Unit 2 Maths Methods (CAS) Exam

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Name: Teacher: Unit Maths Methods (CAS) Exam 1 014 Monday November 17 (9.00-10.

1 Name: Teacher: Unit Maths Methods (CAS) Exam Monday November 17 ( am) Reading time: 15 Minutes Writing time: 90 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 /0 /40 /60 1

2 Section A Multiple choice questions (0 marks) Question 1 A straight line has the equation 4y 3x = 8. Another form of this equation is: a) y = 3x b) y = 8x + 3 c) y = 3x 4 + d) y = 3x e) y = 3x Question The equation of the linear function graphed here is: a) y = x + 7 b) y = x + 7 c) y = x 7 + d) y = 7x e) y = x 7

3 Question 3 The graph of a cubic function is shown below. Which of the following equations describes the graph? a) y = x (x + 4) b) y = x(x 4) c) y = x (x + 4) d) y = x (x 4) e) y = x (x 4) Question 4 In a mouse infested area, the population of mice is increasing by 5% every month. The time that it takes for the population to double is found by which equation? a) log 1.5 = x b) log 0.5 = x c) log 1.5 = x d) log 0.5 = x e) log = x Question 5 3 Which of the following is equal to x? 3 a) x 3 b) x c) x 3 d) 1 x 3 e) 1 x 3 3

4 Question 6 Which of the sets correctly describes the interval shown on the number-line below? a) ( 5, ) (3, ) b) [ 5, ] [3, ] c) [ 5, ) [3, ) d) ( 5, ] (3, ] e) ( 5, ] [3, ) Question 7 The function y = x 4 has the inverse: a) y = x 4 b) y = 4x c) y = x + d) y = x 4 e) y = x + 4 Question 8 Which of the following pairs of functions correctly describes the circle x + y = 5? a) y = 5 x and y = 5 + x b) y = 5 x and y = 5 x c) y = x 5 and y = x 5 d) y = 5 x and y = x 5 e) y = 5 x and y = 5 + x 4

5 Question 9 The angle θ is in the second quadrant ( π < θ < π). Which of the following statements about θ is incorrect? a) sinθ > 0 b) tanθ > 0 c) sin θ + π < 0 d) tan θ + π > 0 e) sin ( π θ) > 0 Question 10 The graph shown here could be described by the equation: a) y = 3cos 4x b) y = 3cos 4x 1 c) y = 3sin 4x 1 d) y = cos 4x 1 e) y = 3cos 4x 1 Question 11 An angle θ has a value sinθ = 0.9. Which of the following is the value of cosθ? a) cosθ = 0.1 b) cosθ = ±0.19 c) cosθ = ± 0.19 d) cosθ = ± 0.9 e) cosθ = 0.9 5

6 Question 1 The graph here shows the price of a particular share over 8 months from the start of January to the end of August. The average change in the price over the time period was: a) $1.5 / month b) $3.13 / month c) $10 / month d) $5 / month e) $45 / month Question 13 At the point x =, the gradient of the curve y = 3x + 3 is equal to: a) 0 b) 7 c) 9 d) 1 e) 7 Question 14 The cubic function y = x3 3 3x ( ) and 3, 3 1 a) 0,1 b) 1,0 ( ) and 3, 3 1 ( ) and 3,3 1 c) 0, 1 d) 0,1 ( ) and 3,3 1 e) 0,1 ( ) and 3, has stationary points at: 6

7 Question 15 Which of the following graphs correctly shows the derivative of the function shown here? a) b) c) d) e) 7

8 Question 16 For the matrix multiplication shown below, what is the value of x? a) b).5 c) 3 d) 3.5 e) 17 7 x 1 = [ 17] Question 17 Which one of the following operations can be performed for the matrices shown below? A = a) B -1 b) A c) A+B d) AB e) BA 1 3 and B = Question 18 What is the determinant of the matrix a) Undefined b) 60 c) -60 d) 36 e) ? Question 19 The chance of tossing 10 heads from 10 coin tosses is closest to: a) 0.1% b) 1% c) 10% d) 90% e) 99.9% 8

9 Question 0 The Venn diagram shown here gives the results of a survey of 50 passengers on the 5.38 am train to Melbourne. Passengers were either in first or economy class, male or female. Which of the following statements about the results is incorrect? a) There were 0 passengers in first class. b) 8 of the first class passengers were female. c) 16 of the passengers were males in economy class. d) There were 8 males in first class. e) of the passengers were female. 9

10 Section B Short answer questions (40 marks) Full workings must be shown. Question 1 Simplify the following expressions. (4 marks) a) a 3 (5 a) b) 1 3 log 7 ( ) 1 log 36 ( ) 10

11 Question A population of rabbits can increase by 0% per month. At the start of the year, the number of rabbits in an area was 500. a) Write an equation that describes the relationship between rabbit numbers and time. ( marks) b) Calculate the number of rabbits at the end of the sixth month. ( marks) c) Use a CAS calculator or other means to find the time (in months) at which the population of rabbits reaches 100. ( marks) 11

Question 3 The water level (in metres) at a harbour dock on a particular day is modelled by the equation: h(t) = 1.5cos 4πt 5 + 4 (Time is measured from 1 midnight.

12 Question 3 The water level (in metres) at a harbour dock on a particular day is modelled by the equation: h(t) = 1.5cos 4πt (Time is measured from 1 midnight.) a) Calculate the time period between successive high tides. ( marks) b) State the minimum and maximum heights that the water level reaches. (1 mark) Minimum: m Maximum: m c) Calculate the time of the first low tide. ( marks) d) Boats can only leave the harbour if the water level is above 3.0 m. Use a CAS calculator or another method to find the time periods when boats are stuck in the harbour. ( marks) 1

13 Question 4 Two matrix transformations as shown are applied to a point (x,y) x y a) Find the single ( x ) matrix that can be used to describe the combined transformations. ( marks) b) Find the new co-ordinates if the point (3,6) undergoes this transformation. (1 mark) 13

14 Question 5 The graph below shows the area under the curve y = x + 6x 5 between the two x intercepts (x=1 and x=5). Use integration to calculate the area between the curve and between the two x intercepts. (3 marks) 14

15 Question 6 A stuntman is preparing for a stunt jump in a car. He has used a quadratic equation to model the path he will take, where h(x) is the height above the ramp and x is the horizontal distance covered. (Both distances are measured in metres.) h(x) = x 40 + x a) Find the horizontal distance (a) covered by the jump. (1 mark) b) Find the derivative dh. (1 mark) dx 15

16 c) Use calculus to show that the maximum height (b) occurs at x = 0m. ( marks) d) Calculate the maximum height reached during the jump. ( marks) e) Find the gradient of the launch ramp (when x = 0). ( marks) f) Calculate the angle of the launch ramp (c) above the horizontal. (1 mark) 16

17 Question 7 Two cards are dealt (without replacement) from a standard deck of 5. a) What is the probability that the first card dealt is a king? (1 mark) b) If the first card dealt is a king, what that is the chance that the second card is also a king? (1 mark) c) What is the chance that of the two cards dealt, exactly one is a king? ( marks) 17

18 Question 8 The probability of a particular football team winning its next game is 75% if it won the previous game and 60% if it lost the previous game. a) Draw the transition matrix for these outcomes. (1 mark) b) If the team was successful in the opening game of the season, calculate the probability that it will win the third game of the season. ( marks) c) What percentage of games would the team be expected to win over the whole season? (1 mark) 18

19 Answer sheet for section A 1. a b c d e. 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 1. 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 0. a b c d e 19

20 Answer sheet for section A Unit Maths Methods (CAS) Exam 014 Solutions 1. a b c d e. 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 1. 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 0. a b c d e 1

21 Section B Short answer questions (40 marks) Full workings must be shown. Unit Maths Methods (CAS) Exam 014 Solutions Question 1 Simplify the following expressions. (4 marks) a) a 3 (5 a) = a a = 5 5 a or a = 315a b) 1 3 log ( 7) 1 log ( 36) = log log 36 = log ( 3) log ( 6)= log 3 6 = log 1 = log ( ) 1 = 1 Question A population of rabbits can increase by 0% per month. At the start of the year, the number of rabbits in an area was 500. a) Write an equation that describes the relationship between rabbit numbers and time. ( marks) P = 500 (1. x ) b) Calculate the number of rabbits at the end of the sixth month. ( marks) P = 500 (1. 6 )=1493 c) Use a CAS calculator or other means to find the time (in months) at which the population of rabbits reaches 100. ( marks) 100 = 500 (1. x ) x = log x = 4.8 months

22 Unit Maths Methods (CAS) Exam 014 Solutions Question 3 The water level (in metres) at a harbour dock on a particular day is modelled by the equation: h(t) = 1.5cos 4πt (Time is from 1 midnight) a) Calculate the time period between successive high tides. ( marks) T = π k T = π 4π 5 = 5 = 1.5h b) State the minimum and maximum heights that the water level reaches. (1 mark) Minimum:.5 m Maximum: 5.5 m c) Calculate the time of the first low tide. ( marks).5 = 1.5cos 4πt 4πt + 4, 1 = cos 5 5 π = 4πt 5, t = 5π = 6.5 h = 6.15 am 4π d) Boats can only leave the harbour if the water level is above 3.0 m. Use a CAS calculator or another method to find the time periods when boats are stuck in the harbour. ( marks) 3 = 1.5cos 4πt + 4 gives the times when the water level is at 3.0 m. 5 There are four solutions: t=4.58 or t=7.9 or t=17.08 or t= am am and 5.05 pm pm 3

23 Unit Maths Methods (CAS) Exam 014 Solutions Question 4 Two matrix transformations as shown are applied to a point (x,y) x y a) Find the single ( x ) matrix that can be used to describe the combined transformations. ( marks) = b) Find the new co-ordinates if the point (3,6) undergoes this transformation. (1 mark) = 30 3 Question 5 The graph below shows the area under the curve y = x + 6x 5 between the two x intercepts. Use integration to calculate the area between the curve and between the two x intercepts (x=1 and x=5). (3 marks) 5 1 x + 6x 5 = 1 3 x 3 + 3x 5x 5 1 = = 3 3 4

24 Unit Maths Methods (CAS) Exam 014 Solutions Question 6 A stuntman is preparing for a stunt jump in a car. He has used a quadratic equation to model the path he will take, where h(x) is the height above the ramp and x is the horizontal distance covered. (Both distances are measured in metres.) a) Find the horizontal distance (a) covered by the jump. (1 mark) h(x) = x x 40 = x(1 x ) x = 0m and x = 40m 40 b) Find the derivative dh. (1 mark) dx dh dx = 1 x 0 c) Use calculus to show that the maximum height (b) occurs at x = 0m. ( marks) 0 = 1 x 0, 1 = x 0, x = 0m d) Calculate the maximum height reached during the jump. ( marks) h(0) = = 10m e) Find the gradient of the launch ramp (when x = 0). ( marks) dh dx = = 1 f) Calculate the angle of the launch ramp (c) above the horizontal. (1 mark) θ = tan 1 (1) = 45 5

25 Unit Maths Methods (CAS) Exam 014 Solutions Question 7 Two cards are dealt (without replacement) from a standard deck of 5. a) What is the probability that the first card dealt is a king? (1 mark) 4 5 = 1 13 b) If the first card dealt is a king, what that is the chance that the second card is also a king? (1 mark) 3 51 = 1 17 c) What is the chance that of the two cards dealt, exactly one is a king? ( marks) Pr (First card only is a king) = = = 16 1 Pr (Second card only is a king) = = 16 1 Pr (Only one king) = = 3 1 Question 8 The probability of a particular football team winning its next game is 75% if it won the previous game and 60% if it lost the previous game. a) Draw the transition matrix for these outcomes. (1 mark) 75% 60% 5% 40% b) If the team was successful in the opening game of the season, calculate the probability that it will win the third game of the season. ( marks) 75% 60% 5% 40% 1 0 =71.5% c) What percentage of games would the team be expected to win over the whole season? (1 mark) 60% 85% =70.6% 6

Unit 1 & 2 Maths Methods (CAS) Exam

Name: Teacher: Unit 1 & 2 Maths Methods (CAS) Exam 2 2017 Monday November 20 (1.00pm - 3.15pm) Reading time: 15 Minutes Writing time: 120 Minutes Instruction to candidates: Students are permitted to bring