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Semester One Examination, 2016

Question/Answer Booklet

Student Name:_____________________________Teacher’s Name:____________________________Student Name:_____________________________Teacher’s Name:____________________________MATHEMATICS

SPECIALIST

UNIT 1

Section One:

Calculator-free

In words

Time allowed for this section

Reading time before commencing work: five minutes

Working time for section: fifty minutes

Materials 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 fluid/tape, eraser, 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 examinations are detailed in the school handbook. Sitting this examination implies that you agree to abide by these rules.

Write your answers in this Question/Answer Booklet.

You must be careful to confine your response to the specific question asked and to follow any instructions that are specified 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.

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 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.

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

Section One: Calculator-free 35% (49 Marks)

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

Working time for this section is 50 minutes.

Question 1 (7 marks)

(a) Evaluate

(i) . (2 marks)

(ii) . (3 marks)

(b) Determine the values of a and b given (2 marks)

Question 2 (8 marks)

Given and , determine

(a) . (1 mark)

(b) . (1 mark)

(c) . (2 marks)

(d) a unit vector in the same direction as . (2 marks)

(e) the scalar projection of b onto . (2 marks)

Question 3 (7 marks)

(a) Prove that the opposite angles of a cyclic quadrilateral are supplementary. (4 marks)

(b) Determine, with reasons, the size of in the diagram below. (3 marks)

Question 4 (7 marks)

Consider the vectors , and .

(a) Determine the vector projection of a onto b. (3 marks)

(b) Express c in the form . (4 marks)

Question 5 (8 marks)

(a) An equilateral triangle of side 2a circumscribes a circle, as shown in the diagram below. Express the exact radius of the circle in terms of a. (4 marks)

(b) Two circles touch internally at B, as shown below. AB, AC and AD are tangents, and . Determine, with reasons, the size of . (4 marks)

Question 6 (6 marks)

OABC is a trapezium with . M lies on the diagonal OB so that and N lies on the diagonal CA so that . Let and .

By determining a vector for MN, or otherwise, prove that ABNM is a parallelogram.

Question 7 (6 marks)

(a) Show that when but not when . (2 marks)

(b) Prove by contradiction that, for every positive real number x, . (5 marks)

Additional working space

Question number: _________

Additional working space

Question number: _________

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