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Semester Two Examination, 2021

Question/Answer booklet

MATHEMATICS
SPECIALIST
UNITS 1&2

Section Two:
Calculator-assumed

Teacher

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, which can include scientific, graphic and Computer Algebra System (CAS) calculators, are permitted in this ATAR course 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 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 Two: Calculator-assumed 65% (92 Marks)

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

Working time: 100 minutes.

Question 9 (6 marks)

(a) Determine the vector projection of 5-5 on -12. (2 marks)

(b) Determine the value(s) of t so that the vectors t-3 and 3t+81 are

(i) parallel. (2 marks)

(ii) perpendicular. (2 marks)

Question 10 (7 marks)

(a) Five-digit odd numbers are to be made using the digits 1, 2, 3, 4, 5, 6 and 7. Determine how many such numbers exist if the number must exceed 50 000 and no digit may be used more than once in a number. (3 marks)

(b) The library in a small guesthouse has 32 different books, of which 21 are non-fiction and the remainder fiction. Determine the number of different ways that a guest can select four books if they want

(i) the same number of fiction and non-fiction books. (2 marks)

(ii) more fiction than non-fiction books. (2 marks)

Question 11 (7 marks)

Two transformation matrices are M=0110 and N=22-44.

Triangle PQR has an area of 39 cm2, with vertices at P5, 3, Q-2, 8 and R-5, -1.

(a) Determine the coordinates of PQR after the triangle has been transformed by matrix M.

(3 marks)

(b) Use the geometric transformation to explain why the determinant of M is 1. (1 mark)

(c) Use the geometric transformation to explain why M2=I, where I is the 2×2 identity matrix. (1 mark)

(d) Determine the area of PQR after the triangle has been transformed by matrix N. (2 marks)

Question 12 (8 marks)

(a) Write the converse of the true statement 'if a figure is a square then it has four congruent sides' and use an example or counter-example to briefly discuss the truth of the converse.

(2 marks)

(b) Points A, B, C and D lie as shown on a circle
with centre O so that AD is a diameter,
AB=CD and ∠COD=42°.

Determine the size of

(i) ∠CAD. (1 mark)

(ii) ∠BAD. (1 mark)

(iii) ∠CAB. (1 mark)

(c) Two chords of a circle, LM and PQ, intersect at N so that LM=41 cm, NM=20 cm and PQ=44 cm. Determine all possible lengths of QN. (3 marks)

Question 13 (8 marks)

In triangle OAB, P is the midpoint of OA and M is the midpoint of PB. Let OA=a and OB=b.

(a) Show that OM=14a+12b. (2 marks)

The position vector of A is 64, position vector of B is 7-4 and O is the origin.

(b) Determine a unit vector u in the same direction as OM. (2 marks)

(c) Show that OA is perpendicular to PM. (2 marks)

(d) Determine the size of ∠AOB. (2 marks)

Question 14 (7 marks)

(a) In trapezium ABCD, AC and BD are diagonals, and AB is parallel to CD. Use a vector method to prove that AC+DB=AB+DC. (3 marks)

(b) In rectangle OPQR, let OP=p and OR=r. Use a vector method to prove that if the diagonals OQ and PR are perpendicular, then OPQR is a square. (4 marks)

Question 15 (6 marks)

Consider the following statement:

For two integers a,b if 3a2-2b2 is a multiple of 4 then at least one of a,b is even.

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

(b) Prove that the statement is true. (5 marks)

Question 16 (8 marks)

The height of the tide, h cm, of the sea above the mean level at time t hours after midnight one day is given by

ht=28cosπt6+45sinπt6 .

(a) Express h in the form acos(bt-θ), where a, b>0 and 0≤θ≤2π. (3 marks)

(b) Determine, to the nearest minute, the time of the first high tide. (2 marks)

(c) Sketch the graph of the height of the tide on the axes below. (3 marks)

Question 17 (7 marks)

Three forces F1, F2 and F3 act on a small body, where F1=4i-10j N, F2=-8i+16j N and
F3=9i-15j N.

(a) Determine the magnitude of the resultant force and the angle between the resultant force and the vector i. (3 marks)

(b) Two of the forces, F2 and F3, can be multiplied by scalars λ and μ respectively so that the three forces are in equilibrium. Determine the value of λ and the value of μ. (4 marks)

Question 18 (7 marks)

(a) 90 people are asked to choose two different letters from those in the word GAMBLER and write them down in order. Use the pigeonhole principle to prove that at least three people will write the same pair of letters in the same order. (3 marks)

(b) Three character codes, such as TCU, are made using three different letters chosen from the word DISCOUNT. Determine the proportion of all possible codes that start with a D or end with a T. (4 marks)

Question 19 (7 marks)

Airport B lies 170 km due east of airport A, and in the region of the airports a wind of 25 km/h is blowing from the northeast.

A small plane, with a cruising speed of 120 km/h, leaves airport A. The pilot, not aware of the wind and intending to fly to airport B, steered the plane on a bearing of 090°.

Assuming that the pilot does not realise their mistake, determine how close the plane will come to airport B if it continues to fly for several hours on the same bearing.

Question 20 (7 marks)

Triangle ABC has vertices A2, 2, B(-4, 6) and C(4, 8).

ABC is rotated clockwise 60° about the origin to form triangle A'B'C'.

(a) Determine the coordinates of C'. (2 marks)

ABC is reflected in the line y=2x to form triangle A''B''C''.

(b) Determine the coordinates of B''. (2 marks)

(c) Determine matrix T that will transform A'B'C' to A''B''C''. (3 marks)

Question 21 (7 marks)

The diagram, not drawn to scale, shows
vertices A, B and D of an isosceles triangle
lying on a circle so that AD=AB.

Lines CE and CF are tangential to the
circle at D and B respectively.

ABCE is a cyclic quadrilateral.

Let ∠FCE=x.

(a) Determine, with reasons, the size of ∠AEC in terms of x. (5 marks)

(b) Hence determine the range of values for the size of ∠AEC in degrees. (2 marks)

Supplementary page

Question number: _________

Supplementary page

Question number: _________

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