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Rossmoyne Senior High School

Year 11 Examination, 2015

Question/Answer Booklet

SOLUTIONSSOLUTIONS

MATHEMATICS

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 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 examinations are detailed in the school handbook. 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 eight (8) questions. Answer all questions. Write your answers in the spaces provided.

Working time for this section is 50 minutes.

Question 1 (5 marks)

(a) If tanA=12, determine the exact value of tan2A. (2 marks)

(b) Solve cos2x-10°=0.5, 0°≤x≤180°. (3 marks)

Question 2 (6 marks)

If , state whether the following are true or false. If false, clearly explain your reasoning.

False – cannot add different size matrices.False – cannot add different size matrices.(a) . (1 mark)

True.True.(b) . (1 mark)

(c) . (1 mark)

False, as and do not exist.False, as and do not exist.

False, result is the matrix [13].False, result is the matrix [13].(d) . (1 mark)

True.True.(e) . (1 mark)

(f) . (1 mark)

False, matrix C must be square to have an inverse.False, matrix C must be square to have an inverse.

Question 3 (7 marks)

Two vectors are given by and . Determine

(a) a vector parallel to a-b of magnitude 25. (3 marks)

(b) a in terms of d and e, where and . (4 marks)

Question 4 (8 marks)

(a) Evaluate . (1 mark)

(b) Determine the number of different permutations of the letters in the word NEEDLED.

(2 marks)

(c) A password is formed using all seven of the characters $, %, @, Y, Z, 8 and 9 just once. Determine the number of different passwords that are possible in which all the symbols are adjacent, all the letters are adjacent and all the digits are adjacent. (3 marks)

(d) Determine the least number of randomly chosen integers between 10 and 99 required to be certain that the difference of the digits in at least two of the integers is the same. (For example, the difference of the digits in the integer ). (2 marks)

There are 10 possible differences (from 0 to 9), which give us 10 pigeonholes to fill.If more than 10 numbers are chosen, then at least one pigeonhole must contain two or more numbers. So at least 11 numbers must be chosen.There are 10 possible differences (from 0 to 9), which give us 10 pigeonholes to fill.If more than 10 numbers are chosen, then at least one pigeonhole must contain two or more numbers. So at least 11 numbers must be chosen.

Question 5 (5 marks)

A proposition states that for any integer n, if is even, then n is odd.

(a) Write the contrapositive of this proposition. (1 mark)

If n is not odd, then is not even.If n is not odd, then is not even.

(b) Use the contrapositive statement to prove the proposition is true. (4 marks)

If n is not odd: Hence, as the contrapositive has been shown to be true, the original proposition must also be true.If n is not odd: Hence, as the contrapositive has been shown to be true, the original proposition must also be true.

Question 6 (7 marks)

(a) Sketch the graph of y=2cosec(x+90) for . (3 marks)

(b) Prove the identity cotA+tanA=secAcosecA. (4 marks)

Question 7 (7 marks)

(a) Matrix A represents a rotation of 180º about the origin. Determine

(i) matrix A. (1 mark)

(ii) the exact coordinates of the point (-2, 3) after transformation by matrix A. (1 mark)

(iii) the determinant of matrix A. (1 mark)

11

(b) Matrix . Describe the transformation represented by B and calculate its determinant. (2 marks)

B is a reflection in the y-axis.detB=-1B is a reflection in the y-axis.detB=-1

(c) Use an example to show that two non-singular square matrices C and D exist such that the determinant of their sum is equal to the sum of their determinants. (2 marks)

Question 8 (7 marks)

The complex number , where is a real constant.

(a) Show that and that . (4 marks)

(b) Determine the value(s) of when . (3 marks)

Additional working space

Question number: _________

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