Mathematics. Pre-Leaving Certificate Examination, Paper 1 Higher Level. Time: 2 hours, 30 minutes. 300 marks. Name. School. Teacher.

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1 Name School Teacher Pre-Leaving Certificate Examination, 2015 Mathematics Paper 1 Higher Level Time: 2 hours, 30 minutes 300 marks For examiner Question Mark 1 2 Centre Stamp 3 Running Total Total Grade page running 1

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. You may 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. You will lose marks 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: 2

3 Section A Concepts and Skills 150 marks Answer all six questions from this section Question 1 (a) The equation, where p is a positive constant, has equal roots. (25 marks) (i) Find the value of p. (ii) For this value of p, solve the equation. (b) Find the equation whose roots are three times those of. page running 3

4 Question 2 (25 marks) (a) If and, find the exact value of arg(w). (b) Show, using De Moivre s Theorem, that. 4

5 Question 3 (25 marks) (a) An athlete prepares for a race by completing a practice run on each of 11 consecutive days. On each day after the first day, he runs further than he ran on the previous day. The lengths of his 11 practice runs form an arithmetic sequence with first term a km and common difference d km. He runs 9 km on the 11th day, and he runs a total of 77 km over the 11 day period. Find the value of a and the value of d. (b) (i) Explain what is meant by proof by contradiction. Give an example in your answer. (ii) Prove, using mathematical induction, that 8 is a factor of, for. page running 5

6 Question 4 (25 marks) The curve C with equation,, passes through the point (3, ). Given that, (a) Find. (b) Verify that. (c) Find an equation for the tangent to C at the point ( 2, 5), giving your answer in the form where a, b and c are integers. 6

7 Question 5 (a) If both 11 2 and 3 3 are factors of the number smallest possible value of a? Explain your answer fully., then what is the (25 marks) (b) If x, y, z are chosen from the three numbers and 2, what is the largest possible value of the expression? (c) If and find x and y. page running 7

8 Question 6 (25 marks) (a) (i) Given that, complete the table below, giving the values of y to 2 decimal places (ii) Use the trapezoidal rule, with all the values of y from your table, to find an approximate value for (iii) The diagram above shows a sketch of part of the curve with equation. 8

9 At the points A and B on the curve, x = 2 and x = 3 respectively. The region S is bounded by the curve, the straight line through B and (2, 0), and the line through A parallel to the y-axis. The region S is shown as the shaded region. (c) Use your answer to part (b) to find an approximate value for the area of S. page running 9

10 Section B Contexts and Applications 150 marks Answer all three questions from this section. Question 7 (45 marks) (a) The graph below can be used as a model to predict the number of bacteria at the end of each week in a contaminated pond, if their growth is not controlled. y is the predicted number of bacteria, in thousands, in the contaminated pond and x is the number of weeks since the number of bacteria was first recorded. (i) Find the increase in the number of bacteria from the end of the first week (x = 1) to the end of the fourth week if their growth rate is not controlled. 10

11 (ii) At the end of the first week, a scientist adds some chemical to the pond. The chemical reduces the growth rate of the bacteria by 80% each week for 5 weeks. How many bacteria would be expected to be in the pond at the end of 5 weeks of treatment? (b) The diagram shows a container with a width of (h + 3) metres, and length (2h + 1) metres. The length of the diagonal of the base of the container is d metres. Show that the height is given by. page running 11

12 Question 8 (55 marks) (a) Write down the present value P of an investment with future value F in n years with an interest rate of r% per annum. (b) What sum of money, to the nearest euro, must a person invest at 4% per annum to give a lump sum of 60,000 in seven years? (c) A new company, PiedPiper, develops a new music app for the Smartphone market based on a newly developed high compression formula. A rival company offers to buy PiedPiper for 120 million now or pay 200 million for it in seven years time. (i) If the interest rate is 5% should the owners of PiedPiper sell now or not? Explain your answer fully. 12

13 (ii) At what interest rate would it make no difference for PiedPiper to sell now or in seven years time. Give you answer to the nearest whole number. (d) Write the recurring decimal as an infinite geometric series and hence as a fraction. page running 13

14 Question 9 (50 marks) (a) A cuboid has a rectangular cross-section where the length of the rectangle is equal to twice its width, x cm. The height of the rectangle is h cm. The volume of the cuboid is 81 cubic centimetres. (i) Find an expression in x for the height of the cuboid. (ii) Show that the total length, L cm, of the twelve edges of the cuboid is given by cm (iii) Use calculus to find the minimum value of L, and prove that this value is indeed a minimum. 14

15 (b) The pesticide DDT was used in Ireland until its ban in DDT is toxic to a wide range of animals and aquatic life, and is suspected to cause cancer in humans. The halflife of DDT is 15 years. Half-life is the amount of time it takes for half of the amount of a substance to decay. Scientists and environmentalists worry about such substances because these hazardous materials continue to be dangerous for many years after their disposal. The mass of DDT remaining, y, can be modelled by the function, a = the initial amount before the decay begins r = decay rate x = the number of intervals (i) Complete the table below, writing your y values correct to 1 decimal place (ii) Sketch the graph of y against x for and. page running 15

16 (c) and, where are two functions. Show that f(x) and g(x) are inverse functions. 16

17 You may use this page for extra work. page running 17

18 You may use this page for extra work. 18

19 You may use this page for extra work. page running 19

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Mathematics (Project Maths Phase 3)

2012. M329 S Coimisiún na Scrúduithe Stáit State Examinations Commission Leaving Certificate Examination, 2012 Sample Paper Mathematics (Project Maths Phase 3) Paper 1 Higher Level Time: 2 hours, 30 minutes