Name: Teacher: Unit 1 Maths Methods (CAS) Exam 2012 Thursday June 7-1.45 pm Reading time: 10 Minutes Writing time: 80 Minutes Instruction to candidates: Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a single bound exercise book containing notes and class-work, CAS calculator. Materials Supplied: Question and answer booklet, detachable multiple choice answer sheet at end of booklet. Instructions: Write your name and that of your teacher in the spaces provided. Answer all short answer questions in this booklet where indicated. Always show your full working where spaces are provided. Answer the multiple choice questions on the separate answer sheet. Section A Section B Total exam /20 /30 /50 1
Section A Multiple choice questions (20 marks) Question 1 Which of the following correctly shows the transformation of the equation PV = nrt to make n the subject? a) n = PV RT b) n = RT PV c) n = PV + RT d) n = PV! RT e) n = PVRT Question 2 A straight line has the equation 4y! 3x = 8. Another form of this equation is: a) y = 3x 8 + 4 b) y = 8x + 3 c) y = 3x 4 + 2 d) y = 3x 4 + 4 e) y = 3x 4 + 8 Question 3 A straight line segment joins the points (1,5) and (6,10). The distance between the points is: a) 5 2 b) 2 5 c) 274 d) 50 e) 1 2
Question 4 The equation of the linear function graphed here is: a) y = 2x + 7 b) y =!2x + 7 c) y = 2x 7! 2 d) y =! 2x 7 + 2 e) y = 7x 2! 2 Question 5 The straight lines y =! 3x 2 a) (1,2) b) (2,1) c) (1,3) d) ( 5 2,2) + 4 and y = 2x! 3 intersect at the point: e) (7, 1 2 ) 3
Question 6 Which one of the following lines is perpendicular to the line y = x 5 + 3? a) y = x 3 + 5 b) y = 5! x 3 c) y = x 15! 3 d) y = 5x e) y =!5x + 3 Question 7 The quadratic function y = 2x 2 + 4x + 4 has a turning point at: a) (2,4) b) (!2,!4) c) (4,2) d) (!1,2) e) (2,1) Question 8 The equation x 2! kx + 9 = 0 has only one solution if: a) k > 6 b) k >!6 c) k! 6 d) k! "6 e) k = ±6 Question 9 What are the x intercepts of the quadratic function y = x 2 + ( f! g)x! fg? a) x = f and x = g b) x = f and x =!g c) x =! f and x = g d) x =! f and x =!g e) x = f g and x = g f 4
Question 10 A parabola is shown below. Which of the following quadratic equations correctly defines the parabola? a) y =!2(x +1)(x + 3) b) y = 2(x!1)(x! 3) c) y = 2(x +1)(x + 3) d) y = (x!1)(x! 3) e) y = (x +1)(x + 3) Question 11 Which of the following correctly lists all of the factors of the cubic function y = 2x 3 + 3x 2! 23x!12? a) (x!12) 3 b) (x! 3)(x + 4)(2x +1) c) (x! 4)(2x!1)(x + 3) d) (x! 2)(x! 3)(x! 23)!12 e) (x + 2)(x + 3)(x + 23) +12 Question 12 The complete set of solutions to the equation 8 =!125x 3 is: a) x = 2 5 b) x =! 5 2 c) x =! 2 5 d) x = 2 5 and x =! 2 5 e) x = 2 5 and x = 5 2 5
Question 13 The graph of a cubic function is shown below. Which of the following equations describes the graph? a) y =!x 2 (x + 4) b) y =!x(x! 4) 2 c) y = x 2 (x + 4) d) y =!x 2 (x! 4) e) y = x 2 (x! 4) 6
Question 14 The graph of a cubic function is shown below. Which of the following equations describes the graph? a) y =! 1 4 (x!1)3 + 2 b) y = 1 4 (x + 2)3 +1 c) y = 1 4 (x +1)3 + 2 d) y = (x +1) 3 + 2 e) y = (x!1) 2 + 2 7
Question 15 The expression log 7 x 3! log 7 x 2 a) log 7 x b) 1 c) 7 3! 7 2 d) log 7 x 5 can be simplified as: e) 7 x Question 16 In a mouse infested area, the population of mice is increasing by 25% every month. The time that it takes for the population to double is found by which equation? a) log 2 1.25 = x b) log 2 0.25 = x c) log 1.25 2 = x d) log 0.25 2 = x e) log 2.25 1.25 = x Question 17 The concentration of a particular drug remaining active in a patient halves every 2 hours. After 12 hours, the concentration has dropped to the fraction: 1 a) 2 b) c) d) e) 1 6 1 32 1 64 1 128 8
Question 18 The expression a) b) c) d) e) ( xy) 2 ( ) 3 2xy 4xyz xy ( xy) 3 8z 8z ( xy) 3 ( xy) 2 2xy! xy 4xyz 2z xy ( ) 3 ( xy) 2 ( ) 2xy 4xyz can be simplified as: 3 xy Question 19 x! 13 can be also be expressed as: a) b) x x 3 x 3 x c)! x x 3 d) 3 1 x e)! 1 3 x 9
Question 20 The graph of the function y = a x is shown here. Which of the following graphs below correctly shows the function y = a!x! 2? a) b) c) d) e) 10
Section B Short answer questions (30 marks) Question 1 Shifty car rentals charge $60.00 per day to hire a car, with an extra cost of 10 cents per kilometre. a) Write a linear relation that describes the cost in dollars (c) as a function of the distance travelled in kilometres (x). b) Calculate the total cost of hiring the car for a day & driving the following distances: (1 mark) Distance Cost 100 km 200 km 500 km (3 marks) c) Use this data to draw an appropriate graph of the cost vs distance. (3 marks) 11
Low Budget rentals offer the same car at a fixed cost of $80 per day and 5 cents per kilometre. d) Draw this cost on the graph. e) Calculate the distance at which the Low Budget rental becomes cheaper than Shifty. Question 2 Use long division to find the three linear factors of the cubic function p(x) = 2x 3! x 2! 5x! 2, given that p(2) = 0. (4 marks) 12
Question 3 Recognising the strength of the arch, engineers are building a parabolic shaped mine tunnel. The tunnel is 5 m wide at the bottom and 4 m high in the middle. The tunnel is designed for one single lane of traffic in one direction. (0,0) a) The equation used to model the curve of the tunnel is y = k ( x 2! 5x), where the point (0,0) is at the lower left intersection of the wall and the ground. Show that the value of k =! 16 25. b) Trucks moving through the tunnel are 3 m high. Using a CAS calculator or other means to find the widest truck that can fit through the tunnel. 13
Question 4 The shape of a slide is defined by the cubic function y =! 1 250 (x3! 40x 2 + 500x! 2000), where y is the height above ground level and x is the horizontal distance. The slide starts at x = 0 and ends where the curve goes below ground level. a) Calculate the initial starting height of the curve. b) Calculate the height of the slide where x = 10. c) Using a CAS calculator or other means, find the value of x (to two decimal places) where the height of the slide is 4 m. d) Using a CAS calculator or other means, find the value of x where the slide ends. (1 mark) (1 mark) 14
Question 5 The value of a computer was found to drop according to the equation V(t) = 1200! 0.65 t, where t is the time (measured in years). a) Find the initial value of the computer at the time of purchase. b) What percentage of value does the computer lose every year? (1 mark) c) Find the value (to the nearest dollar) one year after purchase. (1 mark) (1 mark) d) Find the time (to the nearest month) at which the value of the computer had halved. 15
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Answer sheet for section A 1. a b c d e 2. a b c d e 3. a b c d e 4. a b c d e 5. a b c d e 6. a b c d e 7. a b c d e 8. a b c d e 9. a b c d e 10. a b c d e 11. a b c d e 12. a b c d e 13. a b c d e 14. a b c d e 15. a b c d e 16. a b c d e 17. a b c d e 18. a b c d e 19. a b c d e 20. a b c d e 17
Answer sheet for section A Unit 1 Maths Methods (CAS) Exam 2012 Solutions 1. a b c d e 2. a b c d e 3. a b c d e 4. a b c d e 5. a b c d e 6. a b c d e 7. a b c d e 8. a b c d e 9. a b c d e 10. a b c d e 11. a b c d e 12. a b c d e 13. a b c d e 14. a b c d e 15. a b c d e 16. a b c d e 17. a b c d e 18. a b c d e 19. a b c d e 20. a b c d e 1
Section B Short answer questions (30 marks) Unit 1 Maths Methods (CAS) Exam 2012 Solutions Question 1 Shifty car rentals charge $60.00 per day to hire a car, with an extra cost of 10 cents per kilometre. a) Write a linear relation that describes the cost in dollars (c) as a function of the distance travelled in kilometres (x). C (x) = 0.1x + 60 b) Calculate the total cost of hiring the car for a day & driving the following distances: (1 mark) Distance Cost 100 km C (x) = 10 + 60 = $70 200 km C (x) = 20 + 60 = $80 500 km C (x) = 50 + 60 = $110 (3 marks) c) Use this data to draw an appropriate graph of the cost vs distance. Cost ($) Shifty Rentals $110 $100 Low Budget $90 $80 $70 $60 100 200 300 400 500 Distance (km) (3 marks) 2
Low Budget rentals offer the same car at a fixed cost of $80 per day and 5 cents per kilometre. d) Draw this cost on the graph. Unit 1 Maths Methods (CAS) Exam 2012 Solutions e) Find the distance at which the Low Budget rental becomes cheaper than Shifty. 80 + 0.05x = 60 + 0.1x 20 = 0.05x x = 400 km Question 2 Use long division to find the three linear factors of the cubic function p(x) = 2x 3! x 2! 5x! 2, given that p(2) = 0. 2x 2 + 3x + 1 x! 2 2x 3! x 2! 5x! 2 2x 3! 4x 2 3x 2! 5x! 2 3x 2! 6x x-2 x-2 0 The other two factors from 2x 2 + 3x +1 are (2x + 1) and (x + 1). (4 marks) 3
Unit 1 Maths Methods (CAS) Exam 2012 Solutions Question 3 Recognising the strength of the arch, engineers are building a parabolic shaped mine tunnel. The tunnel is 5 m wide at the bottom and 4m high in the middle. The tunnel is designed for one single lane of traffic in one direction. a) The equation used to model the curve of the tunnel is y = k ( x 2! 5x), where the point (0,0) is at the lower left intersection of the wall and the ground. Show that the value of k =! 16 25. y = kx(x! 5) 4 = k( 5 )( 5! 5) = k( 5 )(! 5 ) = k(! 25 ) 2 2 2 2 4 k =! 16 25 b) Trucks moving through the tunnel are 3 m high. Using a CAS calculator or other means to find the widest truck that can fit through the tunnel. 3 =! 16 (x 2! 5x) 25 0 =! 16 x 2 + 16 x! 3 25 5 0 =! 16 25 x 2 + 80 x! 75 =! 1 (16x 2! 80x + 75) 25 25 25 x = 5 and 15 =1.25 m and 3.75 m. 4 4 The widest truck is 2.5 m. 4
Unit 1 Maths Methods (CAS) Exam 2012 Solutions Question 4 The shape of a slide is defined by the cubic function y =! 1 250 (x3! 40x 2 + 500x! 2000), where y is the height above ground level and x is the horizontal distance. The slide starts at x = 0 and ends where the curve goes below ground level. a) Find the initial starting height of the curve. y =! 1 (!2000) = 8m 250 b) Find the height of the slide where x = 10. y =! 1 (10 3! 40 " 10 2 + 500 " 10! 2000) = 0m 250 c) Using CAS calculator or other means, find the value of x (to two decimal places) where the height of the slide is 4 m. 4 =! 1 (x 3! 40x 2 + 500x! 2000) 250 x = 2.45m d) Using CAS calculator or other means, find the value of x where the slide ends. (1 mark) 0 =! 1 (x 3! 40x 2 + 500x! 2000) 250 x = 10m or 20m (The end must be 20m) (1 mark) 5
Unit 1 Maths Methods (CAS) Exam 2012 Solutions Question 5 The value of a computer was found to drop according to the equation V(t) = 1200! 0.65 t, where t is the time (measured in years). a) Find the initial value of the computer at the time of purchase. V (0) = 1200! 0.65 0 = 1200! 1 = $1200 b) What percentage of value does the computer lose every year? (1 mark) 100%! 65% = 35% c) Find the value (to the nearest dollar) one year after purchase. (1 mark) V (1) = 1200! 0.65 1 = 1200! 0.65 = $780 (1 marks) d) Find the time (to the nearest month) at which the value of the computer had halved. $600 = $1200! 0.65 t 0.5 = 0.65 t t = log 0.65 0.5 = 1.6 years = 19 months 6