SPECIALIST MATHEMATICS

Transcription

1 Victorian Certificate of Education 07 SUPERVISOR TO ATTACH PROCESSING LABEL HERE Letter STUDENT NUMBER SPECIALIST MATHEMATICS Section Written examination Monday 3 November 07 Reading time: 3.00 pm to 3.5 pm (5 minutes) Writing time: 3.5 pm to 5.5 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 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 07

2 07 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 implied domain of f( x) = cos A. R B. [, ] C. (, ] [, ) {} {} D. R \ 0 E. [, ]\ 0 is x Question cosec for x ( 0, π ) \ { } The solutions to cos( x) > ( x) 4 A. x π 5π 5π 3π 7π,,, B. x π 5π 3π 7π,, C. x π 5π 3π π 3π,,, D. x π 3π 7π,, π E. x π 5π 3π π 7π,,, π π π π are given by Question 3 The number of distinct roots of the equation (z 4 )(z + 3iz ) = 0, where z C, is A. B. 3 C. 4 D. 5 E. 6 SECTION A continued

3 3 07 SPECMATH EXAM Question 4 The solutions to z n = + i, n Z + are given by A. B. C. D. E. n n π πk cis +, 4n n k R π cis + π k, k Z 4n n n k cis π π +, 4 n k R π πk cis +, k Z 4n n n π πk cis +, k Z 4n n Question 5 On an Argand diagram, a point that lies on the path defined by z + i = z 4 is A. 3, B. 3, C. 3, 3 D. 3, 3 E. 3, Question 6 Given that dy x = e arctan ( y), the value of d y dx dx A. 3 B. 8 at the point (0, ) is C. D. 4 E. π 8 SECTION A continued TURN OVER

4 07 SPECMATH EXAM 4 Question 7 With a suitable substitution x x dx can be expressed as A u 4u + u du B u 4u + u du C u + 4u u du 0 D u 4u + u du E u 4u u du Question 8 Let f (x) = x 3 mx + 4, where m, x R. The gradient of f will always be strictly increasing for A. x 0 B. m x 3 C. m x 3 D. m x 3 E. m x 3 Question 9 Consider dy = x + x+, where y() = y dx 0 =. Using Euler s method with a step size of 0., an approximation to y(0.8) = y is given by A B..48 C..6 D..4 E..85 SECTION A continued

5 5 07 SPECMATH EXAM Question 0 A function f, its derivative f ꞌ and its second derivative f ꞌꞌ are defined for x R with the following properties. f (a) =, f ( a) = f (b) =, f ( b) = ( x+ a) ( x b) and f ( x) =, gx ( ) where gx () < 0 The coordinates of any points of inflection of f( x) are A. ( a, ) and (b, ) B. (b, ) C. ( a, ) and (b, ) D. ( a, ) E. (b, ) Question The vectors a = i + 3j+ d k, b= i + j 4k and c= i + j k, where d is a real constant, are linearly dependent if A. d = 0 B. d R\ { 4} C. d = 4 D. d R\ { 0} E. d R Question ( ) + Let r() t = asin( t) i cos( t) j for t 0 and a, b R + be the path of a particle moving in the b cartesian plane. The path of the particle will always be a circle if A. ab = B. a b = C. ab D. ab = E. a b Question 3 Given the vectors a = 3i 4j+ k and b= i + j k, the vector resolute of a in the direction of b is A. 4 3 B i j k ) C i j k ) D. 4 3 E i 4 j k ) SECTION A continued TURN OVER

6 07 SPECMATH EXAM 6 Question 4 Two particles with mass m kilograms and m kilograms are connected by a taut light string that passes over a smooth pulley. The particles sit on smooth inclined planes, as shown in the diagram below. m m θ θ If the system is in equilibrium, then m m sec( θ ) A. B. sec(θ) C. cos(θ) D. E. is equal to Question 5 A body has displacement of 3i + j metres at a particular time. The body moves with constant velocity and two seconds later its displacement is i + 5 j metres. The velocity, in ms, of the body is A. i + 6j B. i + j C. 4i + 4j D. 4i 4j E. i + 3j Question 6 An object of mass 0 kg, initially at rest, is pulled along a rough horizontal surface by a force of 80 N acting at an angle of 40 upwards from the horizontal. A friction force of 0 N opposes the motion. After the pulling force has acted for 5 s, the magnitude of the momentum, in kg ms, of the object is closest to A. 0 B. 40 C. 60 D. 0 E. 400 SECTION A continued

7 7 07 SPECMATH EXAM Question 7 Forces of 0 N and 8 N act on a body as shown below. 0 N 60 8 N The resultant force acting on the body will, correct to one decimal place, have A. magnitude 5.6 N and act at 6.3 to the 0 N force. B. magnitude 9. N and act at 49. to the 0 N force. C. magnitude 5.6 N and act at 33.7 to the 0 N force. D. magnitude 9. N and act at 70.9 to the 0 N force. E. magnitude 5.6 N and act at 49. to the 0 N force. Question 8 U and V are independent normally distributed random variables, where U has a mean of 5 and a variance of, and V has a mean of 8 and a variance of. The random variable W is defined by W = 4U 3V. In terms of the standard normal variable Z, Pr(W > 5) is equivalent to 9 7 A. Pr Z > 7 B. Pr(Z <.8) C. 9 7 Pr Z < 7 D. Pr(Z > 0.) E. Pr(Z >.8) SECTION A continued TURN OVER

8 07 SPECMATH EXAM 8 Question 9 A confidence interval is to be used to estimate the population mean μ based on a sample mean x. To decrease the width of a confidence interval by 75%, the sample size must be multiplied by a factor of A. B. 4 C. 9 D. 6 E. 5 Question 0 In a one-sided statistical test at the 5% level of significance, it would be concluded that A. H 0 should not be rejected if p = 0.04 B. H 0 should be rejected if p = 0.06 C. H 0 should be rejected if p = 0.03 D. H 0 should not be rejected if p 0.05 E. H 0 should not be rejected if p = 0.0 END OF SECTION A

9 9 07 SPECMATH EXAM CONTINUES OVER PAGE TURN OVER

10 07 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 ( marks) x Let f : D R, f( x) =, where D is the maximal domain of f. + 3 x a. i. Find the equations of any asymptotes of the graph of f. mark ii. Find f (x) and state the coordinates of any stationary points of the graph of f, correct to two decimal places. marks iii. Find the coordinates of any points of inflection of the graph of f, correct to two decimal places. marks SECTION B Question continued

11 07 SPECMATH EXAM x b. Sketch the graph of f( x)= from x = 3 to x = 3 on the axes provided below, marking + 3 x all stationary points, points of inflection and intercepts with axes, labelling them with their coordinates. Show any asymptotes and label them with their equations. 3 marks y x c. The region S, bounded by the graph of f, the x-axis and the line x = 3, is rotated about the x-axis to form a solid of revolution. The line x = a, where 0 < a < 3, divides the region S into two regions such that, when the two regions are rotated about the x-axis, they generate solids of equal volume. i. Write down an equation involving definite integrals that can be used to determine a. marks ii. Hence, find the value of a, correct to two decimal places. mark SECTION B continued TURN OVER

12 07 SPECMATH EXAM Question (0 marks) A helicopter is hovering at a constant height above a fixed location. A skydiver falls from rest for two seconds from the helicopter. The skydiver is subject only to gravitational acceleration and air resistance is negligible for the first two seconds. Let downward displacement be positive. a. Find the distance, in metres, fallen in the first two seconds. marks b. Show that the speed of the skydiver after two seconds is 9.6 ms. mark After two seconds, air resistance is significant and the acceleration of the skydiver is given by a = g 0.0v. c. Find the limiting (terminal) velocity, in ms, that the skydiver would reach. mark d. i. Write down an expression involving a definite integral that gives the time taken for the skydiver to reach a speed of 30 ms. marks ii. Hence, find the time, in seconds, taken to reach a speed of 30 ms, correct to the nearest tenth of a second. mark SECTION B Question continued

13 3 07 SPECMATH EXAM e. Write down an expression involving a definite integral that gives the distance through which the skydiver falls to reach a speed of 30 ms. Find this distance, giving your answer in metres, correct to the nearest metre. 3 marks SECTION B continued TURN OVER

14 07 SPECMATH EXAM 4 Question 3 (0 marks) A brooch is designed using inverse circular functions to make the shape shown in the diagram below. y 0 x The edges of the brooch in the first quadrant are described by the piecewise function x 3arcsin, 0 x f( x) = x 3arccos, < x a. Write down the coordinates of the corner point of the brooch in the first quadrant. mark b. Specify the piecewise function that describes the edges in the third quadrant. mark SECTION B Question 3 continued

15 5 07 SPECMATH EXAM c. Given that each unit in the diagram represents one centimetre, find the area of the brooch. Give your answer in square centimetres, correct to one decimal place. 3 marks d. Find the acute angle between the edges of the brooch at the origin. Give your answer in degrees, correct to one decimal place. 3 marks e. The perimeter of the brooch has a border of gold. Show that the length of the gold border needed is given by a definite integral of the form b a x dx, where a, b R. Find the values of a and b. marks SECTION B continued TURN OVER

16 07 SPECMATH EXAM 6 Question 4 (0 marks) a. Express 3i in polar form. mark b. Show that the roots of z + 4z + 6 = 0 are z = 3 i and z = + 3. i mark c. Express the roots of z + 4z + 6 = 0 in terms of 3i. mark d. Show that the cartesian form of the relation z = z ( 3 i) is x 3y 4= 0. marks SECTION B Question 4 continued

17 7 07 SPECMATH EXAM e. Sketch the line represented by x 3y 4= 0 and plot the roots of z + 4z + 6 = 0 on the Argand diagram below. marks Im(z) O Re(z) f. The equation of the line passing through the two roots of z + 4z + 6 = 0 can be expressed as z a z b =, where a, b C. Find b in terms of a. mark g. Find the area of the major segment bounded by the line passing through the roots of z + 4z + 6 = 0 and the major arc of the circle given by z = 4. marks SECTION B continued TURN OVER

18 07 SPECMATH EXAM 8 Question 5 (0 marks) On a particular morning, the position vectors of a boat and a jet ski on a lake t minutes after ( ) + ( + ) they have started moving are given by r B () t = cos( t ) i 3 sin () t j and r J () i j t = ( sin( t ) ) cos( ) + ( t ) respectively for t 0, where distances are measured in kilometres. The boat and the jet ski start moving at the same time. The graphs of their paths are shown below. 5 y x a. On the diagram above, mark the initial positions of the boat and the jet ski, clearly identifying each of them. Use arrows to show the directions in which they move. marks b. i. Find the first time for t > 0 when the speeds of the boat and the jet ski are the same. marks ii. State the coordinates of the boat at this time. mark SECTION B Question 5 continued

19 9 07 SPECMATH EXAM c. i. Write down an expression for the distance between the jet ski and the boat at any time t. mark ii. Find the minimum distance separating the boat and the jet ski. Give your answer in kilometres, correct to two decimal places. mark d. On another morning, the boat s position vector remained the same but the jet skier considered starting from a different location with a new position vector given by r() t = ( sin( t) ) i + ( a cos( t) ) j, t 0, where a is a real constant. Both vessels are to start at the same time. Assuming the vessels would collide shortly after starting, find the time of the collision and the value of a. 3 marks SECTION B continued TURN OVER

20 07 SPECMATH EXAM 0 Question 6 (9 marks) A dairy factory produces milk in bottles with a nominal volume of L per bottle. To ensure most bottles contain at least the nominal volume, the machine that fills the bottles dispenses volumes that are normally distributed with a mean of 005 ml and a standard deviation of 6 ml. a. Find the percentage of bottles that contain at least the nominal volume of milk, correct to one decimal place. mark Bottles of milk are packed in crates of 0 bottles, where the nominal total volume per crate is 0 L. b. Show that the total volume of milk contained in each crate varies with a mean of ml and a standard deviation of 6 0 ml. marks c. Find the percentage, correct to one decimal place, of crates that contain at least the nominal volume of 0 L. mark SECTION B Question 6 continued

21 07 SPECMATH EXAM d. Regulations require at least 99.9% of crates to contain at least the nominal volume of 0 L. Assuming the mean volume dispensed by the machine remains 005 ml, find the maximum allowable standard deviation of the bottle-filling machine needed to achieve this outcome. Give your answer in millilitres, correct to one decimal place. 3 marks e. A nearby dairy factory claims the milk dispensed into its L bottles varies normally with a mean of 005 ml and a standard deviation of ml. When authorities visit the nearby dairy factory and check a random sample of 0 bottles of milk, they find the mean volume to be 004 ml. Assuming that the standard deviation of ml is correct, carry out a one-sided statistical test and determine, stating a reason, whether the nearby dairy s claim should be accepted at the 5% level of significance. marks END OF QUESTION AND ANSWER BOOK

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23 Victorian Certificate of Education 07 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 07

24 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 4 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)

25 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

26 SPECMATH EXAM 4 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