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Semester Two Examination, 2017

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. Additional working space pages at the end of this Question/Answer booklet are for planning or continuing an answer. If you use these pages, indicate at the original answer, the page number it is planned/continued on and write the question number being planned/continued on the additional working space page.

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 (6 marks)

(a) Determine the values of the real constants b and c if z=2-4i is a solution of the equation z2+bz+c=0. (3 marks)

(b) Express the real quadratic polynomial z2-6z+25 as a product of its linear factors.

(3 marks)

Question 2 (6 marks)

(a) A set of real numbers is given by 5, π, 325, 5.56 . Clearly show that one of the numbers in the set is rational. (3 marks)

(b) Show that if n is two more than a multiple of five, then n2 will be one less than a multiple of five, where n∈Z. (3 marks)

Question 3 (4 marks)

The complex numbers u and v are shown in the complex plane below.

Plot and label the following complex numbers:

(a) z1=u-v. (1 mark)

(b) z2=2v+u. (1 mark)

(c) z3=u. (1 mark)

(d) z4=u-v-u-v. (1 mark)

Question 4 (8 marks)

Relative to the origin O, the points A, B and C have position vectors a=5i-6j, b=i-3j and c=-8i+15j respectively.

(a) Determine in Cartesian form

(i) the vector AB. (1 mark)

(ii) a vector d, parallel to AB and of magnitude 5. (3 marks)

(b) If c=λa+μb, determine the values of the constants λ and μ. (4 marks)

Question 5 (9 marks)

Let A=4735 and B=-5446.

(a) Determine

(i) A-3B. (2 marks)

(ii) AB. (2 marks)

(iii) A-1. (2 marks)

(b) Use a matrix method to solve the system of equations 4x+7y=9 and 3x+5y=5.

(3 marks)

Question 6 (8 marks)

(a) Determine the acute angle θ in each of the following cases:

(i) cosθ=sin38°. (2 marks)

(ii) secθ=cosec100°. (2 marks)

(b) Prove that tanx+secx=cosx1-sinx. (4 marks)

Question 7 (5 marks)

Cyclic quadrilateral ABCD has diagonals AC and BD that intersect at M. Given that AM=6 cm, CM=8 cm and BD=16 cm, determine the smallest possible length of BM.

Question 8 (6 marks)

Let z1 and z2 be complex numbers such that 2z1+3z2=12 and z1+iz2=5+5i.

Determine z1 and z2 in the form z=a+bi, where a, b∈Z.

Additional working space

Question number: _________

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