# Mathematics (Project Maths Phase 3)

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1 *B6* Pre-Leaving Certificate Examination, 2014 Triailscrúdú na hardteistiméireachta, 2014 Mathematics (Project Maths Phase 3) Paper 1 Ordinary Level 2½ hours 300 marks Name: School: Address: Class: Teacher: For examiner Question Mark Total Page 1 of 24

2 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 3 questions Answer all nine questions. Write your answers 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 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

3 Section A Concepts and Skills 150 marks Answer all six questions from this section. Question 1 (25 marks) (a) Solve the equations and verify your solutions. 3x+ 2y = 7 5x + 37= 3y Page 3 of 24

4 (b) Solve the inequality 3x+ 5 6x 10, where x. (c) Show the solution set to part (b) on the number line below. Page 4 of 24

5 Question 2 (25 marks) The width of a rectangular vegetable patch is x metres. Its length is 5 metres longer than its width. (a) Write an expression in x to represent the length of the vegetable patch. (b) Show that the area of the patch can be expressed in the form 2 x + 5x. Page 5 of 24

6 (c) If the area of the patch is 36 m, find the dimensions of the garden. Page 6 of 24

7 Question 3 (25 marks) (a) (i) Simplify 4( 2 3i) 2( i 4) i( 5 2i) +. (ii) If 4+ ti = 5, find two possible values for tt,. (b) Identify each complex number, z 1, z 2 and z 3, on the complex plane below given that: z z 1 = z 2 = i z 2 1 ( ) 3 1 Im( z) Re( z) Page 7 of 24

8 Question 4 (25 marks) (a) Differentiate 3 2 3x 2x 5x with respect to x. (b) As a plane travels down a runway its distance from 2 a marker is given by s() t = 2t + 5t 20, where s is measured in metres and t is measured in seconds. (i) How far does the plane travel in the first 10 seconds? Page 8 of 24

9 (ii) Find the speed of the plane in terms of t. (iii) After how many seconds is the speed of the plane 85 m/s? Page 9 of 24

10 Question 5 (25 marks) The graph of the function ( ) 2 f x = x 7x+ 10 is shown below in the domain 1 x 6. (a) Using the graph write down the roots of f ( x ) = 0. (b) Find f '( x ), the derivative of f ( x ). Page 10 of 24

11 (c) For what value of x will f '( x ) = 0. (d) Which of the graphs below represent f '( x ), the derivative of ( ) Explain your answer fully. f x. Answer: Reason: Page 11 of 24

12 Question 6 (25 marks) A woman has a gross yearly income of 60,000. Her standard rate cut-off point is 27,000 and she has tax credits of 3,200. The standard rate of tax is 20% of income up to the standard rate cut-off point and 41% on all income above the standard rate cut-off point. (a) Calculate the tax payable at the standard rate of tax. (b) Calculate the amount of gross tax payable for the year. Page 12 of 24

13 (c) Calculate the woman s net yearly income. (d) The woman saves 11% of her net income each month. Calculate her yearly savings correct to the nearest euro. Page 13 of 24

14 Section B Contexts and Applications 150 marks Answer all three questions from this section. Question 7 (50 marks) An indoor cycling race has a rolling start. The speed of two competing cyclists, in metres per second, as they crossed the starting line each lap was recorded as follows. Cyclist A has a starting speed of 2 m/s and accelerates uniformly by 1.5 m/s 2 during the first 5 seconds. Cyclist B has a starting speed of 1.5 m/s and accelerates uniformly by 2.5 m/s 2 during the first 5 seconds. (a) Complete the following table to show the speed of each cyclist during the first 5 seconds of the race. Time (seconds) Cyclist A Cyclist B (b) Display the change in speed of both cyclists on a suitable graph. Page 14 of 24

15 Page 15 of 24

16 (c) Use your graph to find: (i) the speed of both cyclists after 4.5 seconds. Cyclist A Cyclist B (ii) the time when cyclist B overtakes cyclist A. Page 16 of 24

17 (d) Find the equations that model the speeds of both cyclists. Cyclist A Cyclist B (e) Find algebraically the time when the speeds of the cyclists are equal. Page 17 of 24

18 Question 8 (60 marks) In the first month after opening, a mobile phone shop sold 300 phones. A model for future trading assumes that sales will increase by x phones per month for the next 35 months. (a) Write down an expression to represent the number of phones expected to be sold in the second month. (b) Write down an expression to represent the number of phones expected to be sold in the third month. (c) Write down an expression to represent the number of phones, P n, expected to be sold in the n th month. Page 18 of 24

19 (d) Paul states the sequence is arithmetic, Jake disagrees. Is Paul correct? Explain your answer fully. (e) Using this model with x = 6, calculate the number of phones sold in the 36 th month. Page 19 of 24

20 (f) Calculate the total number of phones sold over the 36 months. (g) The shop sets a sales target of 20,000 phones to be sold over the 36 months. Find the least value of x required to achieve this target. Page 20 of 24

21 Question 9 (40 marks) An open-top box is cut from a sheet of cardboard with dimensions as shown. A square of side x cm is cut from each corner. (a) Write down the length of the side of the box in terms of x. (b) Write down the width of the side of the box in terms of x. Page 21 of 24

22 (c) Write down an expression in x to represent the volume of the box. (d) Find, correct to two decimal places, the value of x that will maximise the volume of the box. Page 22 of 24

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