Specialist Mathematics 2017 Sample paper

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1 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 You may write on page 5 if you need more space Allow approximately 90 minutes Examination information Materials Question Booklet (Part ) Question Booklet (Part ) one SACE registration number label Reading time 0 minutes You may make notes on scribbling paper Writing time 3 hours Show all working in the question booklets State all answers correct to three significant figures, unless otherwise instructed Use black or blue pen You may use a sharp dark pencil for diagrams Approved calculators may be used complete the box below Total marks 50 SACE Board of South Australia 07 Graphics calculator For office use only Attach your SACE registration number label here. Brand Model Supervisor check Re-marked. Brand Model

2 This sample Specialist Mathematics paper shows the format of the examination for 07. page of 5

3 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 SPECIALIST MATHEMATICS Circular Functions sin Acos A tan cot Asec A Acosec A sina B sin Acos B cos Asin B cosa B cosacos B sin Asin B tan A tan B tana B tanatanb sin A sin Acos A cos Acos Asin A cos A sin A tan A tan A tan A sin Acos BsinABsinAB cos Acos B cosa BcosAB sin Asin BcosABcosAB sina sin B sin A Bcos A B cos Acos Bcos ABcos A B cos Acos Bsin A Bsin A B Matrices and Determinants If A a b then det A A adbc and c d A A d c b. a Quadratic Equations b b 4ac If ax + bx + c = 0 then x. a Distance from a Point to a Plane The distance from x, y, z to Ax By Cz D 0 is given by AxByC z D. A B C Derivatives f x y arcsin x arccos x arc tan x fx x x x dy dx 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 dx gx g x d dx f g x f g x g x Integration by Parts d f x g x x f x g x f x g x dx Volumes of Revolution b About x axis y dx, where y is a function of x. a d About y axis x dy, where y is a one-to-one c function of x. Mensuration Area of sector r Arc length r, where is in radians In any triangle ABC B c a A b C Area of triangle absin C a b c sin A sin B sin C a b c bccos A page 3 of 5

4 This sample Specialist Mathematics paper shows the format of the examination for 07. page 4 of 5

5 The examination questions begin on page 6. This sample Specialist Mathematics paper shows the format of the examination for 07. page 5 of 5 PLEASE TURN OVER

6 PART (Questions to 9) (75 marks) Question (8 marks) The parametric equations below describe a Bézier curve. xt 4t 4t0 3 y t 8t 4t 4 where t 0, (a) Find all points on the curve where the x-value is 0. ( marks) (b) Find the leftmost point on the curve. (3 marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 6 of 5

7 (c) Find the length of the curve. (3 marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 7 of 5 PLEASE TURN OVER

8 Question (8 marks) x (a) On the axes in Figure, graph the function f( x). x x Clearly show the behaviour of the function near the asymptotes. 4 f( x) x 3 Figure (6 marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 8 of 5

9 (b) On the axes in Figure, graph the function f f 4 x x x 3 Figure ( marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 9 of 5 PLEASE TURN OVER

10 Question 3 (8 marks) In the following system of equations, k is a real constant. xyz x3yz k x5ykz (a) Write the system in augmented matrix form and, stating all row operations used, show that this leads to the reduced system: : 0 : k. 0 0 k 3 : k 3 (3 marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 0 of 5

11 (b) (i) State the value of k for which the system has an infinite number of solutions. ( mark) (ii) Solve the system for the value of k stated in your answer to part (b)(i). (3 marks) (iii) Give a geometrical interpretation of the system of equations for this value of k. ( mark) This sample Specialist Mathematics paper shows the format of the examination for 07. page of 5 PLEASE TURN OVER

12 Question 4 (9 marks) (a) Find an approximate value for 3 3 x x d x. 3 x x x Give your answer to four significant figures. ( marks) (b) (i) Show that x x 3 x x x x x. ( marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page of 5

13 (ii) Show that 3 3 x x dx ln3 3. x x x 6 (5 marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 3 of 5 PLEASE TURN OVER

14 Question 5 (8 marks) (a) Let a (i) Find a.,, 4 and b 5, 3,. ( mark) (ii) Find a b. ( mark) (iii) Verify that ab a b a b. ( marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 4 of 5

15 (b) Use the vector property cd cd c d to prove that, for any vectors c and d, cd c d c d. ( marks) (c) If c 0, d 5, and c d 5, find: (i) c d. ( mark) (ii) c d. ( mark) This sample Specialist Mathematics paper shows the format of the examination for 07. page 5 of 5 PLEASE TURN OVER

16 Question 6 (8 marks) Let v 68i and w 7i. (a) On the Argand diagram in Figure 3, plot and label the points corresponding to v and w. Im z Re z Figure 3 ( mark) (b) On the Argand diagram in Figure 3, draw the set of complex numbers z such that z 0. ( marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 6 of 5

17 (c) Verify that v is a member of the set of complex numbers that you drew in part (b). ( mark) (d) Let u be a complex number such that ui ww. Find u in Cartesian form. ( marks) (e) On the Argand diagram in Figure 3, mark the set of complex numbers z that satisfies both z 0 and zu zv. ( marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 7 of 5 PLEASE TURN OVER

18 Question 7 (8 marks) (a) Use mathematical induction to prove that n 3n n n for n, where n is an integer. (5 marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 8 of 5

19 (b) (i) Show that the sum of n terms n n n Sn 3 can be written as the cubic polynomial function 3 f n3n 4. 5n. 5n. ( marks) (ii) For what value of n will S n 3065? ( mark) This sample Specialist Mathematics paper shows the format of the examination for 07. page 9 of 5 PLEASE TURN OVER

20 Question 8 (8 marks) (a) Use integration by parts to show that where c is a constant. x xsin xdx sin x cos xc 4 (4 marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 0 of 5

21 (b) Figure 4 shows the graph of y xsin x, where 0 x. y x 3 Figure 4 Show that the area of one of the regions bounded by the curve and the x-axis is exactly three times the area of the other region. (4 marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page of 5 PLEASE TURN OVER

22 Question 9 (0 marks) Figure 5 shows OA, the parabolic arc y x, for 0 x. Figure 6 represents the three-dimensional shape formed by rotating OA around the y-axis. y y A 4 A 3 3 y = x O x O x Figure 5 Figure 6 Figure 7 shows a thin rectangular strip of length x cm and width dy cm, drawn from a point x, y on OA. Figure 8 represents the thin disc of radius r cm and height h cm that is created when the rectangular strip is rotated around the y-axis. y y A 4 A 3 x dy 3 r = x h = dy O x O x Figure 7 Figure 8 This sample Specialist Mathematics paper shows the format of the examination for 07. page of 5

23 (a) (i) State the volume of the thin disc shown in Figure 8 in terms of x and y. ( mark) (ii) Hence write a definite integral in terms of y to represent the exact volume of the three-dimensional shape shown in Figure 6. (3 marks) (b) (i) A glass manufacturer makes drinking glasses in the three-dimensional shape shown in Figure 6. One of these glasses is filled with water to a depth of k cm, where 0 k 4. Find the volume of water in the glass in terms of k. ( marks) Question 9 continues on page 4. This sample Specialist Mathematics paper shows the format of the examination for 07. page 3 of 5 PLEASE TURN OVER

24 (ii) An empty glass is placed under a tap from which water is flowing at the rate of cm 3 s. At what rate is the depth changing when the glass is half full? (4 marks) This sample Specialist Mathematics paper shows the format of the examination for 07. page 4 of 5

25 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 4(b)(ii) continued ). This sample Specialist Mathematics paper shows the format of the examination for 07. page 5 of 5 end of question booklet

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