WACEhub

Semester One Examination, 2015

Question/Answer Booklet

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 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 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 2015. 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 (52 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 (8 marks)

(a) Two vectors, and , are shown on the grid below.

Draw and label the vectors on the grid, where , and . (3 marks)

(b) Determine a unit vector perpendicular to the vector . (2 marks)

(c) The point P divides the line segment from M(-3, 3) to N(13, -9) in the ratio 1:3. Determine the position vector of point P. (3 marks)

Question 2 (6 marks)

The statement 'if two rectangles are congruent then they have the same area' is true.

(a) Write the inverse of the statement and explain whether or not the inverse is also true.

(2 marks)

(b) Write the contrapositive of the statement and explain whether or not the contrapositive is also true. (2 marks)

(c) Write the converse of the statement and explain whether or not the converse is also true. (2 marks)

Question 3 (7 marks)

(a) In the diagram below, and .

Determine the sizes of

(i) . (1 mark)

(ii) . (1 mark)

(iii) . (1 mark)

(b) Determine, with reasons, the sizes of the angles marked and in the diagram below.

(4 marks)

Question 4 (8 marks)

(a) Simplify . (2 marks)

(b) Prove that . (3 marks)

(c) If and , determine

(i) . (1 mark)

(ii) . (2 marks)

Question 5 (8 marks)

The vectors are given by and .

(a) Determine

(i) . (1 mark)

(ii) . (1 mark)

(iii) the vector projection of onto . (2 marks)

(b) Determine the vectors if and . (4 marks)

Question 6 (7 marks)

(a) Prove that it is possible to draw a circle through the points , , and shown below. (3 marks)

(b) Prove by contradiction that it is impossible to draw a circle through the vertices of the quadrilateral shown below. (4 marks)

Question 7 (8 marks)

(a) A bag contains 17 identical cubes except for their colour, with four coloured orange, six coloured blue and seven coloured white.

(i) How many different arrangements of coloured cubes are possible when three cubes are drawn from the bag and placed in a line? (1 mark)

(ii) How many different combinations of coloured cubes are possible when three cubes are drawn from the bag? (2 marks)

(iii) Determine the least number of cubes that should be removed from the bag to ensure that the resulting selection contains at least three cubes of one colour. Justify your answer. (2 marks)

(b) Show that if 50 different integers are selected from the set {1, 2, 3, ..., 97, 98}, there will be at least two integers whose sum is 99. (3 marks)

Additional working space

Question number: _________

This examination paper may be freely copied, or communicated on an intranet, for non-commercial purposes within educational institutes that have purchased the paper from WA Examination Papers provided that WA Examination Papers is acknowledged as the copyright owner. Teachers within Perth Modern School may change the paper provided that WA Examination Paper's moral rights are not infringed.

Copying or communication for any other purposes can only be done within the terms of the Copyright Act or with prior written permission of WA Examination papers.

Published by WA Examination Papers

PO Box 445 Claremont WA 6910