Maths Specialist 3C3D Calc Free 2012

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Western Australian Certicate of Education

Examination, 2012

Question/Answer Booklet

Copyright School Curriculum and Standards Authority 2012

MATHEMATICS:

SPECIALIST

3C/3DSection One:Calculator-free

Time allowed for this sectionReading time before commencing work: ve minutesWorking time for this section: fty minutes

Materials required/recommended for this sectionTo be provided by the supervisorThis Question/Answer BookletFormula Sheet

To be provided by the candidate

Standard items: pens (blue/black preferred), pencils (including coloured), sharpener,correction tape/uid, eraser, ruler, highlighters

Special items: nil

Important note to candidatesNo other items may be taken into the examination room. It is yourresponsibility to ensure thatyou do not have any unauthorised notes or other items of a non-personal nature in theexamination room. If you have any unauthorised material with you, hand it to the supervisorbeforereading any further.

*MAS3CD-S1*MAS3CD-S1

Number of additionalanswer booklets used(if applicable):

Ref: 12-101


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CALCULATOR-FREEMATHEMATICS: SPECIALIST 3C/3D 2

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Section

Section One:Calculator-free

Section Two:Calculator-assumed

Number ofquestionsavailable

7

12

Number ofquestions tobe answered

7

12

Working

time

(minutes)

50

100

Total

Marksavailable

50

100

150

Percentage

of total exam

33

66

100

Instructions to candidates

1. The rules for the conduct of Western Australian external examinations are detailed inthe Year 12 Information Handbook 2012. Sitting this examination implies that youagree to abide by these rules.

2. Answer the questions according to the following instructions.

Section One: Write answers in this Question/Answer Booklet. Answer all questions.

Show all your working clearly.Your working should be in sufcient detail to allow your

answers to be checked readily and for marks to be awarded for reasoning. Incorrectanswers given without supporting reasoning cannot be allocated any marks. For anyquestion or part question worth more than two marks, valid working or justication isrequired to receive full marks. If you repeat an answer to any question, ensure that youcancel the answer you do not wish to have marked

It is recommended that you do not use pencil, except in diagrams.

3. You must be careful to conne your responses to the specic questions asked and tofollow any instructions that are specic to a particular question.

4. Spare pages are included at the end of this booklet. They can be used for planning yourresponses and/or as additional space if required to continue an answer. Planning: If you use the spare pages for planning, indicate this clearly at the top

of the page. Continuing an answer: If you need to use the space to continue an answer,

indicate in the original answer space where the answer is continued, i.e. give thepage number. Fill in the number of the question that you are continuing to answerat the top of the page.

5. The Formula Sheet is nothanded in with your Question/Answer Booklet.

Structure of this paper

13

23


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CALCULATOR-FREE 3 MATHEMATICS: SPECIALIST 3C/3D

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Question 1 (6 marks)

(a) Evaluate sin x dx .

(3 marks)

(b) Determine 6x2ex3+9 dx . (3 marks)

Section One: Calculator-free (50 Marks)

This section has seven (7)questions. Answer allquestions. Write your answers in the spacesprovided.

Spare pages are included at the end of this booklet. They can be used for planning your

responses and/or as additional space if required to continue an answer. Planning: If you use the spare pages for planning, indicate this clearly at the top of the page. Continuing an answer: If you need to use the space to continue an answer, indicate in the

original answer space where the answer is continued, i.e. give the page number. Fill in thenumber of the question(s) that you are continuing to answer at the top of the page.

Working time: 50 minutes.


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Let the 22 matrix A=1 0

1 1 .

(a) Evaluate A2and A3. (2 marks)

(b) Given your answers to (a), conjecture the form of the matrix Anwhere nis a positiveinteger. Use proof by mathematical induction to establish the truth of your conjecture.

(4 marks)


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(c) Determine the effect of applying the matrixA2to the rectangle with vertices (1,0), (3,0),(3,1) and (1,1). What type of transformation does the matrix A2represent? (2 marks)


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(a) Evaluatecos 3x + 2sin 2x

6sin 3x 8cos 2x0

dx3 . (3 marks)

(b) Use the substitution u = lnxto determinex dx

lnx + lnx . (4 marks)


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In the triangle ABCthe sides BCand AChave midpoints Mand Nrespectively. Linesperpendicular to these sides are drawn through their respective midpoints and intersect at thepoint O.

Denote the line segments OA =a, OB =band OC=c.

B

b

ac

C

O

A N

M

(a) Express ON

in terms of aand cand OM

in terms of band c. (1 mark)

(b) Show that |a|=|b|=|c|. (2 marks)

(c) If Pis the midpoint of side AB, show that OP

is perpendicular to

AB . (3 marks)


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A mining drill moves through a cross-section of rock such that its coordinatesxand ysatisfy:

dxdt

dydt=2sin t, =6sin 2t

wherexand yare measured in metres and tis the time expressed in minutes.Initiallyx= 2and y=3.

(a) Determinedydx when t =

6 . (2 marks)

(b) Determine the position of the drill when t =. (3 marks)

(c) Determine the Cartesian equation of the path of the drill. (3 marks)


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(d) Sketch the path of the drill. Label clearly the points on the path corresponding to the twotimes t=0and t=. (3 marks)


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Describe geometrically the set of points r=(x, y, z) that satisfy each of the following vectorequations in which adenotes a non-zero constant vector.

(a) |ra| =3 (2 marks)

(b) (ra) a=0 (2 marks)

(c) (ra) r=0 (3 marks)


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(a) A square matrix is said to be singular if it has no inverse.

Show that the matrix X=cos 72 sin 18sin 72 cos 18 is singular. (2 marks)

(b) Let Abe any square matrix such that A3=0. Prove by contradiction that Amust be singular. (3 marks)

End of questions


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Additional working space

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Published by the School Curriculum and Standards Authority of Western Australia

27 Walters Drive

This examination paper apart from any third party copyright material contained in it may be freely copied, or communicated on anintranet, for non-commercial purposes in educational institutions, provided that it is not changed and that the School Curriculum andStandards Authority is acknowledged as the copyright owner.

Copying or communication for any other purpose can be done only within the terms of the Copyright Act or with prior writtenpermission of the Authority. Copying or communication of any third party copyright material can be done only within the terms of the

Copyright Act or with permission of the copyright owners.

Maths Specialist 3C3D Calc Free 2012

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