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Semester One Examination 2018

Question/Answer Booklet

MATHEMATICS SPECIALIST

UNIT 1

Section One:

Calculator-free

Time allowed for this section

Reading time before commencing work: five minutes

Working time for paper: fifty minutes

Material required/recommended for this section

To be provided by the supervisor

This Question/Answer booklet

Formula Sheet

To be provided by the candidate

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

Special Items: nil

Important note to candidates

No other items may be taken into the examination room. It is your responsibility to ensure that you do not have any unauthorised notes or other items of a non-personal nature in the examination room. If you have any unauthorised material with you, hand it to the supervisor before reading any further.

Structure of this paper

Instructions to candidates

The rules for the conduct of Western Australian external examinations are detailed in the Year 12 Information Handbook 2018. Sitting this examination implies that you agree to abide by these rules.

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 sufficient detail to allow your

answers to be checked readily and for marks to be awarded for reasoning. Incorrect

answers given without supporting reasoning cannot be allocated any marks. For any

question or part question worth more than two marks, valid working or justification is

required to receive full marks. If you repeat an answer to any question, ensure that you

cancel the answer you do not wish to have marked.

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

You must be careful to confine your responses to the specific questions asked and to follow any instructions that are specific to a particular question.

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 the number of the question that you are continuing to answer at the top of the page.

The Formula Sheet is not handed in with your Question/Answer Booklet.

Section One: Calculator–free 50 marks

This section has seven (7) questions. Attempt all questions. Write your answers in the spaces provided.

Working time: 50 minutes

Question 1 (6 marks)

Points A and B have position vectors a=2i+j and b=i+3j respectively.

(a) Show that a is perpendicular to b-a. (3 marks)

(b) Given that OABC is a square, with O as the origin, state:

(i) the position vector of vertex C. (1 mark)

(ii) AC (2 marks)

Question 2 (8 marks)

(a) Evaluate each of the following.

(i) 13!+12!13!-12! (2 marks)

(ii) 10C4 8C4 (2 marks)

(b) Show that knk=nn-1k-1 is true for all integers n,k with n>k. (4 marks)

Question 3 (7 marks)

Consider the following statements.

A: If m>n, with m,n∈R, then m2>n2.

B: If the triangle has three equal angles, then the triangle is equilateral.

C: ∀ p∈N,∃ q∈R such that q=p

D: If n is divisible by 6, then n is divisible by 3.

(a) Provide a counter example for statement A. (1 mark)

(b) Write down the contrapositive of statement B.

Is the contrapositive always true? Explain. (2 marks)

(c) Write down the converse of statement D.

Is the converse always true? Explain. (2 marks)

(d) Rewrite statement C in words. (2 marks)

Question 4 (10 marks)

Consider the vectors a and b shown below.

(a) Find a vector u that is parallel to a+b with a magnitude of 210 units.

Sketch u on the same grid above. (4 marks)

Question 4 – Continued

Vector c=-4α where α∈R.

(b) Find α given that:

(i) a and c are parallel. (2 marks)

(ii) c-b=3a (4 marks)

Question 5 (7 marks)

The diagram shows parallelogram OABC.

Let: OA=a and OC=c.

(a) Use the fact that AC and OB are perpendicular to prove that OABC is a rhombus. (3 marks)

(b) Show that the sum of the squares of the lengths of the diagonals of a parallelogram is

equal to the sum of the squares of the lengths of the sides. (4 marks)

Question 6 (8 marks)

Consider the circle shown below with centre at O and diameter AD, with TS tangent to the circle at C.

Points B and E lie on the circumference with EA parallel to BC, ∠ADE = 60° and ∠ABC = 40°.

Determine the size of the following angles giving reason(s) to justify your answer.

(a) ∠AED (2 marks)

(b) ∠ABE (2 marks)

(c) ∠CAE (2 marks)

(d) ∠TCE (2 marks)

Question 7 (4 marks)

Points A, B and C have position vectors a=xi-j, b=i+3j and c=4i+yj respectively,

where x,y∈R.

Given that AB:BC = 2:1, determine the value(s) of x and y. (4 marks)

End of Section One

Additional working space

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Additional working space

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