Name: Teacher: Unit 2 Math Methods (CAS) Exam 1, 2015 Tuesday November 6-1.50 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 detachable answer sheet. Section A Section B Total exam /20 /30 /50 1
Section A Multiple choice questions (20 marks) Question 1 Which of the sets correctly describes the interval shown on the number-line below? a) (, 5] b) [5, ) c) [5, ) d) [, 5) e) [5, 0) ( ) Question 2 a) y = 3( x 2) 2 1 b) y = ( x 3) 2 + 1 c) y = ( x + 3) 2 + 1 d) y = ( x 3) 2 1 e) y = 2( x 3) 2 + 1 2
Question 3 The angle 30! a) 30π 140 b) π π c) 6 π d) 30 π e) 180 in radians is equal to: Question 4 If sin = 0.6,(0 < x < 2π ) a) 0.6 b) 0.6 c) d) e) 0.6 2π x find sin( 2π x) tan(θ) > 0 ( ) = sin( θ) sin π + θ Question 5 The graph shown here could be described by the equation: a) y = 2 sin(2x) + 1 3π 2 b) y = cos( x) 1 c) y = 3π 2 sin(x) 1 d) y = 2 sin(x) 1 e) y = 2cos(2x) 1 3
Question 6 Solve the following equation for θ : sin(θ) = 3 2 a) 4 π 5, π 3 3 π 5π b), 6 6 c) 2 π, π 3 3 π 5π d), 6 6 e) 2π, π 3 3 for θ [ π,π ]: 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=2 and t=12. a) 0 m/s b) 2.5 m/s c) 1.7 m/s d) 1.9 m/s e) 1.8 m/s Question 8 3 2 The cubic function p = 2t 5t 4t + 13 a) t =-2 b) t = 2 (for t>0) has a stationary point at: c) t = 1 / 3 d) t = -1 e) t = 1 4
Question 9 Find the midpoint of the line segment joining A(2,6) and B(-3,-4) a) ( 1 2,1) b) (1, 1 2 ) c) ( 1 2,1) d) (0,1) e) ( 1 2, 1) Question 10 The derivative of the function f (x) = x2 + 3x x dy a) dx = x2 + 3x dy b) dx = 2x + 3 c) d) e) dy dx = x + 3 dy dx =1 dy dx = 0, x 0 is: Question 11 Determine the gradient of the line passing through the points (3,2) and (5,7): a) m = 7 2 b) m = 5 c) m = 5 2 d) m = 2 e) m = 3 2 Question 12 The derivative of the function y = 2 x 3 is: a) 3x 2 b) 1 3x 2 5
c) d) 3x 2 e) 2x 2x 1 Question 13 Calculate the distance EF (to two decimal places): a) 7.81 b) 8.06 c) 8.98 d) 9.16 e) 9.89 Question 14 Find an anti-derivative of the function f (x) = 3x 2 + 4x 3 + 3 : a) f (x) = 3x 2 + 4x 3 b) f (x) = x 3 + x 4 + 5x + c c) f (x) = x 3 + x 4 + 3x d) f (x) = x 3 + x 4 + 3 e) f (x) = x 3 + x 4 + 3 c 6
Question 15! If A = # 2 4 " 3 6 a) b) c) d) e)! # "! # "! # "! # "! # " 22 33 11 12 40 36 5 3 2 3 $ & % $ & % $ & % $ & % $ & % $! & and B = # 5 % " 3 $ &, find AB % Question 16 Which one of the following multiplications of matrices shown below cannot be performed? " A = $ # a) AB b) DA c) DB d) CD e) DC 4 3 4 7 % " ' B = $ 4 & # 3 " % ' C =! " 1 0 1 # $ $ D = & $ # $ 4 4 9 8 6 0 % ' ' &' Question 17 Find the derivative of f (x) = 3x 3 6x 2 +1and hence find f!(1) : a) f!(1) = 3 b) f!(1) = 2 c) f!(1) = 3 d) f!(1) = 2 e) f!(1) =1 Question 18 Solve x 3 2x 2 5x + 6 = 0 a) x =1, x = 3, x = 2 b) x = 1, x = 3, x = 2 c) x = 6, x = 5, x = 2 d) x = 6, x = 5, x = 2 7
e) no real solutions Question 19 Solve the following equation for x; x 2 3x +1= 0 a) b) 3 + 5 3 5 x = and x = 2 2 3 ± 5 x = 2 c) x = 3 2 d) x = 3+ 5 2 e) no real solutions Question 20 Simplify: " 6 % 2log 10 3+ log 10 16 2log 10 $ ' # 5& a) 0.5 b) log c) 2 d) 10 e) Cannot be simplified 8
Section B Short answer questions (30 marks) Question 1 (total 4 marks) For the function f (x) = 3 2x : a) Sketch the graph of the function, including labels on any endpoints and intercepts. (2 mark) b) State the implied domain of the function. (1 mark) c) State the range of the function. (1 mark) 9
Question 2 (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) On the following axes, draw the graph of y = h(t) for 0 t 24 (3 marks) t b) What was the height of the tide at 2am? (1 mark) c) A boat can only cross the harbour bar when the tide is at least 2.5 metres above mean sea level. When could the boat cross the harbour bar on this day? (2 marks) 10
Question 3 (total 6 marks) A square sheet of cardboard has edges of length 20cm. 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 (1 mark) b) What is the range of values that x can take? (1 mark) c) Show that the volume of the open rectangular box can be expressed as!! = 400! 80!! + 4!! (2 marks) d) Find the volume of the box when x=6 (1 mark). e) Find the derivative of V(x) (1 mark) f) Hence, find the value(s) of x that give a maximum volume for the box. (1 mark) 11
Question 4 (2 marks) Question 5 (total 4 marks) Let A and B be the matrices 2 A = 0 1 1 and 4 B = 8 1 2 a) Find the determinant of each matrix ( det(a) and det(b) ). (2 marks) b) Hence find the inverse of each matrix (A -1 and B -1 ), if they exist. (2 marks) Question 6 (total 6 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 (2) t a) What is the initial population of E.coli in the dish? (1 mark) b) What is the population of E.coli after 4 hours of incubation? (1 mark) 12
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 mark) 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? (1 mark) e) Find the time at which the population of E.coli exceeds the population of Lysteria in the dish. Answer to the nearest hour. (2 marks) 13
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 14