2005 Academic Challenge

Transcription

1 005 Academic Challenge MATHEMATICS TEST - STATE FINALS Mathematics Test Production Team Tom Ferry, Illinois State University Author/Team Coordinator Kevin Boyer, Illinois State University Author Sandra Cox, Kaskaskia College Reviewer Don Wayman, WYSE Coordinator of Test Production GENERAL DIRECTIONS Please read the following instructions carefully. This is a timed test; any instructions from the test supervisor should be followed promptly. The test supervisor will give instructions for filling in any necessary information on the answer sheet. Some Academic Challenge test sites may use answer sheets that require you to fill in the letter of the answer you choose on the blank line after each question number. Others will ask you to indicate your answer to each question by marking an oval that corresponds to the correct answer for that question. Only one oval should be marked to answer each question. Multiple ovals will automatically be graded as an incorrect answer. Be sure ovals are marked as. Not as,,, etc. Test Question - 1. Which of the following was formed by the New Madrid Earthquakes of ? a. Lake of the Ozarks (MO) b. Little Grassy Lake (IL) c. Kentucky Lake (KY) d. Reelfoot Lake (TN) e. Crater Lake (OR) Answer Sheet - 1. A B C E If you wish to change an answer, erase your first mark completely before marking your new choice. You are advised to use your time effectively and to work as rapidly as you can without losing accuracy. Do not waste your time on questions that are too difficult for you. Go on to the other questions and come back to the difficult ones later if you can. *** TIME: 40 MINUTES *** DO NOT OPEN TEST BOOKLET UNTIL YOU ARE TOLD TO DO SO! 005 Worldwide Youth in Science and Engineering WYSE, Worldwide Youth in Science and Engineering and the WYSE Design are service marks of and this work is Copyright 005 Board of Trustees of the University of Illinois at Urbana - Champaign. All rights reserved.

2 WYSE Academic Challenge Mathematics Test (State Finals) What is the product of the slope and y intercept of the slant asymptote listed below: a) -5 b) -3 c) -1 d) 1 e) 3 3x 3 4x x 5x 6 1. Eight pet owners each brought their sick pets in to the vet. Two of the owners brought their pets in at 10 AM, two brought their pets in at noon, two brought their pets in at PM, and two brought their pets in at 4PM. I. Ben and his tabby cat were at the vet at 4PM along with his sister and her dog. II. Amy brought her terrier at the same time as a boy and his bulldog. III. George and his cocker spaniel were there hours after his brother and his hound dog. IV. Emily brought her calico cat in before Helen brought her collie in. V. Candace was at the vet before Dan and his dog, but after Frank and his dog. Who was at the vet at the same time as Dan and his dog? a) George and his cocker spaniel b) Emily and her calico cat c) Helen and her collie d) Amy and her terrier e) Situation can t exist as written. 3. Instead of buying one, Smiley made his own tent. Not very good with numbers or a sewing machine, the tent turned out a bit lopsided. After being set-up, the pup style tent with exposed ends had a floor width of 5 feet and a depth of 8 feet. One side of the tent is 6 feet while the other side is 7 feet. What is the volume of Smiley s tent? a) 4 feet 3 b) 4 6 feet 3 c) 48 6 feet 3 d) feet 3 e) 5 14 feet State Finals

3 Mathematics 4. Three ellipses are all centered on the origin. All three ellipses have an eccentricity of 0.8. The area between the middle ellipse and the smallest ellipse is equal to the area between the middle ellipse and the largest ellipse. If the foci of the smallest ellipse are 0 meters apart, and the foci of the largest ellipse are 140 meters apart, how far apart are the foci of the middle ellipse? (Round your answer to the nearest meter.) a) 57 meters b) 69 meters c) 71 meters d) 80 meters e) 100 meters 4 5. Rewrite cos x as a sum of first powers of the cosine of several angles. a) (1/4)(1cosxcos4x) b) (1/4)(1cosx4cos8x) c) (1/4)(cosxcos4x) d) (1/8)(1cosx4cos8x) e) (1/8)(34cosxcos4x) 6. You are given the following three matrices: (x, y, and z are all distinct, non-zero integers.) x z y z y [ A ] = y z x [ B ] = x y z [ C ] x z y x z y x x = y z z x y y z x You are asked to take two of these matrices, add them together, and then take the inverse of the sum. Are there any pairs of these matrices where this cannot be done? a) None of the possible pairs ( [ A ] [ B], [ A ] [ C], and [ B ] [ C] b) [ A ] [ B] has an inverse, but [ A ] [ C] and [ B ] [ C] do not. c) [ B ] [ C] has an inverse, but [ A ] [ B] and [ A ] [ C] do not. d) [ A ] [ C] and [ B ] [ C] have inverses, but [ A ] [ B] does not. e) All of the possible pairs ( [ A ] [ B], [ A ] [ C], and [ B ] [ C] 7. Determine the derivative of the product function listed below: 3 f ( x) = (3x ) a) f (x) = 7x 4-36x 1 b) f (x) = 7x 5-36x 3 1x c) f (x) = 16x 4-16x 7 d) f (x) = 16x 5-16x 3 7x e) f (x) = 16x 5 ) have inverses. ) have inverses. 005 State Finals

4 Mathematics 3 8. What is the distance between the polar points ( 8,60 ) and ( 1,10 )? (Please round your answer to the nearest tenth of a unit.) a) 4.1 units b) 10.0 units c) 10.6 units d) 14.4 units e) 17.4 units 9. A four sided die is labeled 1,, 3 and 4. The probability that the die lands on the is twice as likely as it landing on the 1. The probability that the die lands on the 3 is twice as likely as it landing on the. The probability that the die lands on the 4 is twice as likely as it landing on the 3. What is the probability of the die landing on an odd number? a) 1/6 b) 1/5 c) 1/4 d) 1/3 e) 1/ 10. Scientists have discovered examples of math problems done by an unknown civilization. The scientists have determined that the problems all appear to be addition problems. The following problems have solutions given: = = = = The scientists later discover the following unfinished problem: What would be the solution to this problem? a) b) c) = d) e) 005 State Finals

5 Mathematics Determine the inverse of the function listed below: 4x 1 f ( x) = x 3 a) f 1 (x) = b) f 1 (x) = c) f 1 (x) = d) f 1 (x) = e) f 1 (x) = 1 4x 3 x 3x 1 x 4 4x 1 3 x x 3 4x 1 4x 1 3 x 1. Circles A, B, and C are tangent as shown in the diagram below. Circle A and Circle B both have diameters of 30 meters, and Circle C has a diameter of 0 meters. What is the area of the region bound by the three circles? (Round your answer to the nearest square meter.) a) 7 square meters b) 35 square meters c) 43 square meters d) 70 square meters e) 7 square meters 13. A lookout in a lighthouse 75 feet above sea level observes two ships approaching the lighthouse in a straight line (one right behind the other). The angles of depression to the ships are 5 degrees and 8 degrees. How far apart are the ships? (Round your answer to nearest foot.) a) 1,187 feet b) 1,179 feet c) 1,156 feet d) 1,093 feet e) 1,057 feet 005 State Finals

6 Mathematics The third degree function f has two complex roots and an integer root. If we add all three roots together, we get 8. If we multiply all three roots together, we get 0. If we multiply just the two complex roots, we get 10. What is the correct equation for f? a) b) c) d) e) 3 x 8x 10x 0 3 x 8x 10x 0 3 x 8x x 0 3 x 8x x 0 3 x 8x 8x A chemist has a one liter jar filled with a mixture that is 40% acid and 60% water. He carefully adds a solution that is 0% base and 80% water. Each unit of base neutralizes one unit of acid, since they combine to form water. How much base solution should the chemist add in order to end up with a solution that is 5% acid and 95% water? (Round your answer to the nearest tenth of a liter.) a) 1.0 liters b) 1.3 liters c) 1.4 liters d) 1.7 liters e) 1.8 liters 16. What is the length of the cross product of the vectors 4 i 3 j k and 3 i j k? (Round your answer to the nearest tenth of a unit.) a) 0 b) 1.0 c) 1. d) 1.7 e) Simplify 3 3 a) 8/81 b) 43/81 c) 81/97 d) 10/81 e) x 1 x 18. Solve for x: 3 = 9 3. (Round to the nearest tenth of a unit.) a) -0.3 b) 0.1 c) 0.3 d) 1 e) all real numbers 005 State Finals

7 19. The decay model for carbon-14 is A Mathematics t = A0e. A paint contained 3% of the original carbon-14. About how old is the paint? (Round your answer to the nearest year.) a) 957 years b),675 years c) 1,146 years d) 104,995 years e) 1,06,00 years 0. The number of wolves in a region with a fixed food supply can be modeled by the equation A 1 49e P( t) 0. 0t =, where t represents the number of years after a few wolves arrive in the region, and A is the maximum number of wolves that the food supply can support. How many years does it take for the number of wolves to reach half of the maximum possible? (Round your answer to the nearest year.) a) 1 year b) 50 years c) 66 years d) 195 years e) 30 years 1. Determine the derivative of the expression below: (x 1)dx a) 0 b) 11/3 c) x 1 d) 6 e) 8 1. A radio telescope is composed of a dish with a parabolic cross section and a receiver on a stem coming out of the center of the dish. The sensor part of the receiver is put at the focus of the parabola. If a radio telescope is to have a dish 4 feet across and 3 feet deep, how far away from the surface of the center of the dish should the sensor part of the receiver be placed? a) 1/16 inch b) 1/4 inch c) 1 inch d) 3 inches e) 3 feet 3. P( Q) = 1/ 6, P( T ) = 5 /1, and P( Q T ) P( T Q) = 7 /10 then P( Q T ) =? a) 1/30 b) 1/4 c) 1/18 d) 1/1 e) 1/ State Finals

8 Mathematics 7 4. The position of a particle orbiting around the origin can be described as ( 5 sin 8t 4,3cos 8t ), where t is a time in seconds. How long does it take the particle to make a single orbit? (Round your answer to the nearest tenth of a second.) a) 0.4 seconds b) 0.8 seconds c) 1.3 seconds d).4 seconds e).5 seconds 5. A ball is tossed into the air. The function s ( t) = 16t 140t 30 represents the ball s height above ground. What is the velocity of the ball when t = 1.5? a) 0 b) 40 c) 60 d) 80 e) The fifth number in an arithmetic sequence is 45. The eighth number in the same sequence is 69. The eleventh number of this sequence is 93. What is the sum of the first ten numbers in this sequence? a) 85 b) 90 c) 410 d) 490 e) Consider an oblique triangle. Leg AB is 4 inches long, leg BC is 7 inches long and the length of leg CA is unknown. The angle formed between legs AB and BC is 34 degrees. What is the area of this triangle? (Round to the nearest hundredth.) a) 4.8 square inches b) 6.45 square inches c) 7.83 square inches d) 8.34 square inches e) 9.04 square inches 005 State Finals

9 Mathematics 8 8. You are given the following four matrices: 1 [ A ] = [ B ] = [ C ] = [ D ] = What would be the matrix X that would solve the equation AX B = CX D? 3 1 = 1 7 / 3 = 5/ /10 1/ = 3/ 5 1 / 5 4 /5 = 9/10 17/10 / 3 = 5/ 4 1 a) [ ] X X b) [ ] X c) [ ] X d) [ ] X e) [ ] 1 9. Approximate t in the expression below: a) 8 b) 9 c) 10 d) 11 e) 1 3( 1 t 4 ) 5 = A cyclist usually takes about 0 minutes to go from point A to point B on a particular stretch of trail. One day when the cyclist is /3 of the way from A to B, she passes a pedestrian. Upon arriving at point B, the cyclist turns around and heads back to point A. When she is 1/5 of the way from point B back to point A, she meets the same pedestrian. If the pedestrian has been going at a constant speed, how long would it take him to walk the distance from point A to point B? (Round your answer to the nearest minute.) a) 33 minutes b) 49 minutes c) 60 minutes d) 80 minutes e) 100 minutes 005 State Finals

10 Mathematics 9 d y 31. Determine if dx y = (3 x) 1 a) (3 x) b) (3 x) c) 6(3 x) 3 d) 8(3 x) 3 e) 1(3 x) 3 3. An automated quality control test tends to correctly identify acceptable part as acceptable 95% of the time, and it tends to correctly identify defective pieces as defective 90% of the time. Inspections done by hand have shown that the parts created by a casting machine tend to be defective 5% of the time. If the automated test says that one of the parts created by the casting machine is defective, what is the probability that the piece is actually acceptable? (Round your answer to the nearest percent.) a) 5% b) 8% c) 10% d) 1% e) 51% 33. Solve the radical equation below: a) 3 b) 5 c) 7 d) 9 e) x 4 3 = A cone with an equilateral triangle cross section and a circular base has a volume of 30π cubic meters. What is the height of this cone? (Round your answer to the nearest tenth of a meter.) a).6 meters b) 3.1 meters c) 3.8 meters d) 4.5 meters e) 6.5 meters 35. A class grade is based on 4 exams. The first exam is worth 10% of the class grade, the second exam is worth 0%, the third exam is worth 30% and the forth exam is worth 40%. What is the class grade if the four exam grades are 61, 7, 83, and 7 respectively? a) 70 b) 7 c) 73 d) 74 e) State Finals

11 Mathematics When a clothing store buys a shirt, they determine the retail price by adding a $10 markup to the wholesale price. This gives the store a good profit on each shirt. The store later puts the shirt on sale by taking 0% off the retail price. The store later puts the shirt on clearance by taking an additional 5% off the sale price. The store now receives approximately 8% profit on each shirt sold at the clearance price. What was the original wholesale price? a) $10.38 b) $1.50 c) $14.80 d) $16.67 e) $ Two thirds of a number is equal to the number divided by six, plus one. What is the reciprocal of the number? a) 1/5 b) 1/4 c) 1/3 d) 1/ e) How many different five letter words can be created from the word moonbeam? (One example of a five letter word is oboem.) a) 56 b) 1680 c) 040 d) 3360 e) Box A contains 1 green and red chips, box B contains green and 1 red chip and box C contains an unknown number of green chips and 4 red chips. A box is selected at random and then a chip is selected at random. If the probability of selecting a green chip is 6/15, how many green chips are there in box C? a) 0 b) 1 c) d) 3 e) A regular octagon has an area of 100 square meters. What is its perimeter? (Round your answer to the nearest whole meter.) a) 31 meters b) 3 meters c) 36 meters d) 38 meters e) 40 meters 005 State Finals