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Semester One Examination, 2015

Question/Answer Booklet

SOLUTIONS

MATHEMATICS SPECIALIST

UNIT 1

Section Two:

Calculator-assumed

In words

Your name

Time allowed for this section

Reading time before commencing work: ten minutes

Working time for this section: 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 the WACE examinations

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 Western Australian external examinations are detailed in the Year 12 Information Handbook 2015. 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 Two: Calculator-assumed (98 Marks)

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

Working time for this section is 100 minutes.

Question 8 (5 marks)

Three vectors are given by , and .

(a) Use your calculator to determine the angle between , to the nearest degree.

(2 marks)

Using CAS, angle is 56º.Using CAS, angle is 56º.

(b) Determine all possible values of if and are perpendicular. (3 marks)

Question 9 (8 marks)

(a) A multiple choice test has twelve questions and each question has three possible choices. If all questions are attempted, in how many ways can the test be answered? (2 marks)

(b) A set S contains all the integers between 3 and 102 inclusive. Determine

(i) how many numbers in set S are multiples of 7. (1 mark)

(ii) how many numbers in set S are multiples of 3 or 7. (2 marks)

(iii) how many numbers in set S are multiples of either 3 or 7 but not both. (1 mark)

(c) Ten points are equally spaced around the circumference of a circle. Determine the number of simple (non-self-intersecting) convex polygons that can be formed by joining either three, four or five of these points with straight line segments. (2 marks)

Question 10 (8 marks)

Three forces are applied to a body. One has magnitude 300 N and acts due south. Another has magnitude 250 N and acts on a bearing of 050º.

(a) If all three forces are in equilibrium, determine the magnitude and direction of the third force. (4 marks)

Magnitude is 237 N on bearing 306º. Magnitude is 237 N on bearing 306º.

(b) If the third force has a magnitude of 350 N and acts on a bearing of 250º, determine the magnitude and direction of the resultant force. (4 marks)

Magnitude is 293 N on a bearing of 208º. Magnitude is 293 N on a bearing of 208º.

Question 11 (6 marks)

(a) A triangle has vertices , and . Determine the vector , where is the midpoint of side . (3 marks)

(b) is a triangle with point on side such that . If and , show that . (3 marks)

Question 12 (6 marks)

(a) Vectors have the same magnitude and vectors are perpendicular, where , and Determine the values of and . (2 marks)

(b) Determine the scalar projection of a velocity of 12 m/s on a bearing of 65º onto a velocity of 20 m/s on a bearing of 280º, giving your answer to three significant figures. (2 marks)

(c) The work done, in joules, by a force of Newtons in changing the displacement of an object by metres is given by the scalar product of and .

A force acting on a bearing of 160º does work of 1 200 joules. If the object moved a distance of 350 cm on a bearing of 135º, determine the magnitude of the force. (2 marks)

Question 13 (8 marks)

(a) A triangle has vertices at , and .

(i) Determine the vectors . (2 marks)

(ii) Use a vector method to prove that triangle is right-angled. (2 marks)

(b) Use a vector method to prove that if the diagonals of a parallelogram are perpendicular to each other, then the parallelogram is a rhombus. (4 marks)

Question 14 (8 marks)

(a) Two circles of radius 12.6 cm, with centres as shown below, have a common chord . Determine, with justification, the length . (2 marks)

(b) In the diagram below, . Determine, with justification, the exact length of . (3 marks)

(c) Determine the length, if that the length of chord is 10.5 cm and the length of the tangent is 9.5 cm. (3 marks)

Question 15 (9 marks)

(a) A small body A has position (12, -3) m relative to another small body B. If a third small body C has position (-5, 6) relative to A, determine the position of B relative to C.

(2 marks)

(b) To a cyclist moving with velocity (21, -5) km/h the wind appears to have velocity (-9, 3) km/h. Determine the true speed of the wind. (3 marks)

(c) A small ship is travelling with a constant speed of 14 knots on a bearing of 025º and another, larger ship is travelling with a constant speed of 17 knots on a bearing of 310º.

Determine the velocity of the large ship relative to the small ship. (4 marks)

19.02 knots on bearing 265º.19.02 knots on bearing 265º.

Question 16 (9 marks)

The medians of triangle are , as shown below.

(A median joins a vertex to the midpoint of the opposite side of the triangle).

Let .

(a) Prove that . (4 marks)

(b) The centroid, , is the point of intersection of the medians.

Determine in terms of . (5 marks)

(Hint: Let , and first solve for and )

Question 17 (8 marks)

(a) The diagram shows a triangle with vertices that lie on a circle with centre . Chord passes through . Prove, by contradiction, that angle is acute.

(4 marks)

Assume that is not acute,ie that .Because is a diameter, then .Because is a triangle, then .Given that , and , then the sum of the angles in triangle exceeds 180º, but this contradicts the fact that the sum of the angles in a triangle is 180º.Hence, our assumption that must be false, and so angle is acute.Assume that is not acute,ie that .Because is a diameter, then .Because is a triangle, then .Given that , and , then the sum of the angles in triangle exceeds 180º, but this contradicts the fact that the sum of the angles in a triangle is 180º.Hence, our assumption that must be false, and so angle is acute.

(b) In the diagram below, is the centre of the circle on which points lie. Chord passes through and is parallel to . Prove that the quadrilateral is a rectangle. (4 marks)

(Both angles in semi-circle). (alternate angles). is common side in .Hence are congruent (ASA).Hence . (Co-interior with ).Hence is a rectangle. (Both angles in semi-circle). (alternate angles). is common side in .Hence are congruent (ASA).Hence . (Co-interior with ).Hence is a rectangle.

Question 18 (8 marks)

(a) A small coach has 24 seats, arranged in six rows of four seats each, with two seats in each row on either side of the central aisle. A group passengers consisting of ten males and nine females board the bus.

(i) Determine how many combinations of empty seats are possible once everyone has sat down. (1 mark)

(ii) How many fewer combinations are there if the females all sit on one side of the aisle and the males all sit on the other side? (3 marks)

(b) Determine the number of possible four letter permutations of the letters of the word

(i) RELOAD. (1 mark)

(ii) RELOADED. (3 marks)

All different: 3602 E's, other 2 diff: 2 D's, other 2 diff: 1202 E's and 2 D's: Total: All different: 3602 E's, other 2 diff: 2 D's, other 2 diff: 1202 E's and 2 D's: Total:

Question 19 (7 marks)

A small boat has to travel across a river from to , where and . A uniform current of is flowing in the river and the boat can maintain a steady speed of 4 m/s.

(a) Determine, in the form , the velocity vector the small boat should set to travel directly from to . (5 marks)

(b) Calculate, to the nearest minute and second, how long the journey will take. (2 marks)

Question 20 (8 marks)

(a) In the diagram below, is a tangent to the circle at . If , and , determine the size of angle . (4 marks)

(b) In the diagram below, the line passes through the point , where both circles touch each other. The line is parallel to line . Prove that the points are collinear. (4 marks)

Additional working space

Question number: _________