SPECIALIST MATHEMATICS

Victorian Certificate of Education 00 SUPERVISOR TO ATTACH PROCESSING LABEL HERE STUDENT NUMBER Letter Figures Words SPECIALIST MATHEMATICS Written eamination Monday November 00 Reading time:.00 pm to.5 pm (5 minutes) Writing time:.5 pm to 5.5 pm ( hours) QUESTION AND ANSWER BOOK Section Number of questions 5 Structure of book Number of questions to be answered 5 Number of marks 58 Total 80 Students are permitted to bring into the eamination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set-squares, aids for curve sketching, one bound reference, one approved CAS calculator or CAS software and, if desired, one scientific calculator. Calculator memory DOES NOT need to be cleared. Students are NOT permitted to bring into the eamination room: blank sheets of paper and/or white out liquid/tape. Materials supplied Question and answer book of pages with a detachable sheet of miscellaneous formulas in the centrefold. Answer sheet for multiple-choice questions. Instructions Detach the formula sheet from the centre of this book during reading time. Write your student number in the space provided above on this page. Check that your name and student number as printed on your answer sheet for multiple-choice questions are correct, and sign your name in the space provided to verify this. All written responses must be in English. At the end of the eamination Place the answer sheet for multiple-choice questions inside the front cover of this book. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the eamination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 00

00 SPECMATH EXAM SECTION Instructions for Section Answer all questions in pencil on the answer sheet provided for multiple-choice questions. Choose the response that is correct for the question. A correct answer scores, an incorrect answer scores 0. Marks will not be deducted for incorrect answers. No marks will be given if more than one answer is completed for any question. Take the acceleration due to gravity to have magnitude g m/s, where g = 9.8. Question An ellipse has a horizontal semi-ais length of and a vertical semi-ais length of. The centre of the ellipse is at the point with coordinates (, 5). A possible equation for the ellipse is A. ( + ) + 4(y 5) = 4 B. ( ) + 4(y + 5) = 4 C. 4( + ) + (y 5) = 4 D. 4( ) + (y + 5) = 4 E. 4( ) + (y + 5) = Question Each of the following equations represents a hyperbola. Which hyperbola does not have perpendicular asymptotes? A. ( ) (y + ) = B. y + 4y = 4 C. ( ) (y + ) = 9 D. (y ) ( + ) = E. 4 y 4y = 4 SECTION continued

00 SPECMATH EXAM Question k a Let f( ), where k and a are real constants. If k is an odd integer which is greater than and a < 0, a possible graph of f could be A. y B. y 4 4 6 4 O 4 6 6 4 O 4 6 4 4 C. y D. y 4 4 6 4 O 4 6 6 4 O 4 6 4 4 E. y 4 6 4 O 4 6 4 Question 4 The position vector of a particle at time t 0 is given by r sin t i cos t j. The path of the particle has cartesian equation A. y = B. y = C. y D. y E. y SECTION continued TURN OVER

00 SPECMATH EXAM 4 Question 5 For, the graphs of the two curves given by y = sec () and y 5 tan( ) intersect A. only at the one point (arctan(), 0) B. only at the two points ( arctan (), 0) 5 C. only at the one point arctan, 5 D. only at the two points arctan, 5 E. at the two points arctan,, as well as at the two points (arctan(), 0) Question 6 Let z = cis 5 π 6. The imaginary part of z i is A. i B. C. D. E. i SECTION continued

5 00 SPECMATH EXAM Question 7 A particular comple number z is represented by the point on the following argand diagram. Im(z) O Re(z) All aes below have the same scale as those in the diagram above. The comple number iz is best represented by A. Im(z) B. Im(z) O Re(z) O Re(z) C. Im(z) D. Im(z) O Re(z) O Re(z) E. Im(z) O Re(z) SECTION continued TURN OVER

00 SPECMATH EXAM 6 Question 8 The polynomial equation P(z) = 0 has one comple coefficient. Three of the roots of this equation are z = + i, z = i and z = 0. The minimum degree of P(z) is A. B. C. D. 4 E. 5 Question 9 Given that z 4cis A. B. C. 0 4 7 D. E., it follows that Arg z 5is Question 0 On an argand diagram, a set of points which lies on a circle of radius centred at the origin is A. { zc: zz } B. { zc: z 4} C. { zc:rez Im z 4} D. { zc: z z z z 6} E. { zc: Rez Im z 6} SECTION continued

7 00 SPECMATH EXAM Question A direction field for the volume of water, V megalitres, in a reservoir t years after 00 is shown below. According to this model, for k > 0, dv dt A. kt B. k V C. kv is equal to D. kv E. k V Question Let dy and ( d 0, y 0 ) = (0, ). Using Euler s method, with a step size of 0., the value of y correct to two decimal places is A. 0.7 B. 0.0 C..70 D..7 E..0 SECTION continued TURN OVER

00 SPECMATH EXAM 8 Question The amount of a drug, mg, remaining in a patient s bloodstream t hours after taking the drug is given by the differential equation d 05.. dt The number of hours needed for the amount to halve is 0 A. log e 0 B. log e( ) C. log e (5) D. 5log e E. log e( 00) Question 4 Use a suitable substitution to show that the definite integral A. B. C. D. 0 0 u u u u du du du du E. u du 0 d can be simplified to 0 Question 5 The scalar resolute of a i k in the direction of b i jkis A. B. 8 0 8 9 i j k C. 8 D. E. 4 5 i k 8 SECTION continued

9 00 SPECMATH EXAM Question 6 The square of the magnitude of the vector d5i j 0 k is A. 6 B. 4 C. 6 D. 5. E. 4 Question 7 The angle between the vectors a i k and b i j is eactly A. B. C. D. 6 4 E. Question 8 60 60 5 kg A 5 kg mass is suspended from a horizontal ceiling by two strings of equal length. Each string makes an angle of 60 to the ceiling, as shown in the above diagram. Correct to one decimal place, the tension in each string is A. 4.5 newtons B. 8. newtons C. 4.6 newtons D. 49.0 newtons E. 84.9 newtons SECTION continued TURN OVER

00 SPECMATH EXAM 0 Question 9 An object is moving in a northerly direction with a constant acceleration of ms. When the object is 00 m due north of its starting point, its velocity is 0 ms in the northerly direction. The eact initial velocity of the object could have been A. 0 5 ms B. 5 0ms C. 0 7 ms D. 0 7 ms E. 7 0 ms Question 0 The acceleration, a ms v, of a particle moving in a straight line is given by a log ( e v, where v is the velocity ) of the particle in ms at time t s. The initial velocity of the particle was 5 ms. The velocity of the particle, in terms of t, is given by A. v = e t B. v = e t + 4 C. v e D. v e E. v e t log e ( 5) t(log e 5) t e (log 5 ) Question A light inetensible string passes over a smooth, light pulley. A mass of 5 kg is attached to one end of the string and a mass of M kg is attached to the other end, as shown below. 5 kg M kg The M kg mass accelerates downwards at 7 5 The value of M is ms. A. 5 7 B. 6 5 C. 6 D. 8 5 E. 9 4 5 SECTION continued

00 SPECMATH EXAM Question A particle of mass m moves in a straight line under the action of a resultant force F where F = F (). Given that the velocity v is v 0 where the position is 0, and that v is v where is, it follows that A. v F d v0 m B. v Fd v 0 0 C. v Fd v D. v E. v 0 0 0 m F d v 0 0 F v d m 0 0 END OF SECTION TURN OVER

00 SPECMATH EXAM SECTION Instructions for Section Answer all questions in the spaces provided. Unless otherwise specified an eact answer is required to a question. In questions where more than one mark is available, appropriate working must be shown. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Take the acceleration due to gravity to have magnitude g m/s, where g = 9.8. Question The diagram below shows a triangle with vertices O, A and B. Let O be the origin, with vectors OA a and OB b. B A b ~ a ~ a. Find the following vectors in terms of a and b. O i. MA, where M is the midpoint of the line segment OA ii. BA iii. AQ, where Q is the midpoint of the line segment AB. + + = marks SECTION Question continued

00 SPECMATH EXAM b. Let N be the midpoint of the line segment OB. Use a vector method to prove that the quadrilateral MNQA is a parallelogram. marks Now consider the particular triangle OAB with OA i j k and OB i where, which is greater than zero, is chosen so that triangle OAB is isosceles, with OB OA. c. Show that = 4. d. i. Find OQ, where Q is the midpoint of the line segment AB. mark ii. Use a vector method to show that OQ is perpendicular to AB. + = 4 marks Total marks SECTION continued TURN OVER

00 SPECMATH EXAM 4 Question A block of m kg sits on a rough plane which is inclined at 0 to the horizontal. The block is connected to a mass of 0 kg by a light inetensible string which passes over a frictionless light pulley. 0 a. If the block is on the point of moving down the plane, clearly label the forces which are shown in the diagram. marks b. Write down, but do not attempt to solve, equations involving the forces acting on the block, and the forces acting on the 0 kg mass, for the situation in part a. marks c. Given that the coefficient of friction is 4, show that the mass of the block in kg is given by 80 m 4. marks SECTION Question continued

5 00 SPECMATH EXAM In order to pull the block up the plane, the mass hanging from the string is replaced by a 0 kg mass, and a lubricant is spread on the inclined plane to decrease the coefficient of friction. d. If the block is now on the point of moving up the plane, show that the new value of the coefficient of friction, correct to three decimal places, is = 0.077. marks Before the block actually starts to move up the lubricated plane where = 0.077, the string breaks and the block begins to slide down the plane. e. Find the velocity of the block three seconds after the string breaks. Give your answer in ms correct to one decimal place. marks Total marks SECTION continued TURN OVER

00 SPECMATH EXAM 6 Question The population of a town is initially 0 000 people. This population would increase at a rate of one per cent per year, ecept that there is a steady flow of people arriving at and leaving from the town. The population P after t years may be modelled by the differential equation dp dt P k 00 with the initial condition, P = 0 000 when t = 0, where k is the number of people leaving per year minus the number of people arriving per year. a. Verify by substitution that for k = 800, P = 0 000(4 e 0.0t ) satisfies both the differential equation and the initial condition. marks b. For k = 800, find the time taken for the population to decrease to zero. Give your answer correct to the nearest whole year. marks SECTION Question continued

7 00 SPECMATH EXAM The differential equation which models the population growth can be epressed as dt 00 with P = 0000 when t = 0. dp P 00k c. Show by integration that for k < 0, the solution of this differential equation is P = (0 000 00k)e 0.0t + 00k. marks SECTION Question continued TURN OVER

00 SPECMATH EXAM 8 It can be shown that the solution given in part c. is valid for all real values of k. d. For each of the values i. k = 800 ii. k = 00 iii. k = 00 sketch the graph of P versus t on the set of aes below while the population eists, for 0 t 40. P 0000 0000 O 0 0 0 40 t marks e. i. Find the value of k if the population has increased to 550 after twelve years. ii. Use the definition of k to interpret your answer to part i. in the contet of the population model. + = marks Total marks SECTION continued

9 00 SPECMATH EXAM Working space SECTION continued TURN OVER

00 SPECMATH EXAM 0 Question 4 Consider the function f with rule f () = sin ( ). a. Sketch the graph of the relation y = f () on the aes below. Label the endpoints with their eact coordinates, and label the and y intercepts with their eact values. y O marks b. i. Write down a definite integral in terms of y, which when evaluated will give the volume of the solid of revolution formed by rotating the graph drawn above about the y-ais. ii. Find the eact value of the definite integral in part i. + = marks SECTION Question 4 continued

00 SPECMATH EXAM c. Use calculus to show that f, for (0, a) and find the value of a. d. Complete the following to specify f () as a hybrid function over the maimal domain of f. marks f ' () = for for e. Sketch the graph of the hybrid function f on the set of aes below, showing any asymptotes. marks y 7 6 5 4.0 0.5 O 0.5.0 4 5 6 7 marks Total marks SECTION continued TURN OVER

00 SPECMATH EXAM Question 5 Let u = 6 i and w = + i where u, w C. u w u a. Given that z ( ), show that z 0. iw b. The comple number z can be epressed in polar form as mark z 00 π cis. 4 Find all distinct comple numbers z such that z = z. b Give your answers in the form a n cis c, where a, b, c and n are integers. marks SECTION Question 5 continued

00 SPECMATH EXAM Let the argument of u be given by Arg (u) =. (You are not required to find.) c. By epressing iw in polar form in terms of, show that u iw cis( ). d. Use the relation given in part a. to find Arg (u + w) in terms of. marks marks Total 0 marks END OF QUESTION AND ANSWER BOOK

SPECIALIST MATHEMATICS Written eaminations and FORMULA SHEET Directions to students Detach this formula sheet during reading time. This formula sheet is provided for your reference. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 00

SPECMATH Specialist Mathematics Formulas Mensuration area of a trapezium: curved surface area of a cylinder: volume of a cylinder: volume of a cone: volume of a pyramid: volume of a sphere: area of a triangle: sine rule: cosine rule: a bh π rh π r h π r h Ah 4 π r bcsin A a b c sin A sin B sinc c = a + b ab cos C Coordinate geometry ellipse: h a y k b hyperbola: h a y k b Circular (trigonometric) functions cos () + sin () = + tan () = sec () cot () + = cosec () sin( + y) = sin() cos(y) + cos() sin(y) cos( + y) = cos() cos(y) sin() sin(y) tan( ) tan( y) tan( y) tan( ) tan( y) sin( y) = sin() cos(y) cos() sin(y) cos( y) = cos() cos(y) + sin() sin(y) tan( ) tan( y) tan( y) tan( ) tan( y) cos() = cos () sin () = cos () = sin () tan( ) sin() = sin() cos() tan( ) tan ( ) function sin cos tan domain [, ] [, ] R range π π, [0, ] π, π

SPECMATH Algebra (comple numbers) z = + yi = r(cos θ + i sin θ = r cis θ z y r π < Arg z π z r z z = r r cis(θ + θ ) cis z r θθ z n = r n cis(nθ) (de Moivre s theorem) Calculus d d n n n n n d c, n n d d e a ae a a e d a e a c d log e( ) d d log c e d sin( a) acos( a) sin( a) d cos( a) c d a d cos( a) asin( a) cos( a) d sin( a) c d a d d tan( a) asec ( a) d sin d d cos d ( ) ( ) sec ( a) d tan( a) c a a a d sin ca, 0 a d cos ca, 0 a d tan a ( ) d a d tan a c product rule: quotient rule: chain rule: Euler s method: acceleration: d d uvu dv d v du d u d d v v dy dy du d du d If dy d v du d u dv d f, 0 = a and y 0 = b, then n + = n + h and y n + = y n + h f( n ) d dv a v dv d v dt dt d d constant (uniform) acceleration: v = u + at s = ut + at v = u + as s = (u + v)t TURN OVER

SPECMATH 4 Vectors in two and three dimensions r i yj zk ~ ~ ~ ~ r ~ = y z r ~ r. r ~ = r r cos θ = + y y + z z dr ~ d dy dz r i j k ~ dt dt ~ dt ~ dt ~ Mechanics momentum: p mv ~ ~ equation of motion: R ma ~ ~ friction: F μn END OF FORMULA SHEET