SPECIALIST MATHEMATICS

Transcription

1 Victorian Certificate of Education 06 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Section Written examination Monday 7 November 06 Reading time:.5 am to.00 noon (5 minutes) Writing time:.00 noon to.00 pm ( hours) QUESTION AND ANSWER BOOK Number of questions Structure of book Number of questions to be answered Number of marks A B Total 80 Students are permitted to bring into the examination room: pens, pencils, highlighters, erasers, sharpeners, rulers, a protractor, set squares, aids for curve sketching, one bound reference, one approved technology (calculator or software) and, if desired, one scientific calculator. Calculator memory DOES NOT need to be cleared. For approved computer-based CAS, full functionality may be used. Students are NOT permitted to bring into the examination room: blank sheets of paper and/or correction fluid/tape. Materials supplied Question and answer book of 3 pages. Formula sheet. Answer sheet for multiple-choice questions. Instructions 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. Unless otherwise indicated, the diagrams in this book are not drawn to scale. All written responses must be in English. At the end of the examination Place the answer sheet for multiple-choice questions inside the front cover of this book. You may keep the formula sheet. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 06

2 06 SPECMATH EXAM SECTION A Multiple-choice questions Instructions for Section A 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. Unless otherwise indicated, the diagrams in this book are not drawn to scale. Take the acceleration due to gravity to have magnitude g ms, where g = 9.8 Question The cartesian equation of the relation given by x = 3cosec (t) and y = cot (t) is A. ( y + ) x = 6 9 B. 6( x + 3) ( y + ) = 3 C. x ( y + ) + = 9 6 D. x 3y = 5 ( ) = E. y + 6( x 3) 3 Question x a The implied domain of y = arccos, where b > 0 is b A. [, ] B. [a b, a + b] C. [a, a + ] D. [a, a + bπ] E. [ b, b] Question 3 x The straight-line asymptote(s) of the graph of the function with rule f ( x)= real constant, is given by A. x = 0 only. B. x = 0 and y = 0 only. C. x = 0 and y = x only. D. x = 0, x= a and x= a only. E. x = 0 and y = a only. 3 ax, where a is a non-zero x SECTION A continued

3 3 06 SPECMATH EXAM Question One of the roots of z 3 + bz + cz = 0 is 3 i, where b and c are real numbers. The values of b and c respectively are A. 6, 3 B. 3, C. 3, D., 3 E. 6, 3 Question 5 If Arg ( + ai)= A. 3 B. 3 3 π, then the real number a is C. 3 D. 3 E. 3 Question 6 The points corresponding to the four complex numbers given by π 3π z = z = z3 = π π cis, cis, cis, z = cis 3 3 are the vertices of a parallelogram in the complex plane. Which one of the following statements is not true? A. The acute angle between the diagonals of the parallelogram is 5 B. The diagonals of the parallelogram have lengths and C. zzzz 3 = 0 D. z+ z + z3 + z = 0 E. z for all four of z, z, z 3, z SECTION A continued TURN OVER

4 06 SPECMATH EXAM Question 7 Given that x = sin (t) cos (t) and y A. cos (t) sin (t) B. cos (t) + sin (t) C. sec (t) + cosec (t) D. sec (t) cosec (t) E. ( t ) () sin() t cos cos t = sin ( t), then dy dx in terms of t is Question 8 b 3 x ( ) a Using a suitable substitution, xe dx, where a and b are real constants, can be written as b u ( ) a A. e du b u ( ) a B. e du C. D. E. 8 a u ( e ) du b b u ( e ) a 3 8b 3 8 8a du u ( e ) du Question 9 dy If f( x)= = x x, where y dx 0 = 0 = y(), then y 3 using Euler s formula with step size 0. is A. 0. f () B f (.) C f (.) D f (.3) E f (.) SECTION A continued

5 5 06 SPECMATH EXAM Question 0 y 3 O 3 x The direction field for the differential equation dy + x+ y = 0 is shown above. dx A solution to this differential equation that includes (0, ) could also include A. (3, ) B. (3.5,.5) C. (.5, ) D. (.5, ) E. (.5, ) SECTION A continued TURN OVER

6 06 SPECMATH EXAM 6 Question Let a = 3i + j+ α k and b= i j+ α k, where α is a real constant. 7 If the scalar resolute of a in the direction of b is, then α equals 73 A. B. C. 3 D. E. 5 Question If a = i j+ 3 k and b= mi + j+ k, where m is a real constant, the vector a b will be perpendicular to vector b where m equals A. 0 only B. only C. 0 or D..5 E. 0 or Question 3 A particle of mass 5 kg is subject to forces i newtons and 9j newtons. If no other forces act on the particle, the magnitude of the particle s acceleration, in ms, is A. 3 B.. i + 8. j C.. D. 9 E. 60i + 5j SECTION A continued

7 7 06 SPECMATH EXAM Question Two light strings of length m and 3 m connect a mass to a horizontal bar, as shown below. The strings are attached to the horizontal bar 5 m apart. 5 m horizontal bar 3 m T T m mass Given the tension in the longer string is T and the tension in the shorter string is T, the ratio of the tensions T T A. B. C. D. E is Question 5 A variable force of F newtons acts on a 3 kg mass so that it moves in a straight line. At time t seconds, t 0, its velocity v metres per second and position x metres from the origin are given by v = 3 x. It follows that A. F = x B. F = 6x C. F = x 3 6x D. F = 6x 3 8x E. F = 9x 3x 3 SECTION A continued TURN OVER

8 06 SPECMATH EXAM 8 Question 6 A cricket ball is hit from the ground at an angle of 30 to the horizontal with a velocity of 0 ms. The ball is subject only to gravity and air resistance is negligible. Given that the field is level, the horizontal distance travelled by the ball, in metres, to the point of impact is A. B. C. D. E. 0 3 g 0 g 00 3 g 00 3 g 00 g Question 7 A body of mass 3 kg is moving to the left in a straight line at ms. It experiences a force for a period of time, after which it is then moving to the right at ms. The change in momentum of the particle, in kg ms, in the direction of the final motion is A. 6 B. 0 C. D. 6 E. Question 8 Oranges grown on a citrus farm have a mean mass of 0 grams with a standard deviation of 9 grams. Lemons grown on the same farm have a mean mass of 76 grams with a standard deviation of 3 grams. The masses of the lemons are independent of the masses of the oranges. The mean mass and standard deviation, in grams, respectively of a set of three of these oranges and two of these lemons are A. 76, 3 9 B. 636, C. 76, 33 D. 636, 3 0 E. 636, 33 SECTION A continued

9 9 06 SPECMATH EXAM Question 9 A random sample of 00 bananas from a given area has a mean mass of 0 grams and a standard deviation of 6 grams. Assuming the standard deviation obtained from the sample is a sufficiently accurate estimate of the population standard deviation, an approximate 95% confidence interval for the mean mass of bananas produced in this locality is given by A. (78.7,.3) B. (06.9, 3.) C. (09., 0.8) D. (05.,.8) E. (9, 6) Question 0 The lifetime of a certain brand of batteries is normally distributed with a mean lifetime of 0 hours and a standard deviation of two hours. A random sample of 5 batteries is selected. The probability that the mean lifetime of this sample of 5 batteries exceeds 9.3 hours is A B C D E END OF SECTION A TURN OVER

10 06 SPECMATH EXAM 0 SECTION B Instructions for Section B Answer all questions in the spaces provided. Unless otherwise specified, an exact 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 ms, where g = 9.8 Question (9 marks) x x a. Find the stationary point of the graph of f ( x)= + + 3, x R\{}. 0 Express your answer x in coordinate form, giving values correct to two decimal places. mark b. Find the point of inflection of the graph given in part a. Express your answer in coordinate form, giving values correct to two decimal places. marks SECTION B Question continued

11 06 SPECMATH EXAM x x c. Sketch the graph of f ( x)= + + for x [ 3, 3] on the axes below, labelling the x turning point and the point of inflection with their coordinates, correct to two decimal places. 3 3 marks y 8 3 O 3 x 8 [ ] A glass is to be modelled by rotating the curve that is the part of the graph where x 3, 0. 5 about the y-axis, to form a solid of revolution. d. i. Write down a definite integral, in terms of x, which gives the length of the curve to be rotated. mark ii. Find the length of this curve, correct to two decimal places. mark e. The volume of the solid formed is given by V = a xdy. Find the values of a, b and c. Do not attempt to evaluate this integral. c b mark SECTION B continued TURN OVER

12 06 SPECMATH EXAM Question ( marks) A line in the complex plane is given by z = z+ 3 i, z C. a. Find the equation of this line in the form y = mx + c. marks b. Find the points of intersection of the line z = z+ 3 i with the circle z = 3. marks SECTION B Question continued

13 3 06 SPECMATH EXAM c. Sketch both the line z = z+ 3 i and the circle z = 3 on the Argand diagram below. marks Im(z) 3 3 O 3 Re(z) 3 d. The line z = z+ 3 i cuts the circle z = 3 into two segments. Find the area of the major segment. marks e. Sketch the ray given by Arg ( z)= 3 π on the Argand diagram in part c. mark f. Write down the range of values of α, α R, for which a ray with equation Arg(z) = απ intersects the line z = z+ 3 i. marks SECTION B continued TURN OVER

14 06 SPECMATH EXAM Question 3 ( marks) A tank initially has 0 kg of salt dissolved in 00 L of water. Pure water flows into the tank at a rate of 0 L/min. The solution of salt and water, which is kept uniform by stirring, flows out of the tank at a rate of 5 L/min. If x kilograms is the amount of salt in the tank after t minutes, it can be shown that the differential equation relating x and t is dx x + dt 0 + t = 0. a. Solve this differential equation to find x in terms of t. 3 marks A second tank initially has 5 kg of salt dissolved in 00 L of water. A solution of 60 kg of salt per litre flows into the tank at a rate of 0 L/min. The solution of salt and water, which is kept uniform by stirring, flows out of the tank at a rate of 0 L/min. b. If y kilograms is the amount of salt in the tank after t minutes, write down an expression for the concentration, in kg/l, of salt in the second tank at time t. mark SECTION B Question 3 continued

15 5 06 SPECMATH EXAM c. Show that the differential equation relating y and t is dy dt y t = 3. marks t d. Verify by differentiation and substitution into the left side that y = + 0t+ 900 satisfies 60 ( + t) the differential equation in part c. Verify that the given solution for y also satisfies the initial condition. 3 marks SECTION B Question 3 continued TURN OVER

16 06 SPECMATH EXAM 6 e. Find when the concentration of salt in the second tank reaches kg/l. Give your answer in minutes, correct to two decimal places. marks SECTION B continued

17 7 06 SPECMATH EXAM CONTINUES OVER PAGE SECTION B continued TURN OVER

18 06 SPECMATH EXAM 8 Question (0 marks) Two ships, A and B, are observed from a lighthouse at origin O. Relative to O, their position vectors at time t hours after midday are given by ( ) + ( + ) r A = 5 t i 3 t j r B = ( t ) i + ( 5t ) j where displacements are measured in kilometres. a. Show that the two ships will not collide, clearly stating your reason. marks b. Sketch and label the path of each ship on the axes below. Show the direction of motion of each ship with an arrow. 3 marks y O x 6 SECTION B Question continued

19 9 06 SPECMATH EXAM c. Find the obtuse angle between the paths of the two ships. Give your answer in degrees, correct to one decimal place. marks d. i. Find the value of t, correct to three decimal places, when the ships are closest. marks ii. Find the minimum distance between the two ships, in kilometres, correct to two decimal places. mark SECTION B continued TURN OVER

20 06 SPECMATH EXAM 0 Question 5 (0 marks) A model rocket of mass kg is launched from rest and travels vertically up, with a vertical propulsion force of (50 0t) newtons after t seconds of flight, where t [ 05, ]. Assume that the rocket is subject only to the vertical propulsion force and gravity, and that air resistance is negligible. a. Let v ms be the velocity of the rocket t seconds after it is launched. Write down an equation of motion for the rocket and show that dv 76 = 5. t mark dt 5 b. Find the velocity, in ms, of the rocket after five seconds. marks c. Find the height of the rocket after five seconds. Give your answer in metres, correct to two decimal places. marks SECTION B Question 5 continued

21 06 SPECMATH EXAM d. After five seconds, when the vertical propulsion force has stopped, the rocket is subject only to gravity. Find the maximum height reached by the rocket. Give your answer in metres, correct to two decimal places. marks e. Having reached its maximum height, the rocket falls directly to the ground. Assuming negligible air resistance during this final stage of motion, find the time for which the rocket was in flight. Give your answer in seconds, correct to one decimal place. 3 marks SECTION B continued TURN OVER

22 06 SPECMATH EXAM Question 6 (9 marks) The mean level of pollutant in a river is known to be. mg/l with a standard deviation of 0.6 mg/l. a. Let the random variable X represent the mean level of pollutant in the measurements from a random sample of 5 sites along the river. Write down the mean and standard deviation of X. marks After a chemical spill, the mean level of pollutant from a random sample of 5 sites is found to be. mg/l. To determine whether this sample provides evidence that the mean level of pollutant has increased, a statistical test is carried out. b. Write down suitable hypotheses H 0 and H to test whether the mean level of pollutant has increased. marks c. i. Find the p value for this test, correct to four decimal places. marks ii. State with a reason whether the sample supports the contention that there has been an increase in the mean level of pollutant after the spill. Test at the 5% level of significance. mark SECTION B Question 6 continued

23 3 06 SPECMATH EXAM d. For this test, what is the smallest value of the sample mean that would provide evidence that the mean level of pollutant has increased? That is, find x c such that Pr ( X > x c µ =. )= Give your answer correct to three decimal places. mark e. Suppose that for a level of significance of.5%, we find that x c =. 63. That is, Pr ( X >. 63 µ =. )= If the mean level of pollutant in the river, μ, is in fact. mg/l after the spill, find ( ) Pr X <. 63 µ =.. Give your answer correct to three decimal places. mark END OF QUESTION AND ANSWER BOOK

24

25 Victorian Certificate of Education 06 SPECIALIST MATHEMATICS Written examination FORMULA SHEET Instructions This formula sheet is provided for your reference. A question and answer book is provided with this formula sheet. Students are NOT permitted to bring mobile phones and/or any other unauthorised electronic devices into the examination room. VICTORIAN CURRICULUM AND ASSESSMENT AUTHORITY 06

26 SPECMATH EXAM Specialist Mathematics formulas Mensuration area of a trapezium curved surface area of a cylinder ( a+ b) h π rh volume of a cylinder volume of a cone π r h 3 π r h volume of a pyramid 3 Ah volume of a sphere area of a triangle sine rule 3 π r3 bcsin( A) a b c = = sin( A) sin ( B) sin( C) cosine rule c = a + b ab cos (C ) Circular functions cos (x) + sin (x) = + tan (x) = sec (x) cot (x) + = cosec (x) sin (x + y) = sin (x) cos (y) + cos (x) sin (y) sin (x y) = sin (x) cos (y) cos (x) sin (y) cos (x + y) = cos (x) cos (y) sin (x) sin (y) tan( x) + tan ( y) tan( x+ y) = tan( x)tan ( y) cos (x y) = cos (x) cos (y) + sin (x) sin (y) tan( x) tan ( y) tan( x y) = + tan( x)tan ( y) cos (x) = cos (x) sin (x) = cos (x) = sin (x) tan( x) sin (x) = sin (x) cos (x) tan( x) = tan ( x)

27 3 SPECMATH EXAM Circular functions continued Function sin or arcsin cos or arccos tan or arctan Domain [, ] [, ] R Range π π, [0, ] π π, Algebra (complex numbers) z = x+ iy = r( cos( θ) + isin ( θ) )= r cis( θ ) z = x + y = r π < Arg(z) π z z = r r cis (θ + θ ) z z r = cis θ r θ ( ) z n = r n cis (nθ) (de Moivre s theorem) Probability and statistics for random variables X and Y E(aX + b) = ae(x) + b E(aX + by ) = ae(x ) + be(y ) var(ax + b) = a var(x ) for independent random variables X and Y var(ax + by ) = a var(x ) + b var(y ) approximate confidence interval for μ x z s x z s, + n n distribution of sample mean X mean variance E( X )= µ var ( X )= σ n TURN OVER

28 SPECMATH EXAM Calculus d dx x n ( )= nx n n n+ xdx= x + c, n n + d dx e ax ae ax ax ( )= e dx a e ax = + c d ( log e() x )= dx x x dx = loge x + c d ( sin( ax) )= acos( ax) sin( ax) dx = cos( ax) + c dx a d ( cos( ax) )= asin ( ax) cos( ax) dx = sin ( ax) + c dx a d ( tan( ax) )= asec ( ax) dx d sin ( ( x) )= dx x d cos ( ( x) )= dx x d ( tan ( x) )= dx + x product rule quotient rule chain rule Euler s method acceleration sec ( ax) dx = tan ( ax) + c a x dx = sin ca 0 a x a +, > a x x dx = cos + ca, > 0 a a a x dx x = tan c + a + ( ax b n ) dx an ( ) ( ax b ) n+ + = + + c, n + ( ax + b) dx = loge ax + b + c a d ( dx uv)= u dv dx + v du dx v du u dv d u dx dx dx v = v dy dy du = dx du dx If dy = f( x), x dx 0 = a and y 0 = b, then x n + = x n + h and y n + = y n + h f (x n ) d x dv a v dv d = = = = v dt dt dx dx t arc length + f ( x) dx or x () t y () t dt x x ( ) ( ) + ( ) t Vectors in two and three dimensions Mechanics r= xi+ yj+ zk r = x + y + z = r i dr dx dy dz r = = i+ j+ k dt dt dt dt r. r = rr cos( θ ) = xx + yy + zz momentum END OF FORMULA SHEET equation of motion p= mv R = ma