. All of the water is poured into two conical containers. The first conical container has a radius of 9 cm and a height of 8 cm. If the second conical container has a height of 18 cm, calculate its radius, correct to nearest whole number.

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Question 6 (25 marks) Answer either 6A or 6B. Question 6A (a) Construct the circumcircle of the triangle shown.

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OR Question 6B A, C, D and E are four points on a circle with centre O as shown. [ ]AD bisects CDE∠ . B is the point of intersection of [ ]AD and [ ]CE . Show that ADCΔ and DBEΔ are similar.

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Section B Contexts and Applications 150 marks

Answer both Question 7 and Question 8. Question 7 (75 marks) The first iPad was released by Apple Inc. on April 3 2010. Since its release it has become a desired item by technology enthusiasts worldwide. The table below shows the sales figures (in thousands of units) per quarter of the iPad since its release.

(Source: Apple Inc. Financial statements) (a) Calculate the total number of iPads sold since its release.

2010 2011 2012 2013

Quarter 1 7,331 15,434 22,900

Quarter 2 4,694 11,798 19,500

Quarter 3 3,270 9,246 17,042 14,600

Quarter 4 4,188 11,123 14,036

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(b) Is the data in the table primary or secondary data? Explain your answer fully. (c) Use a suitable graphical display to display the data in the table.

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(d) The fourth generation iPad, the iPad 4, was released on September 12, 2012. Discuss the changes in the sales figures for 2012 with reference to this information.

(e) The first quarter in 2013 saw a record 22.9 million iPads sold by Apple but numbers have

fallen in the subsequent quarters. Suggest a reason why this may have occurred.

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(f) Calculate the mean number of iPads sold per quarter for 2012.

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Question 8 (75 marks) (a) A carpenter is designing a dog house as shown. The front of the dog house is made from

a rectangular sheet of wood. The door is in the shape of a rectangle surmounted by a semicircle with centre K. The roof is manufactured separately and is placed on the dog house when complete.

(i) Calculate the area of the rectangular piece of wood used.

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(ii) Calculate the width of the door in the dog house.

(iii) Calculate BD , correct to the nearest centimetre.

(iv) Calculate the area of the semicircular part of the door.

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(v) Calculate ABD∠ .

(vi) Given that BK is twice the radius of the semicircle, calculate the amount of wood wasted in making the front of the dog house.

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(b) A marksman begins aiming at a clay pigeon when the angle of elevation is 29° .

(i) Calculate AB .

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(ii) The marksman shoots the clay pigeon at the point D, which is 1.5 m above the point B. Calculate the horizontal distance the clay pigeon has travelled, correct to two decimal places.

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