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Victorian Certificate of Education 0 STUDENT NUMBER Figures Words Letter MATHEMATICAL METHOD CAS Trial Written Examination Reading time: 5 minutes Total writing time: hour QUESTION AND ANSWER BOOK Structure of book Number of Number of questions Number of questions to be answered marks 0 0 40 Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers. Students are NOT permitted to bring into the examination room: notes of any kind, blank sheets of paper, white out liquid/tape or a calculator of any type. Materials supplied Question and answer book of 4 pages with a detachable sheet of miscellaneous formulas at the end of this booklet. Working space is provided throughout the booklet. Instructions Detach the formula sheet from the end of this book during reading time. Write your student number in the space provided above on this page. All written responses must be in English. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room.
Mathematical Methods CAS Trial Examination 0 Page 4 Instructions Answer all questions in the spaces provided. A decimal approximation will not be accepted if an exact answer is required to a question. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Question For the linear simultaneous equations x + ky = k+ ( ) k+ x 6y = 0 where k is a real constant, find the value(s) of k, for which there is i. a unique solution. ii. infinitely many solutions. iii. no solution. 4 marks
Mathematical Methods CAS Trial Examination 0 Page 5 Question Let f ( x) = x be a function defined on its maximal domain. Find the inverse function f stating its domain and range. Question 3 a. Differentiate xsin( x ) with respect to x. 3 marks b. Hence find xcos( ) x dx mark marks
Mathematical Methods CAS Trial Examination 0 Page 6 ( ) cos x for 0 x b c. The function f ( x) = 0 otherwise is a probability density function for the continuous random variable X. π i. Show that b =. 4 ii. Find the mean of f ( x ). mark marks iii. Find the median of f ( x ). marks
Mathematical Methods CAS Trial Examination 0 Page 7 Question 4 If d 3 4 log x + bx e dx = 4x 3 + g x ( ), find the value of b and the function g ( x ). 3 marks Question 5 The speed v, in metres per second, of an object moving in a straight line is given by a 7 function of time t, in seconds, where v() t = where t 0. ( 3t + ) Find the distance travelled in metres by the object in the first seconds. 3 marks
Mathematical Methods CAS Trial Examination 0 Page 8 Question 6 Let ( ) 3 f : R R where f x = x k x, where k is a positive real constant. a. Find in terms of k, the subset of R, for which the graph of the function f ( x ) is strictly increasing. marks f x, the x-axis and x = 0andx= k is equal to 64 units, find the value of k. b. If the area bounded by the graph of ( ) marks
Mathematical Methods CAS Trial Examination 0 Page 9 Question 7 π π g : R + R where g x = cos x. Given the functions f :, R where f ( x) = tan ( x) ( ) ( ) and = show that sin ( α ) = ( 5 ) a. If the two functions intersect at x α, b. Show that the functions intersect at right angles. 3 marks 3 marks
Mathematical Methods CAS Trial Examination 0 Page 0 Question 8 A binomial distribution of the random variable X, with six independent trials, is such that Pr X = 3 = Pr X = 4. If p is the probability of a success on any trial, find the value of ( ) ( ) p. 3 marks Question 9 a. Evaluate 4 4 dx x mark
Mathematical Methods CAS Trial Examination 0 Page b. The diagram below shows a shield ABCDEFGH, the shield is symmetrical, with the curve CD being the reflection of AB in the x-axis, and the curve EF the reflection of CD in the y-axis. The function describing the curve AB is given by 4 f : [,4] R where f ( x) =. x y x i. Write down the function which describes the curve EF. mark ii. Find the total area of the shield. mark
Mathematical Methods CAS Trial Examination 0 Page Question 0 x a b x h The transformation T : R R of the plane defined by T y = + c d y k π x maps the equation y = cos( x) into y = cos + + 4. 3 Find the values of a, b, c, d, h and k. 3 marks END OF QUESTION AND ANSWER BOOKLET END OF EXAMINATION
MATHEMATICAL METHODS CAS Written examination FORMULA SHEET Directions to students Detach this formula sheet during reading time. This formula sheet is provided for your reference.
Mensuration Mathematical Methods CAS Formulas area of a trapezium: ( a+ b) h volume of a pyramid: Ah 3 curved surface area of a cylinder: π rh volume of a sphere: volume of a cylinder: 4 3 3 π r r π h area of triangle: bcsin ( A ) volume of a cone: 3π rh Calculus d n n ( x ) nx n n+ = xdx x c, n dx = + n + d ax ax ( e ) ae ax ax = e dx e c dx = + a d ( loge ( x) ) = dx log e x c dx x = + x d ( sin ( ax) ) = a cos( ax) sin ( ax) dx cos( ax) c dx = + a d ( cos( ax) ) = a sin ( ax) cos( ax) dx sin ( ax) c dx = + a d a ( tan ( ax) ) = = asec ( ax) dx cos ( ax) product rule: ( ) Chain rule: d uv = u dv + v du dx dx dx dx dy du du dx du dv v u d u quotient rule: dx dx = dx v v f x+ h f x + h f x dy = approximation: ( ) ( ) ( ) Probability Pr( A) = Pr( A ) Pr( A B) = Pr( A) + Pr( B) Pr( A B) Pr( A B) Pr ( A/ B) = n Pr( B) Transition Matrices S n = T S0 mean: μ = E ( X ) variance: ( ) ( ) ( ) ( ) var X = σ = E X μ = E X μ probability distribution mean variance discrete Pr( X = x) = p( x) μ = x p( x ) σ = ( x μ) p( x) continuous a X b f x dx μ xf x dx σ = x μ f x dx b Pr( ) ( ) < < = ( ) a = ( ) ( )