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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 Two:

Calculator-assumed

In words

Your name

Time allowed for this section

Reading time before commencing work: ten minutes

Working time: one hundred minutes

Materials required/recommended for this section

To be provided by the supervisor

This Question/Answer booklet

Formula sheet (retained from Section One)

To be provided by the candidate

Standard items: pens (blue/black preferred), pencils (including coloured), sharpener, correction fluid/tape, eraser, ruler, highlighters

Special items: drawing instruments, templates, notes on two unfolded sheets of A4 paper, and up to three calculators approved for use in this examination

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 Two: Calculator-assumed 65% (98 Marks)

This section has thirteen (13) questions. Answer all questions. Write your answers in the spaces provided.

Working time: 100 minutes.

Question 9 (6 marks)

An exam has two parts, I and II, containing 15 and 8 questions respectively.

Determine the number of different combinations of questions a candidate could choose if they must answer

(a) 5 questions from part I and 4 questions from part II. (2 marks)

(b) 3 questions, all chosen from the same part. (2 marks)

(c) 3 questions, with at least one question from each part. (2 marks)

Question 10 (6 marks)

(a) The point P(4, -7) is translated by the column vectors xy and -811 to P'(17, -13). Determine the values of the constants x and y. (2 marks)

(b) Determine the single matrix that represents, in order, the composition of a reflection in the line y=3x followed by a rotation of 180° about the origin. Express matrix coefficients in exact form. (4 marks)

Question 11 (6 marks)

(a) A circle property says that if chords of a circle are of equal length then they subtend equal angles at the centre.

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

(ii) Draw a diagram to illustrate the inverse statement and state whether it is true.

(2 marks)

(b) The diagram below shows four points A, B, C and D lying on the circumference of a circle. The line PQ is a tangent to the circle at C, ∠PCD=26°, ∠QCB=37° and ∠ADB=65°.

Determine the size of angles x, y and z. (3 marks)

Question 12 (9 marks)

(a) If p=13i-11j and q=15i+4j determine

(i) the angle between the directions of p and q, to the nearest tenth of a degree.

(2 marks)

(ii) the scalar projection of p on q. (2 marks)

(b) The vector 45i-4aj has a magnitude of 53 and is perpendicular to the vector 3i-5bj. Determine the values of the constants a and b, where a>b. (5 marks)

Question 13 (7 marks)

(a) Point R lies on the circumference of a circle with diameter PQ=51 cm, so that PR=4RQ. Determine the exact length RQ. (3 marks)

(b) Use a vector method to prove that the angle in a semi-circle is a right-angle. (4 marks)

Let OC=c and OB=b.

Question 14 (6 marks)

(a) Prove that sin3A=3sinA-4sin3A. (4 marks)

(b) Hence, or otherwise, solve 3sinA-4sin3A=12, 0≤A≤π3. (2 marks)

Question 15 (8 marks)

In the diagram below, forces F1 and F2 act on a body at the origin.

(a) If F1=85 N, F2=105 N, α=21° and β=35°, determine the magnitude of the resultant force and the angle it makes with the positive x axis. (5 marks)

(b) If F1=145 N and F2=180 N, determine the angles α and β so that the resultant force is directed along the positive x axis and has a magnitude of 310 N. (3 marks)

Question 16 (10 marks)

(a) The graph of y=sinax-b+c is shown below for -π≤x≤π.

Determine the value of the positive constants a, b and c. (3 marks)

(b) On the axes below, sketch the graph of y=3secx-π2, 0≤x≤2π. (3 marks)

(c) The displacement, x cm, of a particle from a fixed point O varies with time, t seconds, according to the model x=2sin(4πt)+3cos4πt, t≥0. Determine

(i) the initial displacement of the particle from O. (1 mark)

(ii) the exact amplitude of the motion. (1 mark)

(iii) the period of motion. (1 mark)

(iv) the first time that the particle passes through O, rounded to two decimal places.

(1 mark)

Question 17 (9 marks)

Triangle ABC has vertices A-1, 5, B2, 7 and C4, 4.

(a) The vertices ABC are transformed to A'B'C' using matrix 0110. Write down the new coordinates of the vertices and describe the transformation. (4 marks)

(b) The vertices ABC are transformed to A''B''C'' using matrix M so that the new coordinates of the vertices are A''25, 2, B''35, -4 and C''20, -8.

(i) Determine the transformation matrix M. (3 marks)

(ii) If the area of triangle ABC is k square units, express the area of triangle A''B''C'' in terms of k. (2 marks)

Question 18 (7 marks)

(a) How many numbers must be chosen from the set of integers between 1 and 2017 inclusive to be certain that one of the numbers chosen is a multiple of 10. (3 marks)

(b) A number is formed using four different digits chosen from those in the number 23 814. Determine how many different numbers can be formed that are

(i) even. (1 mark)

(ii) greater than 8 000. (1 mark)

(iii) even or greater than 8 000. (2 marks)

Question 19 (10 marks)

(a) Trapezium OPQR has parallel sides PQ and OR. M is the midpoint of OQ and N lies on QR so that RN:NQ=4:1.

Given that OP=p, OR=r and PQ=3r, determine the following in terms of p and r.

(i) OM. (2 marks)

(ii) ON. (2 marks)

(iii) NM. (2 marks)

(b) Quadrilateral OABC is shown below, where P, Q, R and S are the midpoints of the sides OA, AB, BC and OC respectively. Let OP=a, AQ=b and OS=c.

Show that PQ=SR. (4 marks)

Question 20 (6 marks)

The diagram shows a semi-circle, with diameter SR and centre O, circumscribed by triangle ABC, in which ∠BAC=48° and ∠BCA=36°.

Determine, with reasons, the size of angles ∠PRO and ∠PQR.

Question 21 (8 marks)

The sum of the first n terms of the sequence 1+11+21+…+(10n-9) is n(5n-4).

(a) Show that this statement is true when n=4. (2 marks)

(b) Use mathematical induction to prove the statement is true for n∈Z ,n≥4. (6 marks)

Additional working space

Question number: _________

Additional working space

Question number: _________

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