Year 2013 VCE Mathematical Methods CAS Trial Examination 2 Suggested Solutions

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1 Year VCE Mathematical Methods CAS Trial Examination Suggested Solutions KILBAHA MULTIMEDIA PUBLISHING PO BOX 7 KEW VIC AUSTRALIA TEL: () FAX: () kilbaha@gmail.com

2 IMPORTANT COPYRIGHT NOTICE This material is copyright. Subject to statutory exception and to the provisions of the relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Kilbaha Multimedia Publishing. The contents of this work are copyrighted. Unauthorised copying of any part of this work is illegal and detrimental to the interests of the author. For authorised copying within Australia please check that your institution has a licence from Copyright Agency Limited. This permits the copying of small parts of the material, in limited quantities, within the conditions set out in the licence. Reproduction and communication for educational purposes The Australian Copyright Act 968 (the Act) allows a maximum of one chapter or % of the pages of this work, to be reproduced and/or communicated by any educational institution for its educational purposes provided that educational institution (or the body that administers it) has given a remuneration notice to Copyright Agency Limited (CAL) under the Act. For details of the CAL licence for educational institutions contact CAL, Level 5, Castlereagh Street, Sydney, NSW, Tel: () Fax: () info@copyright.com.au Web: While every care has been taken, no guarantee is given that these answers are free from error. Please contact us if you believe you have found an error. Kilbaha Multimedia Publishing

3 Mathematical Methods CAS Trial Exam Solutions Section Page SECTION ANSWERS A B C D E A B C D E A B C D E 4 A B C D E 5 A B C D E 6 A B C D E 7 A B C D E 8 A B C D E 9 A B C D E A B C D E A B C D E A B C D E A B C D E 4 A B C D E 5 A B C D E 6 A B C D E 7 A B C D E 8 A B C D E 9 A B C D E A B C D E A B C D E A B C D E Kilbaha Multimedia Publishing

4 Mathematical Methods CAS Trial Exam Solutions Section Page 4 SECTION Question Answer B f x b x b b b f b f b 4 b b f f b b average rate of change f b b b Question a,a f : a, a R, f x a x f a a f a a The range is Answer B y a a a a a a x Question The period T n b b Answer B Question 4 Answer A g xhx loge h x gh gloge h h g g h e h f x g x log e h x using the product rule f x g x h x f Now, 4, and 6 f 4loge e 8 e e Kilbaha Multimedia Publishing

5 Mathematical Methods CAS Trial Exam Solutions Section Page 5 Question 5 Answer A k k k k k k When or k, k there is no unique solution. When k there is no solution. When k there is infinitely many solutions. Question 6 Answer C tan f x x f x cos x x 6 h 8 f tan f 4 cos f x h f x hf x 8 tan Kilbaha Multimedia Publishing

6 Mathematical Methods CAS Trial Exam Solutions Section Page 6 Question 7 f x x x x 4 f x x x 8 x 8x x x Answer A Turning points at x and x To restrict the domain to make f a one-one function we require a Question 8 y log x e Answer E y logex y loge x y x y and x x y y and x x x x T y y y Question 9 Answer C dy b y cosbx bsinbx when x mt bsin dx 6b 6 normal mn 4 b b Question Answer E f x ax bx x ax b, the graph crosses the x-axis at f x ax b. b x and x. a A decreasing function has a negative gradient, f x ax b. so that ax b if a and b b then x. a y b a b a y x x Kilbaha Multimedia Publishing

7 Mathematical Methods CAS Trial Exam Solutions Section Page 7 Question Answer C T catches the train and D drives to work T T.7 T D. D T. D D.8 T D T D long-term drives is.6 5 Question Answer C da da given 6 dx dt A x x dx dx da 6 dx 6 6 cm/min dt da dt x dt x Question Answer D k k e x dx k x k e e k k solving for k gives, k.94 Question 4 Answer E and f x x g x x g x x 5 g x g x x x x f x Peter is incorrect g g x g x x x f x Quentin is correct g g x g x x x f x Sara is incorrect f g x f x x x g x Tanya is correct Kilbaha Multimedia Publishing

8 Mathematical Methods CAS Trial Exam Solutions Section Page 8 Question 5 Answer D Total area under the curve is one k sin x dx k k cosx k cos cos k The mean and median are both symmetrical at x 6 4 D. Is false E X 8 Question 6 Answer E k e Pr X k k! e Pr X e, Pr X e, Pr X! Pr Pr X X X Pr X Pr X Pr X Pr e Pr X e e! e Pr X Kilbaha Multimedia Publishing

9 Mathematical Methods CAS Trial Exam Solutions Section Page 9 Question 7 vt 7 t 4 t 4 dt m/s 7 7 Answer D initial velocity 7 v m/s A. is true. 6 distance travelled in the first two seconds, B. is true. dv d 7 6 a m/s acceleration C. is true. dt dt t4 t xt dt c c t 4 t4 t4 9 9 x c c t t xt 4 t 4 4 t 4 4 t 4 position E. is true, D. is false Question 8 Answer B A cubic with no turning points can be expressed in the form f x ax h k f x a x h k a x x h xh h k ax ahx ah x ah k, and hk, and f x ax bx c a x h x when x h f x ax bx cx d b ah c ah d ah k The point of inflexion is at f real factors, when, thus and this equation has only one root, or for no turning points, no b 4a c 4b ac 4 b ac b ac Kilbaha Multimedia Publishing

10 Mathematical Methods CAS Trial Exam Solutions Section Page Question 9 The area bounded by the graph of Answer A y f x between the x-values of x a and x b, is the same if the area is translated b units to the right, that is the area bounded by the graph of y f x b between the x-values of x a b and x b. This is also the same area if the area is translated b units to the left, that is the area bounded by the graph of y f x b between the x-values of x a b and x a b,b where a, b R and ba. Question Answer D 5, 6 4, f x for x, since X N Z N Z d d x X Z Z Z Z Pr Pr Pr A. is true Z Z Z Z Pr Pr Pr Pr B. is true X X X Pr Pr Pr X X C. is true [Pr 7 Pr 9 by symmetry, E. is true D. is false, all of A. B. C. E. are true X Z Kilbaha Multimedia Publishing

11 Mathematical Methods CAS Trial Exam Solutions Section Page Question Answer E Since it is a discrete random variable, the probabilities add to one, so that a a a 6 a a 6 a a 8 E X xpr X x a a a a 6 E X x Pr X x a 4 9 6a X E X E X var All of A. B. C. and D. are true, E. is false a a E Pr X x a a X x Question Answer B log x 4 f x e x 4 log 4 log 7 log log log 6 f x e e e e e consider four right rectangles, each of width one unit. A log 7 log log log 6 A L e e e e log 76 log 456 L e e END OF SECTION SUGGESTED ANSWERS Kilbaha Multimedia Publishing

12 Mathematics Methods CAS Trial Examination Solutions Section Page SECTION Question a.i. Let R pays by credit card, and C pays using cash C C.5 C R.65 and R R.45 R C.55 Pr usescredit card three times and the first is credit card RRRC RRCR RCRR Pr Pr Pr ii. Pr first and fourth are credit cards Pr RCCR Pr RCRR Pr RRCR Pr RRRR C R alternatively C R fourth is credit card.58 d b.i. X Bi n 6, p? X X Pr 8 Pr p p p p p p 9 p.5 by CAS solving with p gives p.4 or.647 but since E X 8 p.5 so p.647 d ii. X Bi n 6, p.65 X X Pr 8 Pr Kilbaha Multimedia Publishing

13 Mathematics Methods CAS Trial Examination Solutions Section Page d c. X is the time in hours spent shopping, X N?,? Pr X 7.4 Pr X now subtract equations solving gives $4 substituting gives $,55 d.i. Since the function is continuous at t t 6 t ae dt bsin dt 6 ae bsin b 6 Since the total area under the curve is equal to one, substituting b b ae ae 6 t t a e dt esin dt 6 6 t 6e t a e cos 6 6e 6e ae cos cos ae e 6e a a e6e a e Kilbaha Multimedia Publishing

14 Mathematics Methods CAS Trial Examination Solutions Section Page 4 ii. Graph passes through,.97,.6 6, and is continuous G and zero elsewhere y t t Pr X 4 b sin dt 6 iii b t 6b cos cos cos 6 4 e e with e b e iv. PrT T Pr T Pr T t t ae dt e ae dt e t t e e Kilbaha Multimedia Publishing

15 Mathematics Methods CAS Trial Examination Solutions Section Page 5 v. 6 t t E T a t e dt et sin dt with 6 a e ET hours mean time.9 Kilbaha Multimedia Publishing

16 Mathematics Methods CAS Trial Examination Solutions Section Page 6 Question f x x bx cx 6 i. f x x bx c ii. f b c 6 8 4b c 6 4b c 4 f 4b c 4b c Equation of the tangent at P y 4b c 4 4b c x tp x y b c x b c b c tp x y b c x b iii. Solving tpx f x when x tp 6 f tp 4b c 4b 8b c 6 8b c 8 f 6 b c 6 6 b c solving these simultaneous equations, 9b 7 b and c 4 P, 6 iv. so substitute b and c 4 f x x x x tp x x 4 6 and 4 A f x tp x dx x x 4dx v. Equation of the tangent at Q f 6 and f x x 6x 4 f y 6 5x tqx y 5x vi. tqx f x 5x x x 4x 6 solving gives x 5 or x f or 5,6 tq R Kilbaha Multimedia Publishing

17 Mathematics Methods CAS Trial Examination Solutions Section Page 7 5 vii. A tq x f x dx 5 A x x 9x 5dx 7 viii. A 8 4 A 6 A ix. correct graphs, tangents at points P, and Q G 6 y R 5,6 y tpx 4x 4 A 8 A Q, x y tqx 5x -6 - P, 6-8 y x x x Kilbaha Multimedia Publishing

18 Mathematics Methods CAS Trial Examination Solutions Section Page 8 Kilbaha Multimedia Publishing

19 Mathematics Methods CAS Trial Examination Solutions Section Page 9 Question a. Consider the hexagonal base of the vase B B x y A A x x d AB x x y d AB One triangle has area x y x x x 4 or area of one equilateral triangle x sin 6 x x 4 Area of the base is six triangles 6 x x 4 Total surface area is the area of the base and six rectangles of length x and height h. S x 6xh b. Volume is area of the base multiplied by the height V x h c.i. S x S x 6xh 6xh S x S x h x S x V x x h x x V x S x x 8 x V x S x S x S x x 8 and S S so that x or x Kilbaha Multimedia Publishing

20 Mathematics Methods CAS Trial Examination Solutions Section Page dv ii. S 9 x dx 8 dv for a maximum volume S 9 x dx S S x 9 d.i. ii. V x V x h h V x S x 6xh S x 6x 4V 4V 4V S x x x x x x x ds 4 4 V V x x x dx x for a minimum surface area ds 4 4 x V x V dx x 9 4V x 9 Note that from CAS, there are many equivalent forms, for all these answers and equations. e. solving the five non-linear equations using CAS, for four unknowns, S, V, h, x S x 6xh V S x 4 9 x h 4V 5 V 9S 9 x and gives x8 and h 7 cm Kilbaha Multimedia Publishing

21 Mathematics Methods CAS Trial Examination Solutions Section Page Kilbaha Multimedia Publishing

22 Mathematics Methods CAS Trial Examination Solutions Section Page Question 4 a.i. x g x f t dt x g Now f t dt 4 4 f t dt f t dt is the area of a triangle of side lengths and ii. g Now 4 4 f t dt 8 4 iii. g f t dt represents one-quarter of the area of a circle of radius f t dt 6 6 f t dt 6 t 4 dt 6 t 4t 6 9 b. gx f x f x where g x x 4 for x 6 x for x 4 x for x 6 x for x 4 Kilbaha Multimedia Publishing

23 Mathematics Methods CAS Trial Examination Solutions Section Page c. The function gx has a maximum or minimum turning point when gx or f x for x x x 4 4 g f t dt t dx t t 4 x x for 4 6 since x 4 6 x 4 x x g f t dt t dt , is the maximum turning point, is the minimum turning point d. The point, is the inflexion point e. Now g has a domain,4 g and g are endpoints. 8 The range is,, crosses x-axis at and the point, is the inflexion point correct graph over correct domain and range, all stationary points and intercepts,,, G G Kilbaha Multimedia Publishing

24 Mathematics Methods CAS Trial Examination Solutions Section Page 4 5 y x END OF SECTION SUGGESTED ANSWERS Kilbaha Multimedia Publishing

Year 2011 VCE. Mathematical Methods CAS. Trial Examination 1

Year 2011 VCE. Mathematical Methods CAS. Trial Examination 1 Year 0 VCE Mathematical Methods CAS Trial Examination KILBAHA MULTIMEDIA PUBLISHING PO BOX 7 KEW VIC 30 AUSTRALIA TEL: (03) 908 5376 FAX: (03) 987 4334 kilbaha@gmail.com http://kilbaha.com.au IMPORTANT