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Rossmoyne Senior High School

Semester One Examination, 2016

Question/Answer Booklet

SOLUTIONSSOLUTIONSMATHEMATICS

SPECIALIST

UNIT 1

Section One:

Calculator-free

In words

Your name

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% (50 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)

Solution Specific behaviours expands and cancels simplifiesSolution Specific behaviours expands and cancels simplifies

(ii) . (3 marks)

Solution Specific behaviours uses P and C notation correctly cancels like factorials simplifies answerSolution Specific behaviours uses P and C notation correctly cancels like factorials simplifies answer

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

Solution Specific behaviours factors out 8! to obtain a simplifies rest of expression to obtain bSolution Specific behaviours factors out 8! to obtain a simplifies rest of expression to obtain b

Question 2 (6 marks)

Given and , determine

(a) . (1 mark)

Solution Specific behaviours determines vectorSolution Specific behaviours determines vector

(b) . (1 mark)

Solution Specific behaviours determines vectorSolution Specific behaviours determines vector

(c) . (2 marks)

Solution Specific behaviours determines vector determine magnitudeSolution Specific behaviours determines vector determine magnitude

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

Solution Specific behaviours determines magnitude of vector determines unit vectorSolution Specific behaviours determines magnitude of vector determines unit vector

Question 3 (7 marks)

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

Solution∠DOB=2α (angle on arc DCB at centre is twice angle on circumference)∠DOB=2β (angle on arc DAB at centre is twice angle on circumference)2α+2β=360 (angle sum of circle)α+β=180 - opposite angles are supplementarySpecific behaviours labelled diagram uses angle at centre twice circumference uses angle sum at centre completes proofSolution∠DOB=2α (angle on arc DCB at centre is twice angle on circumference)∠DOB=2β (angle on arc DAB at centre is twice angle on circumference)2α+2β=360 (angle sum of circle)α+β=180 - opposite angles are supplementarySpecific behaviours labelled diagram uses angle at centre twice circumference uses angle sum at centre completes proof

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

Solution Specific behaviours uses isosceles triangles determines determines with reasonSolution Specific behaviours uses isosceles triangles determines determines with reason

Question 4 (7 marks)

Consider the vectors , and .

(a) Determine the magnitude of . (3 marks)

Solution Specific behaviours determines resultant uses magnitude formula states correct magnitudeSolution Specific behaviours determines resultant uses magnitude formula states correct magnitude

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

Solution Specific behaviours writes simultaneous equations eliminates one variable and solves solves for other variable writes in required formSolution Specific behaviours writes simultaneous equations eliminates one variable and solves solves for other variable writes in required form

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)

Solution Specific behaviours draws right triangle using tangent and radius places angle and variables on diagram determines relationship between r and a expresses r in terms of aSolution Specific behaviours draws right triangle using tangent and radius places angle and variables on diagram determines relationship between r and a expresses r in terms of a

(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)

SolutionAs B is common to both circles, then as tangents from external point, and . Hence triangles ABC and ACD are both isosceles. Specific behaviours explains why determines determines determines SolutionAs B is common to both circles, then as tangents from external point, and . Hence triangles ABC and ACD are both isosceles. Specific behaviours explains why determines determines determines

Question 6 (7 marks)

(a) In the diagram below, and .

Determine the size of the following angles.

(i) . (1 mark)

Solution Specific behaviours states angleSolution Specific behaviours states angle

(ii) . (1 mark)

SolutionSpecific behaviours states angleSolutionSpecific behaviours states angle

(iii) . (1 mark)

SolutionSpecific behaviours states angleSolutionSpecific behaviours states angle

(b) In the diagram below, AB and AC are tangents to the circle and . Determine, with reasons, the sizes of and . (4 marks)

Solution Specific behaviours angle BDC with reason angle BOC with reasonSolution Specific behaviours angle BDC with reason angle BOC with reason

Question 7 (6 marks)

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

Solution Specific behaviours clearly demonstrates first case is true demonstrates second case is falseSolution Specific behaviours clearly demonstrates first case is true demonstrates second case is false

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

SolutionAssume there exists a positive real number x such that .Since , then and , and so inequality can be multiplied by without reversing inequality direction.Hence This results in the contradiction that and so we must conclude that no such positive real number x exists so that , hence proving that.Specific behaviours writes contra of proof cross multiplies notes no need to reverse inequality simplifies inequality makes conclusionSolutionAssume there exists a positive real number x such that .Since , then and , and so inequality can be multiplied by without reversing inequality direction.Hence This results in the contradiction that and so we must conclude that no such positive real number x exists so that , hence proving that.Specific behaviours writes contra of proof cross multiplies notes no need to reverse inequality simplifies inequality makes conclusion

Additional working space

Question number: _________

Additional working space

Question number: _________

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