MATHEMATICAL METHODS (CAS) PILOT STUDY Written examination 1 (Facts, skills and applications)

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1 MATHEMATICAL METHDS (CAS) PILT STUDY Written eamination 1 (Facts, skills and applications) Friday 7 November 003 Reading time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to am (1 hour 30 minutes) PART I MULTIPLE-CHICE QUESTIN BK This eamination has two parts: Part I (multiple-choice questions) and Part II (short-answer questions). Part I consists of this question book and must be answered on the answer sheet provided for multiple-choice questions. Part II consists of a separate question and answer book. You must complete both parts in the time allotted. When you have completed one part continue immediately to the other part. Structure of book Number of Number of questions Number of questions to be answered marks Students are permitted to bring into the eamination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set-squares, aids for curve sketching, up to four pages (two A4 sheets) of pre-written notes (typed or handwritten) and one approved CAS calculator (memory may be retained) and/or one scientific calculator. For the TI-9, Voyage 00 or approved computer based CAS, their full functionality and/or one scientific calculator may be used, but other programs or files are not permitted. Students are NT permitted to bring into the eamination room: blank sheets of paper and/or white out liquid/tape. Materials supplied Question book of 14 pages, with a detachable sheet of miscellaneous formulas in the centrefold. Answer sheet for multiple-choice questions. Instructions Detach the formula sheet from the centre of this book during reading time. Check that your name and student number as printed on your answer sheet for multiple-choice questions are correct, and sign your name in the space provided to verify this. Unless otherwise indicated, the diagrams in this book are not drawn to scale. At the end of the eamination Place the answer sheet for multiple-choice questions (Part I) inside the front cover of the question and answer book (Part II). You may retain this question book. Students are NT permitted to bring mobile phones and/or any other electronic communication devices into the eamination room. VICTRIAN CURRICULUM AND ASSESSMENT AUTHRITY 003 Victorian Certificate of Education 003

2 MATH METH (CAS) EXAM 1 PT1 Working space PART I continued

3 3 MATH METH (CAS) EXAM 1 PT1 Instructions for Part I Answer all questions in pencil on the answer sheet provided for multiple-choice questions. Choose the response that is correct for the question. A correct answer scores 1, an incorrect answer scores 0. Marks will not be deducted for incorrect answers. No marks will be given if more than one answer is completed for any question. Question 1 A polynomial function p has degree three. A portion of its graph near the point on the graph with coordinates (, 0) is shown below Which one of the following could be the rule for the third degree polynomial p? A. p() = ( + ) B. p() = ( ) 3 C. p() = ( ) D. p() = ( 1)( ) E. p() = ( ) CNTINUED VER PAGE PART I continued TURN VER

4 MATH METH (CAS) EXAM 1 PT1 4 Question 1 The graph of the function f : R + R with rule f ( ) = 3 is most likely to be A. f() B. f() C. f() D. f() E. f() Question 3 Dylan drew the graph of the function f : R functions g and h. The rules for g and h that Dylan could have used are R, f () = by adding the ordinates of the graphs of two A. g () = 3 and h () = 1 B. g () = and h () = 1 C. g () = and h () = D. g () = and h () = 1 E. g () = and h () = 1 PART I continued

5 5 MATH METH (CAS) EXAM 1 PT1 Question 4 If k and P are positive real numbers, which one of the following graphs is most likely to be the graph of the function with equation y = e k + P? A. y B. y C. y D. y E. y PART I continued TURN VER

6 MATH METH (CAS) EXAM 1 PT1 6 Question 5 The graph of the function f is obtained from the graph with equation y = followed by a dilation of units from the -ais. The rule for f is A. f ( ) = B. f ( ) = C. f ( ) = 0. 5 D. f ( ) = 0. 5 E. f ( ) = by a reflection in the y-ais Question 6 The number of solutions of the equation 0.5 cos() = 1, for [, ] is A. 0 B. 1 C. D. 3 E. 4 PART I continued

7 7 MATH METH (CAS) EXAM 1 PT1 Question 7 The diagram shows two cycles of the graph of a circular function. The period of the circular function is A. B. π 3π 4 C. D. 7π 4 E. PART I continued TURN VER

8 MATH METH (CAS) EXAM 1 PT1 8 Question 8 The graph of the function f :[, ] R, f () = P sin(k ) + Q is shown below. The values of P, k, and Q respectively are P k Q A B. 1 C. 1 D E Question 9 If y = cos (), then dy d A. 4sin() cos() B. 4sin() cos() C. 4 sin( ) D. 4 sin( ) E. 4sin() cos() is equal to Question 10 If y = log e (), then the rate of change of y with respect to when = is equal to A. log e () B. 1 C. 1 + log e () D. E. 1 + log (e) PART I continued

9 9 MATH METH (CAS) EXAM 1 PT1 Question 11 Let f :R R be a function such that f ( 1) = 0 and f ( ) > 0 when < 1 and f ( ) > 0 when > 1. At = 1, the graph of f has a A. local minimum. B. local maimum. C. stationary point of inflection. D. point of discontinuity. E. gradient of 1. Question 1 Part of the graph of the function f is shown below. y a b y = f () Let g be a function such that g ( ) = f ( ). n the interval (a, b), the graph of g will have A. negative gradient. B. positive gradient. C. a local minimum value. D. a local maimum value. E. zero gradient. Question 13 If f ( ), then f () could be equal to A. e + 3 B. 4 e + 5 C. 8 e + D. 4 log e () 4 E. log e (8) + 5 PART I continued TURN VER

10 MATH METH (CAS) EXAM 1 PT1 10 Question 14 4 If f ( ) d =, then f ( ) + 3 d is equal to 1 A. B. 4 C. 7 D. 10 E ( ) Question 15 Let g be a continuous function on the interval [0, 5], and f a function such that f ( ) Then 5 0 g( ) d A. g ( 5) g ( 0) B. f (5) C. f ( 5) f ( 0) D. g( 5) g( 0) E. f ( 5) f ( 0) is equal to = g() for all [0, 5]. Question 16 The total area of the regions enclosed by the graph of y = sin() and the -ais, between = 0 and =, is equal to A. 1 B. C. 4 D. 8 E. 16 Question 17 The interval [0, 4] is divided into n equal subintervals by the points 0 1,... n 1, n where 0 = 0 < 1 <... < n 1 < n = 4. Let = i i 1 for i = 1,,..., n. n 0 i 1 Then lim ( i δ) is equal to δ = 0 A. d B C. 0 d D. 4 E. 8 PART I continued

11 11 MATH METH (CAS) EXAM 1 PT1 Question 18 Let f () = e. For all positive real numbers and y, f ( + y) is equal to A. f () + f (y) B. f () f (y) C. f (y) D. (f ()) y E. f ( y ) Question 19 The line with equation y = + k, where k is a real number, intersects the parabola with equation y = + in two distinct points if A. k < B. k > C. k = D. k < E. k Question 0 Let p() = ( + a)( + b)( c) where a, b and c are three distinct positive real numbers. The number of real solutions to the equation p() = 0 is eactly A. 0 B. 1 C. D. 3 E. 4 Question 1 Let f ( ) = The equations of the asymptotes of the graph of the inverse function f 1 are A. = 1 and y = 3 B. = 1 and y = 3 C. = 3 and y = 1 D. = 3 and y = 1 E. = 1 and y = 3 PART I continued TURN VER

12 MATH METH (CAS) EXAM 1 PT1 1 Question If log e () log e ( + ) = 1 + log e (y), then y is equal to A. B. C. D. E ( ) e + ( ) Question 3 The diagram below shows the graphs of two normal distribution curves with means 1 and and standard deviations 1 and respectively. y X1 ~ N( µ 1, σ 1 ) X ~ N( µ, σ ) Which one of the following statements is true? A. 1 > and 1 = B. 1 > and 1 > C. 1 = and 1 > D. 1 = and 1 < E. 1 < and 1 = PART I continued

13 13 MATH METH (CAS) EXAM 1 PT1 Question 4 Which of the following tables could represent the probability distribution of a discrete random variable? I v Pr(V = v) II w Pr(W = w) III Pr(X = ) IV y Pr(Y = y) V z Pr(Z = z) A. I and III B. I and IV C. II and IV D. II, III and V E. I, II and V Question 5 The lifetime, in hours, of a light globe, can be modelled by a continuous random variable with probability density function g( t) = 0.001e 0.001t if t 0 0 if t 0 The probability, correct to three decimal places, that a randomly selected light globe has a lifetime less than 1000 hours is A B C D E PART I continued TURN VER

14 MATH METH (CAS) EXAM 1 PT1 14 Question 6 60 per cent of all tickets sold at a racecourse are Adult tickets and the remaining 40 per cent are Concession tickets. A random sample of 0 tickets is taken. The probability that this sample contains eactly twelve Adult tickets is equal to A C 1 C C0 B. 0 C 1 (0.4) 8 (0.6) 1 C. 0 C 1 (0.4) 1 (0.6) 8 D. (0.4) 8 (0.6) 1 E. (0.4) 1 (0.6) 8 Question 7 A bag contains 1 bread rolls, of which 8 are white and the remainder multigrain. Tony takes bread rolls at random from the bag to eat. The probability that at least one is a multigrain roll is A B. 1 C 1 C 1 C D. 1 E. 8 C 8 1 C 1 1 C C 4 C 1 8 C 1 1 C C 1 END F PART I MULTIPLE-CHICE QUESTIN BK

15 Victorian Certificate of Education 003 SUPERVISR T ATTACH PRCESSING LABEL HERE Figures Words STUDENT NUMBER MATHEMATICAL METHDS (CAS) PILT STUDY Written eamination 1 (Facts, skills and applications) Friday 7 November 003 Reading time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to am (1 hour 30 minutes) PART II QUESTIN AND ANSWER BK This eamination has two parts: Part I (multiple-choice questions) and Part II (short-answer questions). Part I consists of a separate question book and must be answered on the answer sheet provided for multiple-choice questions. Part II consists of this question and answer book. You must complete both parts in the time allotted. When you have completed one part continue immediately to the other part. Structure of book Number of Number of questions Number of questions to be answered marks Letter Students are permitted to bring into the eamination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set-squares, aids for curve sketching, up to four pages (two A4 sheets) of pre-written notes (typed or handwritten) and one approved CAS calculator (memory may be retained) and/or one scientific calculator. For the TI-9, Voyage 00 or approved computer based CAS, their full functionality and/or one scientific calculator may be used, but other programs or files are not permitted. Students are NT permitted to bring into the eamination room: blank sheets of paper and/or white out liquid/ tape. Materials supplied Question and answer book of 8 pages. Instructions Detach the formula sheet from the centre of the Part I book during reading time. Write your student number in the space provided above on this page. All written responses must be in English. At the end of the eamination Place the answer sheet for multiple-choice questions (Part I) inside the front cover of this question and answer book. Students are NT permitted to bring mobile phones and/or any other electronic communication devices into the eamination room. VICTRIAN CURRICULUM AND ASSESSMENT AUTHRITY 003

16 MATH METH (CAS) EXAM 1 PT Working space PART II continued

17 3 MATH METH (CAS) EXAM 1 PT Instructions for Part II Answer all questions in the spaces provided. A decimal approimation will not be accepted if an eact answer is required to a question. In questions where more than 1 mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question 1 Let f : R R, f () = a + b + c, where a, b and c are real numbers and a 0. Suppose that f (1) = 6 and f ( 1) = 4. a. Find the values of a and b in terms of c. 1 0 ( ) = b. Suppose f d 6. Find the value of c. + = 4 marks PART II continued TURN VER

18 MATH METH (CAS) EXAM 1 PT 4 Question ( ) = ( ) Find the eact solutions of the equation sin π 3 cos π, 0 1. marks Question 3 Let f : D R, f () = log e ( + 3 ) + 1, where D is the maimal domain of f. a. State D. b. Find the eact coordinates of the points where the graph of y = f () intersects the - and y-aes. PART II Question 3 continued

19 5 MATH METH (CAS) EXAM 1 PT c. Sketch the graph of y = f () on the aes below. Indicate any asymptote with its equation. y = 6 marks PART II continued TURN VER

20 MATH METH (CAS) EXAM 1 PT 6 Question 4 The graphs with equations y = + 1 and y = 1 e are shown below. The graphs intersect at the point P which has -coordinate k. y P y = 1 e y = + 1 a. Find the value of k, correct to three decimal places. b. Write down a definite integral, the value of which is the area of the shaded region. c. Hence find the area of the shaded region, correct to two decimal places = 3 marks PART II continued

21 7 MATH METH (CAS) EXAM 1 PT Question 5 The probability density function of the continuous random variable X is given by f () = a( ), for < < and 0 elsewhere (where a is a real constant). a. n the aes provided sketch the graph of this probability density function. f() b. Find the value of a. + = 4 marks PART II continued TURN VER

22 MATH METH (CAS) EXAM 1 PT 8 Question 6 Kim goes to the sports centre each evening and either works out in the gym or has a swim. She never has a swim two evenings in a row. If she has a work-out in the gym one evening, then the net evening she is twice as likely to have a swim as she is to have a work-out in the gym. n a particular Monday evening, she works out in the gym. a. What is the probability that she works out in the gym on both the Tuesday and Wednesday evenings of that week? b. What is the probability that she has a swim on the Friday of that week? = 4 marks END F PART II QUESTIN AND ANSWER BK

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24 + ( ) ( ) = + = + + ( ) = = + ( ) = = + ( ) = + ( ) = + ( ) = ( ) = + ( + ) ( ) + ( ) = ( ) ( ) = µ µ µ µ µ µ µ µ = µ = = µ =

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