2016 SPECIALIST MATHEMATICS

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1 2016 SPECIALIST MATHEMATICS External Examination 2016 FOR OFFICE USE ONLY SUPERVISOR CHECK ATTACH SACE REGISTRATION NUMBER LABEL TO THIS BOX Graphics calculator Brand Model Computer software RE-MARKED Friday 18 November: 9 am Time: 3 hours Examination material: one 51-page question booklet one SACE registration number label Pages: 51 Questions: 15 Approved dictionaries, notes, calculators, and computer software may be used. Instructions to Students 1. You will have 10 minutes to read the paper. You must not write in your question booklet or use a calculator during this reading time, but you may make notes on the scribbling paper provided. 2. This paper consists of three sections: Section A (Questions 1 to 9) Answer all questions in Section A. Section B (Questions 10 to 13) Answer all questions in Section B. Section C (Questions 14 and 15) Answer one question from Section C. 75 marks 60 marks 15 marks 3. Write your answers in the spaces provided in this question booklet. There is no need to fill all the space provided. You may write on pages 33, 43, and 48 if you need more space, making sure to label each answer clearly. 4. Appropriate steps of logic and correct answers are required for full marks. 5. Show all working in this booklet. (You are strongly advised not to use scribbling paper. Work that you consider incorrect should be crossed out with a single line.) 6. Use only black or blue pens for all work other than graphs and diagrams, for which you may use a sharp dark pencil. 7. State all answers correct to three significant figures, unless otherwise stated or as appropriate. 8. Diagrams, where given, are not necessarily drawn to scale. 9. The list of mathematical formulae is on page 49. You may remove the page from this booklet before the examination begins. 10. Complete the box on the top right-hand side of this page with information about the electronic technology you are using in this examination. 11. Attach your SACE registration number label to the box at the top of this page. SACE Board of South Australia 2016

2 SECTION A (Questions 1 to 9) (75 marks) Answer all questions in this section. QUESTION 1 (8 marks) A curve has the following parametric equations: xt y t sin 2t cos 2 t where 0t 2. (a) Sketch a graph of the curve on the axes in Figure 1. y x Figure 1 (3 marks) page 2 of 51

3 (b) Show that d y dx 1 tan 2t. 2 (3 marks) (c) (i) What is the slope of the tangent to the curve at t 4? (ii) Draw the tangent to the curve at t on your graph in Figure 1. 4 page 3 of 51 PLEASE TURN OVER

4 QUESTION 2 (8 marks) 2 Let f x x cos 2x. 2 (a) Evaluate f x dx, giving your answer correct to four significant figures. 0 (b) Show that f x is an even function. page 4 of 51

5 (c) Use integral calculus to show that f x dx (d) Hence or otherwise, find A (where A 0) and B such that B x cos 2x dx. A page 5 of 51 PLEASE TURN OVER

6 QUESTION 3 (6 marks) In Figure 2, ABC is circumscribed by a circle with centre O. The line AX is a tangent to the circle at A and it meets BC produced at X. Let CAX. A X O C B Figure 2 (a) Given that ACX 2, prove that AC BC. (3 marks) page 6 of 51

7 4 (b) If AOC,find AXC. 9 (3 marks) page 7 of 51 PLEASE TURN OVER

8 QUESTION 4 (8 marks) 2 Consider the quadratic iteration ziz c, where z and c is a complex number. 0 0 (a) (i) State z 0 and z 1. 2 (ii) Show that z 2 ic c. (b) The iteration will be a two-cycle when z z. 2 0 (i) Find the value of c for which the iteration is a two-cycle. (3 marks) page 8 of 51

9 (ii) Verify that a two-cycle is produced. Show all calculations. page 9 of 51 PLEASE TURN OVER

10 QUESTION 5 (8 marks) Let Px x k 2x 3k 9x 12k 2x10k 10, where k is a real constant. (a) Prove that x 1 is a factor of Px for any value of k. (b) Given that x 2 is also a factor of Px, (i) find the value of k. page 10 of 51

11 (ii) write Px as a product of real factors. (iii) list all the zeros of Px. page 11 of 51 PLEASE TURN OVER

12 QUESTION 6 (9 marks) Figure 3 shows point P on the circumference of a circle with diameter AB. P A B Let AP a and PB b. Figure 3 (a) Prove that ab 0. (b) Explain why ab a b (c) Prove that ab a b. page 12 of 51

13 (d) Prove that the area of the circle is a b., find the exact area of the circle. (e) Given the points A 12,, 4, B 3, 17,, and P 316,, (3 marks) page 13 of 51 PLEASE TURN OVER

14 QUESTION 7 (8 marks) (a) Draw an Argand diagram showing the set of complex numbers such that z 22i 2. (b) On the same Argand diagram, draw the set of complex numbers such that z z2i. page 14 of 51

15 (c) Find in Cartesian form the exact value of all complex numbers that satisfy both the equation in part (a) and the equation in part (b). (4 marks) page 15 of 51 PLEASE TURN OVER

16 QUESTION 8 (9 marks) 2 1 (a) (i) Show that 1 sin x sin x 2 cos x 4 for cos x (ii) Hence prove that 1 sin x sin x sin x 2 cos x 6 for cos x 0. page 16 of 51

17 (b) Use an inductive argument to prove that, for all integers n 1, n n sin x sin x sin x sin x for cos x 0. 2 cos x (3 marks) (c) Hence evaluate 1sin x sin x dx, giving your answer correct to two decimal places. 0 page 17 of 51 PLEASE TURN OVER

18 QUESTION 9 (11 marks) A simple model is used to investigate the formation of ice in the sea. The model is a column with three layers: a thin layer of ice with thickness y between a layer of sea water and a layer of air as shown in Figure 4. air layer y ice layer sea-water layer Figure 4 The heat transferred between the layers is denoted by Q. This can be modelled by Q LDy, where L is a heat-loss constant and D is the constant density of the ice layer; L and D are both positive. The rate of transfer of heat, d Q, from the sea water through the ice layer to the air is given by dt dq ktw Ta, dt y where T w is the temperature of the sea water, T a is the temperature of the air, and k is a positive constant. Close to the ice layer, T w and T a are assumed to be constant. page 18 of 51

19 (a) Show that the rate of change of the thickness of the ice layer, d y, can be written as dt a. dy k Tw T dt LDy (b) Given that Tw Ta, state what happens to the rate of change of the thickness of the ice layer as y gets smaller. page 19 of 51 PLEASE TURN OVER

20 (c) When values for temperature, heat loss, and density are substituted into the differential equation in part (a), the equation becomes dy k dt y with initial condition y Use Euler s method to complete the following table in terms of k, where appropriate, given h 01.. Do not simplify your answers. The equations for Euler s method are tn tn1 h yt yt hy t n n1 n1. n t n yt n You may use this grid for working as you fill in the table. (4 marks) page 20 of 51

21 (d) Using integration, solve the differential equation, dy k Tw Ta dt LDy where k, Tw, Ta, L, and D are assumed to be constant and the initial condition is y (4 marks) page 21 of 51 PLEASE TURN OVER

22 SECTION B (Questions 10 to 13) (60 marks) Answer all questions in this section. QUESTION 10 (15 marks) are three points on plane P 1. Points A 41,, 2, B 5, 1, 4, and C 240,, (a) (i) Find AB. (ii) Find AB AC. (iii) Show that the equation of P1 is 2x2yz 12. page 22 of 51

23 (b) (i) Find the equation of the normal to P 1 through the point D31,, 4. (ii) Plane P 2 has equation 2x2yz 15. The normal to P 1 through D intersects P 2 at the point E. Find the coordinates of E. (iii) Find the distance between P 1 and P 2. page 23 of 51 PLEASE TURN OVER

24 (c) (i) Find the equation of the plane that contains the point T 0, 3, 0 and is parallel to both P 1 and P 2. Figure 5 shows P 1 and P 2 connected by a cylinder of radius 3 units and with a central axis, DE. Point E, which you found in part (b)(ii), and point D31,, 4 are on opposite ends of the cylinder, point S is outside the cylinder, and point R is inside the cylinder. P 2 P 1 E D S R Figure 5 page 24 of 51

25 (ii) Is T inside, outside, or on the surface of the cylinder? Explain your answer. page 25 of 51 PLEASE TURN OVER

26 QUESTION 11 (15 marks) A country s population is growing at a constant annual rate of r. The population growth can be modelled by the differential equation d P rp, where t is time measured in years and t 0. dt Countries with low annual growth rates may accept immigrants. The population growth in these countries can be modelled by the equation dp rp I, dt where I is the average number of immigrants accepted each year and is assumed to be constant. (a) Show that the variables in d P dt 1 rp I can be separated to give P I dp r t d. r (b) Hence solve the differential equation d P rt I rp I to show that Pt Pe dt r e rt 0 1, where P P. 0 0 (4 marks) page 26 of 51

27 (c) For one country in 2016, P million, I million, and r Use P t from part (b) to answer the following questions. (i) Find the population in (1) 5 years time. (2) 50 years time. (ii) State an assumption made by the model and comment on it. (iii) Comment on the reliability of the model in the long term. page 27 of 51 PLEASE TURN OVER

28 (d) The differential equation dp P P I dt represents another model of population growth. Again I represents the average number of immigrants accepted each year and is assumed to be constant. Figure 6 shows the slope field for this differential equation using the value of I that was used previously (I million). (i) Using the initial condition P0 24 million, draw the solution curve on the slope field in Figure 6. P (million) t Figure 6 (3 marks) (ii) Referring to your solution curve, describe the behaviour of P t in the long term. page 28 of 51

29 (e) (i) State how the long-term population predicted by the model dp rp I dt differs from the long-term population predicted by the model used in part (d) dp P P I dt (ii) The model used in part (d) is not a logistic growth model. Referring to your solution curve, state how the long-term population predicted by the model used in part (d) dp P P I dt differs from the long-term population predicted by the logistic growth model dp 68 P 002. P. dt 68 page 29 of 51 PLEASE TURN OVER

30 QUESTION 12 (15 marks) (a) (i) Write 1 i 3 exactly in rcis form, where r 0 and. (ii) Hence find 1 i 3 3. (iii) Solve z 3 8. Write your answer(s) exactly in rcis form. page 30 of 51

31 (b) Show that the solutions to z are 1, cis, cis, cis, and cis z 8z 1 (c) Show that z 2 z z 3z 7z 7z 7z 6z 4. page 31 of 51 PLEASE TURN OVER

32 (d) Use your results from parts (a), (b), and (c) to solve the equation Write your answers exactly in rcis form. z 3z 7z 7z 7z 6z4 0. (e) Plot your solutions from part (d) on the Argand diagram in Figure 7, labelling them z1, z2, z6 anticlockwise from the smallest positive argument. Im z Re z 1 2 Figure 7 (f ) If z a and z b are any two distinct solutions from the Argand diagram in Figure 7: (i) find the minimum value of z z. a b (ii) find the minimum value of z a z. b page 32 of 51

33 You may write on this page if you need more space to finish your answers. Make sure to label each answer carefully (e.g. Question 12(a)(iii) continued ). page 33 of 51 PLEASE TURN OVER

QUESTION 13 (15 marks) A person is attached to bundled elastic ropes (called a bungee cord ) and jumps from a bridge 98.4 m above the surface of a river.

equation d2 x dt 2 9.8 0.14kx, where k is an elasticity constant for the bungee cord and k > 0.

34 QUESTION 13 (15 marks) A person is attached to bundled elastic ropes (called a bungee cord ) and jumps from a bridge 98.4 m above the surface of a river. The bungee cord is 30 m long. (See Figure 8.) x 30 bridge bungee cord 30 m This photo of bungee jumping cannot be reproduced here for copyright reasons. x m river surface Figure 8 Source: Daniel Kerek/Alamy Stock (a) The position of the jumper, x t, t seconds after jumping is given by the differential equation d2 x dt kx, where k is an elasticity constant for the bungee cord and k > 0. This differential equation is found by considering the force of gravity and the force of the bungee cord on the jumper. An approximate solution to this differential equation is of the form k 7.4 t 2.55 sin 24.5 k 7.4 ¹ where t is time measured in seconds and t t 0. x t (i) k t 2.5 cos k k 7.4 Show that at t, ¹ 2.5 seconds the jumper is at x 0. page 34 of 51

35 (ii) Find the velocity of the jumper as she passes through x 0. (3 marks) (b) Air resistance and friction in the bungee cord cause the jumper eventually to come to rest. The following system of differential equations may be used to model the position, x, t and velocity, y, t of the jumper, incorporating these extra forces: x y y x014. y. The initial conditions are x030 and y0 0, which represent the jumper on the bridge with zero velocity. (i) The position of the jumper, x t, is of the form 5 t xt e 74 Acos t Bsin t 17. 1, where A and B are constants. Using the initial conditions, show that A and B correct to three significant figures. (4 marks) page 35 of 51 PLEASE TURN OVER

36 (ii) The graph of x t is shown in Figure x t 20 Figure 9 On the graph is a blue circle and a red horizontal line. Explain, in the context of the question, what the circled section of the graph and the horizontal line indicate. (iii) On the axes in Figure 10, draw the graph of y t y 40 x t for 0 20 t t Figure 10 (3 marks) (iv) From your graph, determine the maximum velocity of the jumper. page 36 of 51

37 SECTION C (Questions 14 and 15) (15 marks) Answer one question from this section, either Question 14 or Question 15. page 37 of 51 PLEASE TURN OVER

38 Answer either Question 14 or Question 15. QUESTION 14 (15 marks) The fish population of a pond is being studied. The size of the population at time t years is denoted by P t. The growth of the population is described by the differential equation where k, m, and N are positive constants. dp P m kp1 1 dt N P (a) Consider the sign of the factors on the right-hand side of the differential equation. (i) Explain why the population is increasing if m P N. (ii) Explain why the population is decreasing if Pm N. page 38 of 51

39 1 (b) Show that P P1 1 N m P N 1 1 N m N P P m. (c) For a particular population, we find that k 0. 8, N 1000, and m100, so that the growth of the population is given by the differential equation dp P 08. P dt 1000 P (i) If the initial population P0 400, explain why P t 400 for t 0. page 39 of 51 PLEASE TURN OVER

40 (ii) Using P0 400, solve the differential equation d P P 08. P to show that dt 1000 P Pt e 1 1 e t 072. t. (5 marks) page 40 of 51

41 (d) (i) In part (c), the initial population P If instead P0 90, explain why Pt for t (ii) Given that P0 90 has the following solution:, the differential equation d P dt Pt e 072. t 91 e 072. t P 08. P P. Hence find the value of t for which P t 0. page 41 of 51 PLEASE TURN OVER

42 (e) Figure 11 shows the graphs of the population described in part (c) for two different initial populations, where t P P(t) when P(0) P(t) when P(0) t years Figure 11 (i) Describe in words the growth of the population when P0 90and when P (ii) State the significance of the constant m in the differential equation dp P m kp1 1 dt N P. page 42 of 51

43 You may write on this page if you need more space to finish your answers. Make sure to label each answer carefully (e.g. Question 14(e)(i) continued ). page 43 of 51 PLEASE TURN OVER

44 Answer either Question 14 or Question 15. QUESTION 15 (15 marks) Figure 12 shows part of the plane 3x5y4z 25and the three-dimensional Bézier curve with the following parametric equations: P t 3 2 xt 12t 15t 6t 3 2 y t 12t 9t 6t5 3 z t 6t 12t where 0t 1. The curve has initial point A, first control point B 2, 3, 4, second control point C 128,,, and final point D, and Q is a point on the curve. z C 1, 2, 8) D B 2, 3, 4) Q A x y Figure 12 page 44 of 51

45 (a) (i) Find the coordinates of points A and D. (ii) Prove that points A, B, C, and D are coplanar on the plane with equation 3x5y4z 25. (b) Point Q is positioned on the curve such that t 1 3. (i) Find exact values for the coordinates of Q. (ii) Find d P dt when t 1 3. page 45 of 51 PLEASE TURN OVER

46 (iii) Find, in exact form, the parametric equations of the line through Q in the direction of dp dt when t 1 3. (iv) On Figure 12 (on page 44), draw the line that has the parametric equations found in part (b)(iii). (c) Does the Bézier curve lie completely in the plane? Show working to support your answer. page 46 of 51

47 (d) Find the minimum distance between the origin and a point on the Bézier curve, giving your answer correct to three significant figures. (e) Find the maximum distance between the origin and a point on the Bézier curve. page 47 of 51 PLEASE TURN OVER

48 You may write on this page if you need more space to finish your answers. Make sure to label each answer carefully (e.g. Question 15(b)(iii) continued ). page 48 of 51

49 You may remove this page from the booklet by tearing along the perforations so that you can refer to it while you write your answers. LIST OF MATHEMATICAL FORMULAE FOR USE IN STAGE 2 SPECIALIST MATHEMATICS Circular Functions 2 2 sin Acos A1 tan 1cot 2 2 A1sec 2 A Acosec 2 A sina B sin Acos B cos Asin B cosa B cosacos B sin Asin B tana tan B tana B 1 tanatanb sin 2A 2sin Acos A cos 2Acos 2 Asin 2 A 2 2cos A1 2 12sin A 2 tan A tan 2A 1 2 tan A 2sin Acos BsinABsinAB 2cos Acos B cosa BcosAB 2sin Asin BcosABcosAB 1 1 sina sin B 2sin 2 A Bcos 2A B 1 1 cos Acos B2cos 2ABcos 2A B 1 1 cos Acos B2sin 2A Bsin 2A B Matrices and Determinants If A a b then det A A adbc and c d A 1 1 d b = A c a. Properties of Derivatives d dx f x g x f x g x f x g x d f x f x g x f x gx 2 dx gx g x d dx f g x f g x g x Quadratic Equations 2 2 b b 4ac If ax + bx + c = 0 then x. 2a Distance from a Point to a Plane The distance from x, y, z to Ax By Cz D 0 is given by Ax1By1C z 1 D A B C Mensuration Area of sector r Arc length r (where is in radians) In any triangle ABC B Area of triangle 1 2 absin C a b c sin A sin B sin C c a A b C Derivatives f x y x n e x ln x log e x sin x cos x tan x fx nx n1 e x 1 x cos x sin x sec 2 x dy dx a b c 2bccos A page 49 of 51 PLEASE TURN OVER

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Specialist Mathematics 2017 Sample paper

Specialist Mathematics 2017 Sample paper South Australian Certificate of Education Specialist Mathematics 07 Sample paper Question Booklet Part (Questions to 9) 75 marks Answer all questions in Part Write your answers in this question booklet