Mathematics (Project Maths Phase 2)

Transcription

1 L.17 NAME SCHOOL TEACHER Pre-Leaving Certificate Examination, 2013 Mathematics (Project Maths Phase 2) Paper 1 Higher Level Time: 2 hours, 30 minutes 300 marks For examiner Question 1 Centre stamp Mark Running total 8 Grade Total 2013 L.17 1/20 Page 1 of 19

2 Instructions There are three sections in this examination paper: Section A Concepts and Skills 100 marks 4 questions Section B Contexts and Applications 100 marks 2 questions Section C Functions and Calculus (old syllabus) 100 marks 2 questions Answer all eight questions. Write your answers in the spaces provided in this booklet. You will lose marks if you do not do so. 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 the make and model of your calculator(s) here: 2013 L.17 2/20 Page 2 of 19 Project Maths, Phase 2

3 Section A Concepts and Skills 100 marks Answer all four questions from this section. Question 1 (25 marks) Let z 1 1 i and z 2 2 2i, where i 2 1. Im (a) Find z 1.z 2 and hence plot z 1, z 2 and z 1.z 2 on an Argand diagram. Re (b) Express z 1, z 2 and z 1.z 2 in polar form. (c) Using your answers to parts (a) and (b), explain what happens when you multiply two complex numbers. (d) Use De Moivre s theorem to evaluate (z 1 ) 6, giving your answer(s) in rectangular form. page running 2013 L.17 3/20 Page 3 of 19 Project Maths, Phase 2

4 Question 2 (25 marks) Future population size can be described using the exponential equation P(t) Ae bt, where A and b are constants. The size of population size P(t) can be determined at various points in time t. The population of a certain village was 1500 in 2000 and 3560 in (a) Find the value of a. Find the value of b, correct to three decimal places. (b) Determine the population size of the village in 2020, correct to three significant figures. (c) During what year will the population of the village reach ? 2013 L.17 4/20 Page 4 of 19 Project Maths, Phase 2

5 Question 3 (25 marks) (a) Solve the simultaneous equations: x y z 2x 3y 2z x 2y 10. (b) (i) Write the following as a single fraction: 1 1. x 2y (ii) Hence, or otherwise, show that 1 1 (x 2y) 4, x 2y given that x, y 0 and x, y Z. page running 2013 L.17 5/20 Page 5 of 19 Project Maths, Phase 2

6 Question 4 (25 marks) (a) Write a polynomial function for the following graph in its simplest form. 160 y x 80 (b) (i) Using the same axis and scales, sketch graphs of the functions f : x x 6 and g : x x 2. (ii) Use your graph to solve the inequality x 6 x 2. (iii) Verify your answer algebraically L.17 6/20 Page 6 of 19 Project Maths, Phase 2

7 Section B Contexts and Applications 100 marks Answer both Question 5 and Question 6. Question 5 (50 marks) A clothing company produces one type of shirt. Market research has found that if the company prices the shirts at 30 each, they will sell 500 units per week. It was also found that if the price was set at 55 each, the company will sell none. The clothing company prices the shirts at x each, where 30 x 55. (a) Draw a straight line graph to represent possible sales per week. Quantity Price ( ) (b) Find an expression for sales per week, in terms of x. page running 2013 L.17 7/20 Page 7 of 19 Project Maths, Phase 2

8 (c) Write an expression for the value in euro of weekly sales, in terms of x. (d) Given that the clothing company s fixed costs are 2000 per week and production costs are 20 for each shirt, find an expression for costs per week. (e) Show that the weekly profit is 20x x L.17 8/20 Page 8 of 19 Project Maths, Phase 2

9 (f) A graph of the clothing company s weekly profits as a function of x is shown below. Use the graph to determine the price that the company should charge in order to maximise profits. 6,000 Profit ( ) 4,000 2, x ,000 (g) Hence, calculate the number of shirts that will sell per week at this price. page running 2013 L.17 9/20 Page 9 of 19 Project Maths, Phase 2

10 Question 6 (50 marks) Lisa has won a major prize in a lottery game. When she goes to collect her prize, she is offered one of the following options: Option A: Option B: Receive a payment of 1500, at the beginning of each month for 25 years. Receive a single payment lump sum immediately. Lisa is unsure of which option to take. Initial exploration: 2013 L.17 10/20 Page 10 of 19 Project Maths, Phase 2

11 (a) Lisa feels that if she takes option A, it may give her a regular income in the future. She plans to put the monthly payment in a bank while she decides what to do. The bank is offering a rate of interest which corresponds to an annual equivalent rate (AER) of 3 5%. Find the rate of interest per month that would, if paid and compounded monthly, correspond to an annual equivalent rate (AER) of 3 5%. (b) After three months, Lisa decides what she wants to do. She plans to continue saving all of the money as part of a pension for the future. Find the present values of the first three monthly payments lodged in her bank account. (c) Show that the total value of Lisa s pension, assuming an annual equivalent rate (AER) of 3 5% over the period of the payments, can be represented by a geometric series. page running 2013 L.17 11/20 Page 11 of 19 Project Maths, Phase 2

12 (d) Alternatively, Lisa could have accepted a single payment lump sum (option B). How much would this payment need to be to match the future value of Lisa s pension plan? (e) Lisa was worried that if she received a large sum of money, she would spend it carelessly and then it would be gone. Assuming that she would spend no more than 750 every month, how much better off would Lisa be if she accepted the single payment lump sum (option B) and save the remainder as a pension under the same conditions? 2013 L.17 12/20 Page 12 of 19 Project Maths, Phase 2

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14 Section C Functions and Calculus (old syllabus) 100 marks Answer both Question 7 and Question 8. Question 7 (50 marks) (a) Let f (x) x 3 6x k, where k Z and x R. Taking x 1 1 as the first approximation of a root of the function f (x) 0 and x 2 2 as the second approximation of this root, use the Newton-Raphson method to find the value of k. (b) (i) Differentiate sin x with respect to x from first principles L.17 14/20 Page 14 of 19 Project Maths, Phase 2

15 (ii) Given that y sin 2x dy, find the value of when x 0. 2(cos x sin x) dx page running 2013 L.17 15/20 Page 15 of 19 Project Maths, Phase 2

16 (c) A curve is defined by the equation x 2 y xy 2 6. (i) Find dy in terms of x and y. dx (ii) Determine whether the tangents are parallel at the point x = L.17 16/20 Page 16 of 19 Project Maths, Phase 2

17 Question 8 (50 marks) (a) Find 4 2 x x 3 dx. 2 x (b) (i) Evaluate 2 1 dx 5 4x x 2. (ii) 3 2x 1 Evaluate dx. x 1 0 page running 2013 L.17 17/20 Page 17 of 19 Project Maths, Phase 2

18 (c) The diagram shows the curve y x 1, y and the line 4x 8y Calculate the area of the shaded region enclosed by the curve and the line. x 2013 L.17 18/20 Page 18 of 19 Project Maths, Phase 2

19 You may use this page for extra work. page running 2013 L.17 19/20 Page 19 of 19 Project Maths, Phase 2

20 Pre-Leaving Certificate 2013 Higher Level Mathematics (Project Maths Phase 2) Paper 1 Time: 2 hours, 30 minutes 2013 L.17 20/20 Page 20 of 19 Project Maths, Phase 2