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Semester Two Examination, 2018

Question/Answer booklet

If required by your examination administrator, please place your student identification label in this boxIf required by your examination administrator, please place your student identification label in this boxMATHEMATICS

SPECIALIST

UNITS 1 AND 2

Section One:

Calculator-free

In words

Your name

Time allowed for this section

Reading time before commencing work: five minutes

Working time: 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 material. If you have any unauthorised material with you, hand it to the supervisor before reading any further.

Structure of this paper

Instructions to candidates

1. 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.

2. Write your answers in this Question/Answer booklet.

3. 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.

4. Supplementary pages for the use of planning/continuing your answer to a question
have been provided at the end of this Question/Answer booklet. If you use these pages to continue an answer, indicate at the original answer where the answer is continued, i.e. give the page number.

5. 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.

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

7. The Formula sheet is not to be handed in with your Question/Answer booklet.

Section One: Calculator-free 35% (52 Marks)

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

Working time: 50 minutes.

Question 1 (7 marks)

Let A=2-1-30 and B=5476.

Determine

(a) AA-1. (1 mark)

(b) 2A+B. (2 marks)

(c) AB. (2 marks)

(d) B-1. (2 marks)

Question 2 (4 marks)

z1 and z2 are the complex solutions to the equation z2-2z+10=0.

The location of z1 is shown in the complex plane below.

Add, with a label, the following complex numbers to the complex plane above.

(a) z2. (1 mark)

(b) u=z1+z2. (1 mark)

(c) v=iz1. (1 mark)

(d) w=u+v. (1 mark)

Question 3 (6 marks)

(a) Draw a neat diagram to illustrate the angle in the alternate segment theorem. (2 marks)

(b) Consider the statement 'if a pentagon is regular, then it has sides of equal length'.

(i) Write the inverse of the statement. (1 mark)

(ii) Write the contrapositive of the statement. (1 mark)

(iii) Write the converse of the statement and neatly sketch a counter-example to show that the converse is not true. (2 marks)

Question 4 (8 marks)

(a) On the axes below, sketch the graphs of y=tanx2 and y=cotx2, where 0≤x≤360°, labelling all asymptotes. Note that tan76°≈4. (5 marks)

(b) Solve the equation sec2(x)=3tan2(x)-1 over the interval 0≤x≤360°. (3 marks)

Question 5 (6 marks)

The position vectors of points P and Q are p=31 and q=1-1 respectively.

(a) Determine the magnitude of the displacement vector PQ. (2 marks)

(b) Determine the values of λ so that p-λq=4. (4 marks)

Question 6 (8 marks)

Let fx=2sinx+2cosx.

(a) Express f(x) in the form rsinx+α where 0<α<π2. (4 marks)

(b) Sketch the graph of y=f(x). (4 marks)

Question 7 (5 marks)

Let OAPB be a parallelogram where OA=a and OB=b.

Use a vector method to prove 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.

Question 8 (8 marks)

(a) Determine all complex solutions to the equation z2-2z+5=0. (3 marks)

(b) Determine the values of the positive real constants p and q so that p-2i is a solution to the equation 2z2-qz+26=0. (5 marks)

Supplementary page

Question number: _________

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