Year 2011 VCE. Mathematical Methods CAS. Trial Examination 1

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1 Year 0 VCE Mathematical Methods CAS Trial Examination KILBAHA MULTIMEDIA PUBLISHING PO BOX 7 KEW VIC 30 AUSTRALIA TEL: (03) FAX: (03) kilbaha@gmail.com

2 IMPORTANT COPYRIGHT NOTICE For individual use only Not to be reproduced Kilbaha Multimedia Publishing 0 ABN This material is copyright. Subject to statutory exception and to the provisions of the relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Kilbaha Pty Ltd. The contents of this work are copyrighted. Unauthorised copying of any part of this work is illegal and detrimental to the interests of the author. For authorised copying within Australia please check that you have a licence from This permits the copying of small parts of the material, in limited quantities, within the conditions set out in the licence. Teachers and students are reminded that for the purposes of school requirements and external assessments, students must submit work that is clearly their own. This file has been purchased for individual use only and reproduction by teachers for their students is not permitted. Schools can purchase a licence to use this material and then distribute this electronic file to the students at the school for their exclusive use. This distribution can be done either on an Intranet Server or on media for the use on stand-alone computers. Schools can purchase a licence to use this material and then distribute this printed file to the students at the school for their exclusive use. To purchase a school licence, contact us at kilbaha@gmail.com KILBAHA MULTIMEDIA PUBLISHING PO BOX 7 KEW VIC 30 AUSTRALIA TEL: (03) FAX: (03) kilbaha@gmail.com Kilbaha Multimedia Publishing

3 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 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.

4 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

5 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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 Find the values of a, b, c, d, h and k. 3 marks END OF QUESTION AND ANSWER BOOKLET END OF EXAMINATION

13 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.

14 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: 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 = ( ) ( )

MATHEMATICAL METHODS (CAS) Written examination 1

Victorian Certificate of Education 2010 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Words MATHEMATICAL METHODS (CAS) Written examination 1 Friday 5 November 2010 Reading time: