Mathematics (Project Maths Phase 3)

*B7* Pre-Leaving Certificate Examination, 2014 Triailscrúdú na hardteistiméireachta, 2014 Mathematics (Project Maths Phase 3) Paper 2 Ordinary Level 2½ hours 300 marks Name: School: Address: Class: Teacher: For examiner Question Mark 1 2 3 4 5 6 7 8 Total Page 1 of 24

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: Page 2 of 24

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? Page 3 of 24

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. Page 4 of 24

Question 3 (25 marks) (a) If the point ( ) 4,t lies on the line 3x+ 2y = 6, find the value of t. (b) Show that the points ( 2,7), ( 4,3 ) and ( 3,8 ) form an isosceles triangle. Page 5 of 24

(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 ax + by + c = 0. Page 7 of 24

Question 4 (25 marks) (a) Match each circle with its correct equation. Equation Circle ( x ) ( y ) 2 2 + 3 + 4 = 4 ( x ) ( y ) 2 2 5 + 4 = 16 ( x ) ( y ) 2 2 5 + + 2 = 1 2 2 x + y = 1 Page 8 of 24

(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 ax + by + c = 0. Page 9 of 24

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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. Page 11 of 24

Question 6 (25 marks) Answer either 6A or 6B. Question 6A (a) Construct the circumcircle of the triangle shown. Page 12 of 24

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 CE. [ AD ] and [ ] Show that Δ ADC and Δ DBE are similar. Page 13 of 24

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. 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 (Source: Apple Inc. Financial statements) (a) Calculate the total number of ipads sold since its release. Page 14 of 24

(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. Page 15 of 24

(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. Page 16 of 24

(f) Calculate the mean number of ipads sold per quarter for 2012. Page 17 of 24

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. Page 18 of 24

(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. Page 19 of 24

(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. Page 20 of 24

(b) A marksman begins aiming at a clay pigeon when the angle of elevation is 29. (i) Calculate AB. Page 21 of 24

(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. Page 22 of 24

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