Unit 1&2 Math Methods (CAS) Exam 2, 2016

Transcription

1 Name: Teacher: Unit 1& Math Methods (CAS) Exam, 016 Thursday November 10 9:05 am Reading time: 15 Minutes Writing time: 10 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: o o o o 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 detachable answer sheet. Section A Section B Total exam /0 /4 /6 1

2 Section A Multiple choice questions ( marks) Question 1 The equation of the straight line with gradient 3 that passes through the point (1,9) is A. y = x + 9 B. y = 3x + 9 C. y = 3x + 6 D. y = -!! x + 1 E. y = -!! x + 6 Question The graph below is correctly defined by which formula A. y = 3( x ) 1 B. y = ( x 3) + 1 C. y = ( x + 3) + 1 D. y = ( x 3) 1 E. y = ( x 3) + 1

3 Question 3 The angle 30! in radians is equal to: A. 30π B. C. D. E. 140 π π 6 π 30 π 180 Question 4 An experiment consists of tossing a coin then rolling a fair six sided die. What is the probability of observing a head and a six? A. ½ B. ¼ C. 1/35 D. 1/1 E. 7/1 Question 5 The graph shown here could be described by the equation: A. y = sin(x) + 1 3π B. y = cos( x) 1 C. y = 3π sin(x) 1 D. y = sin(x) 1 E. y = cos(x) 1 3

4 Question 6 Simplify: " 6 % log log log 10 $ ' # 5& A. 0.5 B. log C. D. 10 E. Cannot be simplified Question 7 The graph of distance travelled (metres) against time (seconds) for the motion of an object is shown. Find the average speed of the object in m/s over the interval from t= and t=1. A. 0 m/s B..5 m/s C. 1.7 m/s D. 1.9 m/s E. 1.8 m/s Question 8 3 The cubic function p = t 5t 4t + 13 A. t =- B. t = (for t>0) has a stationary point at: C. t = 1 / 3 D. t = -1 E. t = 1 4

5 Question 9 Find the midpoint of the line segment joining A(,6) and B(-3,-4) A. ( 1,1) B. (1, 1 ) C. ( 1,1) D. (0,1) E. ( 1, 1) Question 10 The derivative of the function A. B. C. D. E. dy dx = x + 3x dy dx = x + 3 dy dx = x + 3 dy dx =1 dy dx = 0 x + 3x y =, x 0 is: x Question 11 Determine the gradient of the line passing through the points (3,) and (5,7): A. m = 7 B. m = 5 C. m = 5 D. m = E. m = 3 5

6 Question 1 The derivative of the function y = x 3 is: A. 3x B. 1 3x C. D. 3x E. x x 1 Question 13 Calculate the distance EF: A. 3 B. 11 C. 14 D. 33 E. 65 Question 14 3 Find an anti-derivative of the function f '( x) = 3x + 4x + 3: 3 A. f ( x) = 3x + 4x + c B. f (x) = x 3 + x 4 + 5x + c 3 4 C. f ( x) = x + 4x + 3x + c 3 4 D. f ( x) = x + x + 3x + c 3 4 E. f ( x) = x + x c 6

7 Question 15 The quadratic 5!! 10x in turning point form a(x-h) + k, by completing the square, is A. (5! + 1)! + 5 B. (5! 1)! - 5 C. 5(! 1)! - 5 D. 5(! + 1)! - E. 5(! 1)! 7 Question 16 The equation of the asymptote of y = 3log! 5! + is A. x = 0 B. x = C. x = 3 D. x = 5 E. y = Question 17 Find the derivative of f (x) = 3x 3 6x +1and hence find f!(1) : A. f!(1) = 3 B. f!(1) = C. f!(1) = 3 D. f!(1) = E. f!(1) =1 Question 18 Solve x 3 x 5x + 6 = 0 A. x =1, x = 3, x = B. x = 1, x = 3, x = C. x = 6, x = 5, x = D. x = 6, x = 5, x = E. no real solutions 7

8 Question 19 Solve the following equation for x; x 3x +1= 0 A. B. C. 3 x = 3 ± x = x = and D. x = 3+ 5 E. no real solutions 3 5 x = Question 0 Suppose that 57% of the swimmers in a club are female (F), that 3% of the swimmers race butterfly (B), and that 11% of the swimmers in the club are female and race butterfly. Which of the following probability tables correctly summarises the information? A. B B F F B B B F F C B B F F D. B B F F E B B F F

9 Section B Short answer questions (4 marks) Question 1 (total 6 marks) Find the gradient of each of the following lines (a) 3! +! 5 = 0 (1 marks) (b) (c) A line joining points (-1,5) and (,9) (d) 4y = -6x + 1 (e) Which of the above lines are parallel to each other? ( marks) 9

10 Question (total 7 marks) The diagram shows the plans for a new bridge across the Hopkins River of span 50m. The shape of the curve, ABC, is a parabola. The line AC is the water level and B is the highest point of the bridge. (a) Taking A as the origin (0,0) and the maximum height above water as 4.5m, where y is the height of the arch above the water, and x is the horizontal distance from A. Show that the formula is! =! 0.007(! 5)! +4.5 ( marks) (b) Accurately plot this curve on the grid below, label all intercepts and turning point (3 marks)!!!!!!!!!!!!!!!!!!!!!!!! 4!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!! (c) At what horizontal distance from A is the height of the arch above the water equal to 3m, answer to one decimal place (d) What is the height of the arch at a horizontal distance from A of 1m, to one decimal place Question 3 (total 5 marks) The number of bacteria (E.coli) in a petri dish after infection at time t=0 (time in hours) is given by: P(t) = 3000 () t a) What is the initial population of E.coli in the dish? b) What is the population of E.coli after 4 hours of incubation? 10

11 c) The petri dish needs to be disposed of when the population of E.coli exceeds 13,000,000. When must the dish be disposed of (give your answer to the nearest hour)? A second strain of bacteria, Lysteria spp., was also introduced into the dish at time t=0. Its population is given by: L(t) = 100,000 (1.4) t d) What is the population of Lysteria in the dish after 4 hours? e) Find the time at which the population of E.coli exceeds the population of Lysteria in the dish. Answer to the nearest hour. 11

12 Question 4 (total 6 marks) A square sheet of cardboard has edges of length 0cm. Four equal squares of edge length x cm are cut out of the corners and the sides are turned up to form an open rectangular box. a) Find the length of each edge of the base of the box in terms of x b) What is the range of values that x can take? c) Show that the volume of the open rectangular box can be expressed as!! = 400! 80!! + 4!! d) Find the volume of the box when x=6. e) Find the derivative of V(x) f) Hence, find the value(s) of x that give a maximum volume for the box. 1

13 Question 5 (total marks) (a) Simplify! 5! 5!!! (5!!)!! (b) Simplify 1 3 log! 7 1 log!(36) 13

14 Question 6 (total 6 marks) It is suggested that the height, h(t) metres, of the tide above mean sea level during a particular day at Seabreak is given approximately by the rule, t is time after midnight (in hours):! h(t) = 5sin# π " 6 t $ & % a) Find the Period of this function b) Hence, on the following axes, draw the graph of y = h(t) for 0 t 4 ( marks) t b) What was the height of the tide at am? c) A boat can only cross the harbour bar when the tide is at least.5 metres above mean sea level. When could the boat cross the harbour bar on this day? ( marks) 14

15 Question 7 (total 10 marks) Events A and B are such that Pr(A) = 0.6, Pr(B) = 0.5 and Pr(A B) = 0.4. (a) Complete the following probability table B B A A (b) Use the probability table to find Pr(A B ) 1 (c) Find Pr(A B ) (d) Find Pr(A B) (e) Find Pr(B A ) (f) Find Pr(A B) The probability that Jenna goes to the gym on Monday is 0.6. If she goes Monday, the probability she goes to the gym Tuesday is 0.7. If she doesn t go Monday the probability she goes to the gym Tuesday is 0.4. (a) Draw a Venn diagram/tree diagram/karnaugh Map showing Jenna s gym situation ( marks) (b) What is the probability Jenna goes to the gym on both Monday and Tuesday? (c) What is the probability Jenna goes to the gym on Tuesday? 15

16 Answer sheet for section A Name: Teacher: 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 16

17 Formula Sheet Differentiation!! =!!!!,!!!!!! =!!!!!!!! = lim!!!! + h!(!) h Anti Differentiation Quadratic formula Trigonometry!!!!" =!!!!!! + 1 +!! =! ±!! 4!"! Probability Pr!! = Pr! + Pr! Pr!! Pr!(!!) Pr!! = Pr!(!) Pr(A) = 1 Pr(A ) 17

18 Name: Teacher: SOLUTIONS BOT / PEC / THA / VIJ Unit 1& Math Methods (CAS) Exam, 016 Thursday November 10 9:05 am Reading time: 15 Minutes Writing time: 10 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: o o o o 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 detachable answer sheet. Section A Section B Total exam /0 /4 /6 1

19 Section A Multiple choice questions ( marks) Question 1 The equation of the straight line with gradient 3 that passes through the point (1,9) is A. y = x + 9 B. y = 3x + 9 C. y = 3x + 6 D. y = -!! x + 1 E. y = -!! x + 6 Question The graph below is correctly defined by which formula A. y = 3( x ) 1 B. y = ( x 3) + 1 C. y = ( x + 3) + 1 D. y = ( x 3) 1 E. y = ( x 3) + 1

20 Question 3 The angle 30! in radians is equal to: A. 30π B. C. D. E. 140 π π 6 π 30 π 180 Question 4 An experiment consists of tossing a coin then rolling a fair six sided die. What is the probability of observing a head and a six? A. ½ B. ¼ C. 1/35 D. 1/1 E. 7/1 Question 5 The graph shown here could be described by the equation: A. y = sin(x) + 1 3π B. y = cos( x) 1 C. y = 3π sin(x) 1 D. y = sin(x) 1 E. y = cos(x) 1 3

21 Question 6 Simplify: " 6 % log log log 10 $ ' # 5& A. 0.5 B. log C. D. 10 E. Cannot be simplified Question 7 The graph of distance travelled (metres) against time (seconds) for the motion of an object is shown. Find the average speed of the object in m/s over the interval from t= and t=1. A. 0 m/s B..5 m/s C. 1.7 m/s D. 1.9 m/s E. 1.8 m/s Question 8 3 The cubic function p = t 5t 4t + 13 A. t =- B. t = (for t>0) has a stationary point at: C. t = 1 / 3 D. t = -1 E. t = 1 4

22 Question 9 Find the midpoint of the line segment joining A(,6) and B(-3,-4) A. ( 1,1) B. (1, 1 ) C. ( 1,1) D. (0,1) E. ( 1, 1) Question 10 The derivative of the function A. B. C. D. E. dy dx = x + 3x dy dx = x + 3 dy dx = x + 3 dy dx =1 dy dx = 0 x + 3x y =, x 0 is: x Question 11 Determine the gradient of the line passing through the points (3,) and (5,7): A. m = 7 B. m = 5 C. m = 5 D. m = E. m = 3 5

23 Question 1 The derivative of the function y = x 3 is: A. 3x B. 1 3x C. D. 3x E. x x 1 Question 13 Calculate the distance EF: A. 3 B. 11 C. 14 D. 33 E. 65 Question 14 3 Find an anti-derivative of the function f '( x) = 3x + 4x + 3: 3 A. f ( x) = 3x + 4x + c B. f (x) = x 3 + x 4 + 5x + c 3 4 C. f ( x) = x + 4x + 3x + c 3 4 D. f ( x) = x + x + 3x + c 3 4 E. f ( x) = x + x c 6

24 Question 15 The quadratic 5!! 10x in turning point form a(x-h) + k, by completing the square, is A. (5! + 1)! + 5 B. (5! 1)! - 5 C. 5(! 1)! - 5 D. 5(! + 1)! - E. 5(! 1)! 7 Question 16 The equation of the asymptote of y = 3log! 5! + is A. x = 0 B. x = C. x = 3 D. x = 5 E. y = Question 17 Find the derivative of f (x) = 3x 3 6x +1and hence find f!(1) : A. f!(1) = 3 B. f!(1) = C. f!(1) = 3 D. f!(1) = E. f!(1) =1 Question 18 Solve x 3 x 5x + 6 = 0 A. x =1, x = 3, x = B. x = 1, x = 3, x = C. x = 6, x = 5, x = D. x = 6, x = 5, x = E. no real solutions 7

25 Question 19 Solve the following equation for x; x 3x +1= 0 A. B. C. 3 x = 3 ± x = x = and D. x = 3+ 5 E. no real solutions 3 5 x = Question 0 Suppose that 57% of the swimmers in a club are female (F), that 3% of the swimmers race butterfly (B), and that 11% of the swimmers in the club are female and race butterfly. Which of the following probability tables correctly summarises the information? A. B B F F B B B F F C B B F F D. B B F F E B B F F

26 Section B Short answer questions (4 marks) Question 1 (total 6 marks) Find the gradient of each of the following lines (a) 3! +! 5 = 0 (1 marks) (b) y = -3x + 5 y =!!!! + 5 therefore, m =!!! m =!!!!!!!!!! =!!!!!!!! m =!! (c) A line joining points (-1,5) and (,9) m =!!!!!!!!!! =!!!!!!!!! (d) 4y = -6x + 1 m =!! 4y = -6x + 1 y =!!! +!!! therefore, m =!! (e) Which of the above lines are parallel to each other?! ( marks) (a) and (d) are parallel (b) and (c) are parallel 9

27 Question (total 7 marks) The diagram shows the plans for a new bridge across the Hopkins River of span 50m. The shape of the curve, ABC, is a parabola. The line AC is the water level and B is the highest point of the bridge. (a) Taking A as the origin (0,0) and the maximum height above water as 4.5m, where y is the height of the arch above the water, and x is the horizontal distance from A. Show that the formula is! =! 0.007(! 5)! +4.5 ( marks) y = A (x h) + k y = A(x 5) sub in (50,0) 0 = A((50) 5) A = (b) Accurately plot this curve on the grid below, label all intercepts and turning point (3 marks)!!! = 0.007(! 5)!!! + 4.5!!!!!!!!!!!!!!!!!!!! 4 (5, 4.5)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! (0, 0) (50, 0)!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!!!!!!!!!!!!!! (c) At what horizontal distance from A is the height of the arch above the water equal to 3m, answer to one decimal place x = 10.6m and x = 39.4m (d) What is the height of the arch at a horizontal distance from A of 1m, to one decimal place y = 3.3 meters Question 3 (total 5 marks) The number of bacteria (E.coli) in a petri dish after infection at time t=0 (time in hours) is given by: P(t) = 3000 () t a) What is the initial population of E.coli in the dish? 3000 b) What is the population of E.coli after 4 hours of incubation? =

28 c) The petri dish needs to be disposed of when the population of E.coli exceeds 13,000,000. When must the dish be disposed of (give your answer to the nearest hour)? 1 hours A second strain of bacteria, Lysteria spp., was also introduced into the dish at time t=0. Its population is given by: L(t) = 100,000 (1.4) t d) What is the population of Lysteria in the dish after 4 hours? e) Find the time at which the population of E.coli exceeds the population of Lysteria in the dish. Answer to the nearest hour. Solve P(t) > L(t), t t hours or for t 10 hours 11

29 Question 4 (total 6 marks) A square sheet of cardboard has edges of length 0cm. Four equal squares of edge length x cm are cut out of the corners and the sides are turned up to form an open rectangular box. a) Find the length of each edge of the base of the box in terms of x 0 x b) What is the range of values that x can take? 0 < x < 10 c) Show that the volume of the open rectangular box can be expressed as!! = 400! 80!! + 4!! V(x) = H W L = x ( 0 x) = x ( x + 4x ) = 400x 80x + 4x 3 d) Find the volume of the box when x=6. V(6) = 384 cm 3 e) Find the derivative of V(x)!"(!) = 1!! 160! + 400!" f) Hence, find the value(s) of x that give a maximum volume for the box.!"(!)!" = 0 for x =!"!, 10 Exclude x = 10, therefore x =!"! 1

30 Question 5 (total marks) (a) Simplify! 5! 5!!! (5!!)!! = 5 5 a (b) Simplify 1 3 log! 7 1 log!(36) log! 3 log! 6 =! log! 3 6!!!!!!!!!!!!!!!!!!!!!!!!!!!!= log! 1 = -1 13

31 Question 6 (total 6 marks) It is suggested that the height, h(t) metres, of the tide above mean sea level during a particular day at Seabreak is given approximately by the rule, t is time after midnight (in hours):! h(t) = 5sin# π " 6 t $ & % a) Find the Period of this function b) Hence, on the following axes, draw the graph of y = h(t) for 0 t 4 ( marks)!! 6 = 1 t b) What was the height of the tide at am? h() = 5 3 c) A boat can only cross the harbour bar when the tide is at least.5 metres above mean sea level. When could the boat cross the harbour bar on this day? ( marks) Solve ( h(t) =.5, t ) 1am t 5am or 13 t 17 1pm t 5pm 14

32 Question 7 (total 10 marks) Events A and B are such that Pr(A) = 0.6, Pr(B) = 0.5 and Pr(A B) = 0.4. (a) Complete the following probability table B B A A (b) Use the probability table to find Pr(A B ) (c) Find Pr(A B ) Pr(A B ) = 0.5 Pr(A B ) = 0.0 (d) Find Pr(A B) (e) Find Pr(B A ) (f) Find Pr(A B) Pr(A B) =!"!(!!!!)!"!(!) Pr(B A ) =!"!(!!!!!)!"!(!!) =!.!!.! = 0. =!.!!.! = 1.0 The probability that Jenna goes to the gym on Monday is 0.6. If she goes Monday, the probability she goes to the gym Tuesday is 0.7. If she doesn t go Monday the probability she goes to the gym Tuesday is 0.4. (a) Draw a Venn diagram/tree diagram/karnaugh Map showing Jenna s gym situation ( marks).1 M.18 Pr(A B) = Pr(A) + Pr(B) Pr(A B) = = 1.0 T (b) What is the probability Jenna goes to the gym on both Monday and Tuesday? (c) What is the probability Jenna goes to the gym on Tuesday? M M Pr( M T ) = 0.4 Pr( T ) = 0.70 T T T T M M T T

33 Answer sheet for section A Name: Teacher: SOLUTIONS BOT / PEC / THA / VIJ 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 16

34 Formula Sheet Differentiation!! =!!!!,!!!!!! =!!!!!!!! = lim!!!! + h!(!) h Anti Differentiation Quadratic formula Trigonometry!!!!" =!!!!!! + 1 +!! =! ±!! 4!"! Probability Pr!! = Pr! + Pr! Pr!! Pr!(!!) Pr!! = Pr!(!) Pr(A) = 1 Pr(A ) 17