MATHEMATICAL METHODS (CAS) Written examination 1

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1 Victorian Certificate of Eucation 2006 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Wors MATHEMATICAL METHODS (CAS) Written examination 1 Friay 3 November 2006 Reaing time: 9.00 am to 9.15 am (15 minutes) Writing time: 9.15 am to am (1 hour) QUESTION AND ANSWER BOOK Number of questions Structure of book Number of questions to be answere Number of marks Stuents are permitte to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers. Stuents are NOT permitte to bring into the examination room: notes of any kin, blank sheets of paper, white out liqui/tape or a calculator of any type. Materials supplie Question an answer book of 10 pages, with a etachable sheet of miscellaneous formulas in the centrefol. Working space is provie throughout the book. Instructions Detach the formula sheet from the centre of this book uring reaing time. Write your stuent number in the space provie above on this page. All written responses must be in English. Stuents are NOT permitte to bring mobile phones an/or any other unauthorise electronic evices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2006

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3 MATHMETH & MATHMETH(CAS) EXAM 1 Instructions Answer all questions in the spaces provie. A ecimal approximation will not be accepte if an exact answer is require to a question. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise inicate, the iagrams in this book are not rawn to scale. Question 1 Let f (x) = x an g(x) = 2x + 1. Write own the rule of f (g(x)). Question 2 For the function f : R R, f (x) = 3e 2x 1, a. Þn the rule for the inverse function f 1 b. Þn the omain of the inverse function f 1. 2 marks TURN OVER

4 2006 MATHMETH & MATHMETH(CAS) EXAM 1 4 Question 3 a. Let f (x) = e cos (x). Fin f (x) b. Let y = x tan (x). Evaluate y x when x = π 6. 3 marks Question 4 π For the function f :[ π, π ] R, f (x) = 5cos 2 x + 3 a. write own the amplitue an perio of the function 2 marks b. sketch the graph of the function f on the set of axes below. Label axes intercepts with their coorinates. Label enpoints of the graph with their coorinates. y π 1 O π x 3 marks

5 MATHMETH & MATHMETH(CAS) EXAM 1 Question 5 Let X be a normally istribute ranom variable with a mean of 72 an a stanar eviation of 8. Let Z be the stanar normal ranom variable. Use the result that Pr(Z < 1) = 0.84, correct to two ecimal places, to Þn a. the probability that X is greater than 80 b. the probability that 64 < X < 72 c. the probability that X < 64 given that X < marks TURN OVER

6 2006 MATHMETH & MATHMETH(CAS) EXAM 1 6 Question 6 The probability ensity function of a continuous ranom variable X is given by x 1 x 5 f( x)= 12 0 otherwise a. Fin Pr (X < 3). 2 marks b. If Pr (X a) = 5, Þn the value of a. 8 2 marks

7 MATHMETH & MATHMETH(CAS) EXAM 1 Question 7 The graph of f : [ 5, 1] R where f (x) = x 3 + 6x 2 + 9x is as shown. y 3 O x a. On the same set of axes sketch the graph of y = f( x). 2 marks b. State the range of the function with rule y = f( x) an omain [ 5, 1]. Question 8 A normal to the graph of y = x has equation y = 4x + a, where a is a real constant. Fin the value of a. 4 marks TURN OVER

8 2006 MATHMETH & MATHMETH(CAS) EXAM 1 8 Question 9 A rectangle XYZW has two vertices, X an W, on the x-axis an the other two vertices, Y an Z, on the graph of y = 9 3x 2, as shown in the iagram below. The coorinates of Z are (a, b) where a an b are positive real numbers. y Y( a, b) Z(a, b) X O W x a. Fin the area, A, of rectangle XYZW in terms of a. b. Fin the maximum value of A an the value of a for which this occurs. 3 marks

9 MATHMETH & MATHMETH(CAS) EXAM 1 Question 10 Jo has either tea or coffee at morning break. If she has tea one morning, the probability she has tea the next morning is 0.4. If she has coffee one morning, the probability she has coffee the next morning is 0.3. Suppose she has coffee on a Monay morning. What is the probability that she has tea on the following Wenesay morning? 3 marks CONTINUED OVER PAGE TURN OVER

10 2006 MATHMETH & MATHMETH(CAS) EXAM 1 10 Question 11 Part of the graph of the function f : R R, f (x) = x 2 + ax + 12 is shown below. If the shae area is 45 square units, Þn the values of a, m an n where m an n are the x-axis intercepts of the graph of y = f (x). y 12 n O 3 m x 5 marks END OF QUESTION AND ANSWER BOOK

11 MATHEMATICAL METHODS AND MATHEMATICAL METHODS (CAS) Written examinations 1 an 2 FORMULA SHEET Directions to stuents Detach this formula sheet uring reaing time. This formula sheet is provie for your reference. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 2006

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13 3 MATH METH & MATH METH (CAS) Mathematical Methos an Mathematical Methos CAS Formulas Mensuration area of a trapezium: 1 2 a+ b h curve surface area of a cyliner: 2π rh volume of a sphere: ( ) volume of a pyrami: 1 volume of a cyliner: π r 2 h area of a triangle: volume of a cone: 1 2 π r h 3 3 Ah 4 3 π r bcsin A Calculus x x n nx n 1 n 1 n+ 1 ( )= xx= x + c, n 1 n + 1 x e ax ae ax ax 1 ( )= e x a e ax = + c ( log e( x) )= 1 1 x x x x = loge x + c 1 ( sin( ax) )= a cos( ax) sin( ax) x = cos( ax) + c x a 1 ( cos( ax) ) = a sin( ax) cos( ax) x = sin( ax) + c x a a 2 ( tan( ax) ) = = a sec ( ax) x cos 2 ( ax) prouct rule: ( x uv u v v u v u u v )= x + x quotient rule: u x x x v = 2 v chain rule: y x = y u u x approximation: f x + h f x hf x ( ) ( )+ ( ) Probability Pr(A) = 1 Pr(A ) Pr(A B) = Pr(A) + Pr(B) Pr(A B) ( ) ( ) Pr(A B) = Pr A B Pr B mean: µ = E(X) variance: var(x) = σ 2 = E((X µ) 2 ) = E(X 2 ) µ 2 probability istribution mean variance iscrete Pr(X = x) = p(x) µ = x p(x) σ 2 = (x µ) 2 p(x) continuous Pr(a < X < b) = f( x) x b a 2 2 µ = xf( xx ) σ = µ ( x ) f( x) x END OF FORMULA SHEET