Specialist Mathematics 2017 Sample paper

Size: px

Start display at page:

Download "Specialist Mathematics 2017 Sample paper"

Betty Hall
5 years ago
Views:

1 South Australian Certificate of Education The external assessment requirements of this subject are listed on page 20. Question Booklet 2 Specialist Mathematics 2017 Sample paper 2 Part 2 (Questions 10 to 14) 75 marks Answer all questions in Part 2 Write your answers in this question booklet You may write on page 19 if you need more space Allow approximately 90 minutes SACE Board of South Australia 2017 SEQ Copy the information from your SACE label here FIGURES CHECK LETTER SPECIALIST MATHEMATICS BIN Graphics calculator 1. Brand Model 2. Brand Model For office use only Supervisor check Re-marked

2 PART 2 (Questions 10 to 14) (75 marks) Question 10 (15 marks) Let A10,, 3, B3, 1, 1, and C7, 33, be three points. (a) Show that B divides AC internally in the ratio 1:2. (2 marks) Figure 9 shows the plane P 1 through B, with AC as its normal and equation The points D 2x y+ 2z = ,, and E1, 30, are on this plane. P 1 D(3, 5, 2) B C A E(1, 3, 0) Figure 9 (b) (i) If BD BE k AB, find the value of k. (2 marks) page 2 of 20

3 (ii) Find the area of ΔBDE. (2 marks) (c) (i) Find the equation of the plane P 2 that contains the point C 7, 3, 3 the plane P 1. ( ) and is parallel to (ii) Show that the line through points A and D has parametric equations x= 1+ 2t y = 5t z = 3+ 5t. (iii) Show that the line through points A and D intersects P 2 at point F ( 71512,, ). (2 marks) page 3 of 20 PLEASE TURN OVER

4 The line AE continues on to meet the plane P 2 at point G 1, 9, 6 ( ) as shown in Figure 10. P 2 P 1 F(7, 15, 12) D C A B E G(1, 9, 6) Figure 10 (d) (i) Find the area of ΔCFG. (ii) Find the distance between the planes P 1 and P 2. (2 marks) page 4 of 20

5 Figure 11 shows that the lines AB, AD, and AE continue on to meet the plane P 3, which is parallel to P 1 and P 2. The equation of the plane P 3 is 2x y+ 2z =, where λ is a constant. P 2 P 3 P 1 F A B D E C G Figure 11 (e) Find the value of λ if the area of the shaded triangle that is projected onto P 3 is 16 times the area of ΔBDE. (2 marks) page 5 of 20 PLEASE TURN OVER

6 Question 11 (15 marks) Figure 12 shows the graph of f x arcsin 2 x, where f( x) 1 x x Figure 12 (a) (i) On Figure 12, sketch the graph of f 1 x. (ii) Write down the range of f 1 x. page 6 of 20

7 (iii) Find f 1 x. (b) If y arcsin 2 x, then 2x sin y. Hence use implicit differentiation to show dy 1 dx x (4 marks) page 7 of 20 PLEASE TURN OVER

8 1 (c) (i) On the axes in Figure 13, draw the graph of y. 1 2 x 4 Clearly show the behaviour of the function near the asymptotes. 8.0 y x Figure 13 (ii) Find the exact area bounded by the graph of y line x x 2, the x-axis, the y-axis, and the page 8 of 20

9 Question 12 (15 marks) (a) (i) Write 2 i 2 in rcis form, where r 0. ( ) (ii) Hence find 2 i 2 4. (2 marks) (b) Solve z 4 = 16, writing your answers exactly in rcis form. page 9 of 20 PLEASE TURN OVER

10 7 4 3 (c) Show that z + z + 16 z = z z + 1 (d) Use your results from parts (b) and (c) to solve the equation writing your answers exactly in rcis form. z + z + 16z + 16 = 0 page 10 of 20

11 (e) Plot your solutions from part (d) on the Argand diagram in Figure 14, labelling them z, z,..., z anticlockwise from the smallest positive argument Imz Rez 2 Figure 14 (2 marks) (f ) (i) Find the value of z + z + z z (ii) Explain why z + z + z z z + z + z z (iii) Find the value of z + z + z z page 11 of 20 PLEASE TURN OVER

12 Question 13 (15 marks) An ice-skater moves in a path with her position defined by the following parametric equations: xt yt where x and y are in metres and t 0 is in seconds. t 20sin 8 t 12cos 8 (a) On the axes in Figure 15, sketch the path of the ice-skater. y t x t Figure 15 (b) How long does it take for the ice-skater to complete one circuit? page 12 of 20

13 (c) Find the velocity vector v t of the ice-skater at time t seconds and hence show that the speed s (in ms 1 ) of the ice-skater is s 2 t t cos 225. sin 8 8. (d) Show that d dt s 2 t t kcos 8 sin 8, where k is constant. page 13 of 20 PLEASE TURN OVER

14 (e) Hence find the exact time(s) and position(s) at which the ice-skater is moving at the fastest speed during the first circuit. (f ) As the ice-skater moves along the path, she slips at t 6 seconds. (i) Find the velocity vector of the ice-skater at this time. (ii) Find the position of the ice-skater at this time. page 14 of 20

15 Question 14 begins on page 16. page 15 of 20 PLEASE TURN OVER

16 Question 14 (15 marks) A skydiver jumps from an aeroplane and free-falls before opening his parachute. The speed v of the skydiver t seconds after he opens his parachute can be modelled by the differential equation dv = ( v 2 k 2 ) dt where k is a positive constant that is related to the type of parachute, the mass of the skydiver, and gravity. (a) Verify that = 2 2 v k 2k v k. v+ k Source: Ichip/Dreamstime.com (b) Hence show that v 1 1 d v ln k 2k 2 2 v k v k c, where c is a constant of integration. (2 marks) page 16 of 20

17 (c) Using the results of parts (a) and (b), show that the differential equation has a solution kt ( 1+ Ae ) vt ()= k kt 1 Ae where A is a constant. ( ) (d) Let the parachute open at t = 0 and let v( 0)= 10 m s 1. (i) Find the constant A in terms of k. (2 marks) page 17 of 20 PLEASE TURN OVER

18 For a particular skydiver, k 2 = (ii) Find vt (), the speed of this skydiver, solely in terms of t. (iii) Find the limiting speed of this skydiver. (iv) On the axes in Figure 16, graph v() t for the first 1 second after the parachute opens, clearly showing the information that you found in part (d)(iii). 10 vt 5 1 t Figure 16 page 18 of 20

19 You may write on this page if you need more space to finish your answer to any question. Make sure to label each answer carefully (e.g. Question 10(e) continued ). page 19 of 20 PLEASE TURN OVER

20 2017 SAMPLE SPECIALIST MATHEMATICS PAPER The purpose of this sample paper is to show the structure of the Specialist Mathematics examination and the style of questions that may be used. The following extract is from the 2017 subject outline for Specialist Mathematics: EXTERNAL ASSESSMENT Assessment Type 3: Examination (30%) Students undertake a 3-hour external examination. The examination is based on the key questions and key concepts in the six topics. The considerations for developing teaching and learning strategies are provided as a guide only, although applications described under this heading may provide contexts for examination questions. The examination consists of a range of problems, some focusing on knowledge and routine skills and applications, and others focusing on analysis and interpretation. Some problems may require students to interrelate their knowledge, skills, and understanding from more than one topic. Students provide explanations and arguments, and use correct mathematical notation, terminology, and representations throughout the examination. A formula sheet is included in the examination booklet. Students may take two unfolded A4 sheets (four sides) of handwritten notes into the examination room. Students may use approved electronic technology during the external examination. However, students need to be discerning in their use of electronic technology to fi nd solutions to questions/ problems in examinations. All specifi c features of the assessment design criteria for this subject may be assessed in the external examination. Source: Specialist Mathematics 2017 Subject Outline Stage 2, p 41, on the SACE website, page 20 of 20 end of question booklet

Mathematical Methods 2017 Sample paper

South Australian Certificate of Education The external assessment requirements of this subject are listed on page 20. Question Booklet 2 Mathematical Methods 2017 Sample paper 2 Part 2 (Questions 11 to