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Applecross Senior High School

Western Australian Certificate of Education

Semester One Examination, 2018

Question/Answer Booklet

MATHEMATICS

SPECIALIST

UNIT 1

Section One:

Calculator- free

Student’s Name: _______________________________________

As shown on your exam timetable

Student’s Teacher Mrs Waddell

Time allowed for this section

Reading time before commencing work: five minutes

Working time for this 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 unauthorized notes or other items of a non-personal nature in the

examination room. If you have any unauthorized material with you, hand it to the supervisor

before reading any further.

Instructions to candidates

The rules for the conduct of examinations are detailed in the School Examination Rules provided with your exam timetable. 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 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(s) 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 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.

The formula sheet and your notes are not to be handed in with your Question/Answer Booklet.

Section One: Calculator-free 35% (53 Marks)

This section has eight (8) questions. Answer all questions. Write your answers in the spaces provided.

Working time: 50 minutes.

Question 1 (5 marks)

Relative to the origin O, points A and B have position vectors -3i-2j and i-4j respectively.

(a) Determine the unit vector c, where c=AB. (3 marks)

(b) Vector d has magnitude 35, is parallel to c and in the opposite direction. Determine d.

(2 marks)

Question 2 (5 marks)

Let the displacement vectors a, b and c be (11, -4), (5, 14) and (8, m) respectively, where m is a constant.

(a) Determine the vector 3a+2b. (2 marks)

(b) Given that a+b+kc=0, detemine the values of k and m. (3 marks)

Question 3 (8 marks)

Consider the following statement about a simple (no edges that cross) polygon:

If it has an interior angle sum of 360°, then it is a square.

(a) Use a counter-example to explain why the statement is false. (2 marks)

(b) Write the converse statement and state whether it is always, sometimes or never true.

(2 marks)

(c) Write the inverse statement and state whether it is always, sometimes or never true.

(2 marks)

(d) Write the contrapositive statement and state whether it is always, sometimes or never true.

(2 marks)

Question 4 (6 marks)

(a) Determine the value of the constant n, given that the vectors 12i+nj and 5i-8j are perpendicular. (2 marks)

(b) The vectors a and b are such that a=18, b=12 and a∙b=-33. Evaluate

(i) -2a∙3b. (1 mark)

(ii) a+b∙(b-a). (3 marks)

Question 5 (7 marks)

(a) In the diagram below, the vertices of triangle ABC lie on a circle with centre O. Given that ∠ABC=54°, determine the values of ∠AOC and ∠OAC. (2 marks)

(b) The two tangents through a point P exterior to a circle touch it at the points B and D. The chord BD subtends an angle of 130 degrees at a point C on the circumference of the circle as shown below. Determine the angle x between the two tangents. (5 marks)

Question 6 (6 marks)

A drone leaves point P and travels 115 m on bearing of 340° to Q, then 30 m on bearing 070° to R and finally 85 m on bearing 160° to S.

(a) Sketch a neat diagram to show the path of the drone. (2 marks)

(b) The drone is to return directly from S to P. Determine the distance it must fly and on what bearing. (4 marks)

Question 7 (9 marks)

(a) Evaluate 16P11÷ 14P11. (3 marks)

(b) Express 9!+8!+7! in the form a2b!, where a and b are positive integers. (3 marks)

(c) Show that for n∈Z, n≥3, the sum n!+n-1!+n-2! can always be expressed in the form a2b! where a and b are positive integers. (3 marks)

Question 8 (7 marks)

In the diagram below, two chords of a circle, BC and DE, intersect at F. GB is perpendicular to BC at B and GE is perpendicular to DE at E. The line GF intersects chord CD at H.

(a) Explain why GEFB is a cyclic quadrilateral. (1 mark)

(b) Prove that ∠CDE=∠EGF. (3 marks)

(c) Prove that GH is perpendicular to CD. (3 marks)

Additional working space.

Question Number: _________

Additional working space.

Question Number _________