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Semester Two Examination, 2021

Question/Answer booklet

MATHEMATICS
SPECIALIST
UNITS 1&2

Section One:
Calculator-free

Teacher

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 preferably using a blue/black pen.
Do not use erasable or gel pens.

3. You must be careful to confine your answers to the specific question asked and to follow any instructions that are specific to a particular question.

4. 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.

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

6. Supplementary pages for planning/continuing your answers to questions are 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.

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

Section One: Calculator-free 35% (50 Marks)

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

Working time: 50 minutes.

Question 1 (5 marks)

Let matrix A=-2031 and matrix B=2k-112k+1, where k is a constant.

(a) When k=-1, determine

(i) AB. (1 mark)

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

(b) Determine the value(s) of k if matrix B is singular. (2 marks)

Question 2 (6 marks)

(a) Sketch the graph of y=secx2 on the axes below for 0≤x≤4π. (3 marks)

(b) Prove the identity cosec2A-cot2A=tanA. (3 marks)

Question 3 (6 marks)

Let z1=5+3i and z2=5-i. Determine each of the following in the form a+bi.

(a) 2z1-z2. (1 mark)

(b) iz1. (1 mark)

(c) z1×z2. (2 marks)

(d) z1÷z2. (2 marks)

Question 4 (6 marks)

(a) Determine the value(s) of the constant t given that -24t8tt=2t33. (2 marks)

(b) Determine A-1 when A=73-22. (2 marks)

(c) Show use of matrix methods to solve the following system of linear equations:

7x+3y-25=0
2y-2x+10=0

(2 marks)

Question 5 (6 marks)

(a) Using a product identity, or otherwise, evaluate sin5π12+sinπ12. (3 marks)

(b) Solve the equation 2sin22x=3cos2x, 0≤x≤2π. (3 marks)

Question 6 (7 marks)

(a) Determine all complex solutions to the equation z2-10z+27=0. (2 marks)

(b) z1=-4-i is a solution to fz=0, where f(z) is a real quadratic polynomial.

(i) State z2, another solution to fz=0. (1 mark)

(ii) Let z3=z2-z1. Plot and label z1, z2 and z3 in the complex plane below. (2 marks)

(iii) Determine f(z), given that the coefficient of its z2 term is 1. (2 marks)

Question 7 (6 marks)

Use mathematical induction to prove that 25n-5n is divisible by 9 for all integers n≥1.

Question 8 (8 marks)

(a) Points A, B and C lie on a circle.

The tangent to the circle at A
intersects secant BC at point D.

Prove that AD2=BD×CD. (4 marks)

(b) Two unequal circles intersect at P and Q. A common tangent touches one circle at R and the other circle at S. PQ produced intersects RS at X. Prove that X bisects RS. (4 marks)

Supplementary page

Question number: _________

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