Mathematics. Project Maths - Phase 3. Ordinary Level. Paper written by Pat Nevin and S. King

Transcription

1 Leaving Certificate Examination Sample paper prepared by Leamy Maommunity Mathematics Project Maths - Phase 3 Paper 1 Ordinary Level Paper written by Pat Nevin and S. King Leamy Maommunity 300 marks 1

2 Sample 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 questions as follows: In Section A, answer all six questions. In Section B, answer all three 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. You must re- The superintendent will give you a copy of the booklet of Formulae and Tables. turn 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: 2

3 Section A Answer all six questions from this section. Question 1 (a) (i) Solve 5 + 2(x 1) = x + 4(x 3) Concepts and Skills (ii) Verify your answer from part i in the box below. (b) Use simultaneous equations to solve these two equations 2x 5y = 19 3x + 4y = Marks (25 Marks)

4 Question 2 (25 Marks) (a) Find z by completing the pyramids shown below. Each number is calculated by adding the values in the two bricks directly below it. Figure 1:. Figure 2:. (b) Solve the equation z 2 6z + 13 = 0 4

5 (c) Given the complex numbers z = 1 + i, i 2 = 1 and z 4 = 4 complete the table below. Index Notation z 0 z 1 z 2 z 4 z 8 z 16 z 32 Numbers 4 5

6 Question 3 (25 Marks) (a) The Fibonacci Sequence is the series of numbers :0, 1, 1, 2, 3, 5, 8, 13, 21, 34,... The next number is found by adding the two numbers before it. The 2 is found by adding the two numbers before it (1+1). Similarly, the 3 is found by adding the two numbers before it (1+2) and so on. (i) Given this information work out the 11th term (b) The table below shows the hours John worked on Thursday, Friday, Saturday and Sunday. Days Thursday Friday Saturday Sunday Hours Worked h John s basic rate of pay is e15.60 per hour. He is paid one and a half times basic rate for work on Saturday and Sunday. (i) Calculate John s total pay for Thursday, Friday and Saturday. 6

7 (ii) John was paid a total of e702 for the four days work. Find h, the number of hours John worked on Sunday. (iii) The weekly standard rate of tax paid on income is 20% from e0 to e650 and 40% on the balance. John s weekly tax credits are e40. Calculate John s take home pay after tax. 7

8 Question 4 (25 Marks) (a) An airplane leaves Shannon airport. It flies for six and a half hours and lands in JFK New York. The distance between the two airports is 4, 596km. Find the average speed of the airplane in km/h. (b) During the flight, the airplane uses 240 litres of fuel per minute. How many litres of fuel were used in the flight? (c) For emergencies the airplane must carry 20% more fuel than it requires. Find the total amount of fuel carried by the airplane 8

9 (d) If the weight of the aircraft before take-off is 60, 000 tonnes including aviation fuel what would the airplane weigh on landing in JFK. (note: 1 tonne = 1000kg 1000 litres) (e) The plane continues on an internal flight in US, distances are in miles. The aircraft flies 930 miles in 75 minutes. How many miles does it fly in 4 hours 45 minutes to California assuming a constant speed? (f) Convert this distance in miles to kilometres, taking 5 miles to be equal to 8 kilometres. 9

10 Question 5 (25 Marks) (a) In a primary school, students are forming patterns from black and gray discs, their teacher lays out a repeating pattern from black and grey disks as shown below. (i) Draw the next pattern. (ii) Using arithmetic sequences or otherwise find T n, the nth pattern for black discs. 10

11 (iii) How many grey discs are needed for the 10 th pattern? (iv) How many black discs are needed for the 15 th pattern? (v) If there are 590 discs in total, how many complete patterns could be made? 11

12 Question 6 (a) Graph the function f(x) = x 3 3x 2 4x + 12 in the domain 3 x 4 (25 Marks) 12

13 (b) From your graph, identify where f(x) = 0? (c) Using your graph, identify the range of values for which f (x) <

14 Section B Contexts and Applications 150 Marks Answer all three questions from this section. Question 7 (70 Marks) (a) The graph of four functions are shown below. The graphs are labled A, B, C and D. The four functions are listed in the table below the graphs. Match the graphs to the functions, by putting the correct letter beside each one in the Table Function f(x) = x 2 x 2 g(x) = 2x + 4 h(x) = x 3 3x 2 9x + 1 i(x) = 2 x + 1 Graph 14

15 (i) From the graph D above, estimate the max and min points in the form (x, y) (ii) Verify your answer for the max an min using differentiation techniques. (iii) From the graph of the quadratic function, identify its roots. 15

16 (iv) Find the equation of the tangent to the quadratic function at the point (2, 0). (v) What is the area enclosed by the linear function between the x axis, y axis and the origin? 16

17 Question 8 (a) A company sells mp3 players. The function: P (Q) = 3Q Q 93 (80 Marks) Represents the profit to be made by selling Q mp3 players, where Q is the number of mp3 players sold (in 1,000 s). (i) What is the profit from selling 10 mp3 players? (ii) How many mp3 players should the company sell in order to maximise profit? (iii) What is the maximum profit? 17

18 (b) A Norman window has a shape of a rectangle surmounted by a semicircle of a diameter equal to the width of the rectangle. The perimeter of the window is 10 m. (i) Assume that 2x represents the base length of the rectangle part of the window and that y represents the height of the rectangle part. Taking π = 3.14, show that y = x.(Hint: Circumference of a circle l = 2πr) 18

19 (ii) Starting with the expression A = 1 2 πx2 + 2xy, and by taking π = 3.14, show that the area can be written as A = 10x 3.57x 2. (iii) What dimension should the windows have to allow the maximum amount of light in? 19

20 Rough Work 20