Gemini Team Test Mu Alpha Theta State 2009

Transcription

1 Gemini Team Test Mu Alpha Theta State If delicious cake, modeled by a cylinder radius cm and height 3 cm, is to be cut so that the volume of each piece can be modeled as a distinct term in a geometric sequence,, and the smallest positive integral value of r for which the cake can be cut into a whole number of slices is used for this sequence, how many pieces of cake will there be? (The first piece of cake has volume ) a). 4 b). 6 c). 10 d). 12 e). NOTA 2. If the cake from the previous problem needs to instead be cut into an infinite number of tiny pieces, what would the common ratio be if the volume of the n th slice of cake equals the n th term of the series? a). b). c). d). e). NOTA 3. What is the sum of all numbers between 1 and 51, non-inclusive, that are relatively prime to 91? a). 7 b) c) d) e). NOTA 4. Kerrigan has 2100 minerals and 500 vespene gas. If a Zergling costs 25 minerals and a Hydralisk costs 75 minerals and 25 vespene gas, how many Zerglings should Kerrigan buy if she wants to make as many Hydralisks as she can and not have any minerals left over? a). 12 Zerglings b). 15 Zerglings c). 24 Zerglings d). 30 Zerglings e). NOTA 5. Hydralisks and Zerglings are very hard to control. If a Hydralisk is twice as hard to control as a Zergling, and one Overlord can control eight Hydralisks, how many Overlords will Kerrigan need to keep her Hydralisks and Zerglings under control? a). 3 Overlords b). 4 Overlords c). 5 Overlords d). 6 Overlords e). NOTA 6. Evaluate: + +4! a). No solution b). 24 c). 30 d). e). NOTA

2 7. Crono and friends want to make some money using their amazing ability to travel through time. If they travel to 600 AD and invest 125 Gold in Guardia war bonds, which are famous for their exceptionally predictable growth rate of 200% over a period of 40 years, how much will the bonds be worth if they cash them in the year 1000 AD? a) Gold b) Gold c) Gold d) Gold e). NOTA 8. If the market in Guardia collapses (rendering previous war bonds worthless) at some point evenly distributed between 1000 AD and 1800 AD, and the Guardian market records only go back exactly 50 years from any given date, what is the absolute least number of trips that Crono and friends could make into the future to be sure of the date of the crash? a). 4 trips b). 5 trips c). 7 trips d). 10 trips e). NOTA 9. A bit is a binary switch, which can store either a 1 or a 0. One byte contains eight bits. If the datatype long unsigned int uses two bytes to store a non-negative integer, what is the largest integer that could be stored in a long unsigned int? a). 128 b). 256 c) d) e). NOTA (65535) 10. Flying Bison can travel through the air at a rate of 250 miles per hour in a linear path. If the herd wants to fly the 2000 miles north from the Southern Air Temple to reach the Northern Air Temple, and the wind is blowing eastward at a rate of 150 miles per hour, how long will it take the Bison to fly, assuming they take the optimal route? a). 20 hours b). 15 hours c). 12 hours d). 10 hours e). NOTA 11. If the Flying Bison grow tired (it takes a lot of work to fly) and want to slow down, should they change directions, assuming the wind is unchanged and they are still trying to take the optimal route to the Northern Air Temple? a.) No b). Yes, they should turn eastward. c). Yes, they should turn westward. d). Not enough information given. e). NOTA 12. What is the least common multiple of 7, 11, and 13? a). 7 b) c) d) e). NOTA 13. Simon and River Tam are hiding from the Alliance troopers. The Alliance troopers have a.2 chance of finding Simon and a.3 chance of finding River. What is the probability they find Simon, but not River, assuming the two events are independent?

3 a)..12 b)..24 c)..3 d)..6 e). NOTA 14. The number of errors on a Mu Alpha Theta Washington State 2009 test is directly proportional to the length of the test and inversely proportional to the amount of times was proofed. If the calculus BC test, length 30 questions, has 3 errors and was proofed four times, how many errors should there be on this 40 question test, given that it was only proofed twice? (Note: DO NOT assume that any question on this test actually contains an error.) a). 4 b). 5 c). 3 d). 8 e). NOTA 15. How many of the following expressions are equivalent to sin(x)? a). 1 b). 2 c). 3 d). 4 e). NOTA 16. How many distinct prime factors of 50! are there? a). 15 b). 50 c). 14 d). 49! e). NOTA 17. What is the sum of all solutions to the equation?

4 a). 7 b) c) d) e). NOTA 18. In JRR Tolkien s The Lord of the Rings, several Great Rings of Power are constructed. Gandalf describes them thusly: Three Rings for the Elven-kings under the sky, Seven for the Dwarf-lords in their halls of stone, Nine for Mortal Men doomed to die, One for the Dark Lord on his dark throne, In the land of Mordor where the shadows lie. One Ring to rule them all, One Ring to find them, One Ring to bring them all, and in the darkness bind them, In the land of Mordor where the shadows lie. How many Great Rings of Power are there? a). 9 b). 13 c). 23 d). 20 e). NOTA 19. Calculus is the math of change in something over time. Barrack Obama campaigned using Change as defined by a difference over time, but not calculus, and in protest this problem will use neither. Change comes mostly in coins when paying with bills for things that do not cost an even amount of dollars. Though paying with plastic has made change obsolete for a number of people, which of the following combination of coins could not be used to make change a purchase that cost $ 1.44? (Not all of the coins need to be used) a). 400 pennies b). 7 nickels, 2 dimes, 1 penny c). 1 quarter, 3 pennies, 3 dimes, 1 fifty cent piece, 1 dollar coin. d). 1 fifty-cent piece, 1 penny, 1 nickel. e). NOTA 20. What is the greatest prime factor of ? a.) 17 b). 19 c). 11 d). 13 e. NOTA 21) In a room there is a large electronic scale and 32 stacks of 32 coins each. The coins are all identical in appearance, but one stack is made of counterfeit coins. You know that real coins have a mass of 10 grams and counterfeit coins have a mass of 11 grams (no you cannot tell the difference). What is the least number of measurements you can do ensure that you determine which stack the counterfeit coins are in? (Placing a coin on the scale and then systematically adding coins counts as multiple measurements.) A) 1 B) 5 C) 31 D) 32 E) NOTA 22) How many coins would you need to measure the mass of to determine which stack is counterfeit (still in the least number of measurements possible)?

5 A) 31 B) 32 C) 496 D) 528 E) NOTA 23) If a chicken and a half can lay and egg and a half in a day and a half how long (in days) would it take a chicken to lay an egg? A) 2/3 B) 1 C) 4/3 D) 3/2 E) NOTA 24) If Mr. Norris recruits freshman for math team at a rate of 200 per hour and upper classmen convince freshmen to quit at a rate of 150 per hour, and the upper classmen show up to school two hours late, how long will it be (in hours), from the time Mr. Norris arrives, before all 750 freshmen at TJ are on math team (freshmen can join and quit math team an infinite number of times in one day)? A) 3.75 B) 4.5 C) 9 D) 10 E) NOTA 26) In the following equation a distinct number can be substituted for each letter (each occurrence of a letter will be replaced by the same number). What is the value of S + O? M SEND +MORE MONEY A) 5 B) 7 C) 9 D) 10 E) NOTA 27) A hungry goat is tied to a 15-foot chain in the middle of a huge grass field, which is attached to the corner of a building that is an equilateral triangular prism. If the building has a volume of 50 3 cubic feet and a height of two feet, what is the total area in which the goat can graze? A) 225π B) 375π /2 C) 425π /2 D) 1225π /6 E) NOTA 28) Is 36 a perfect, abundant or deficient number? A) 12 B) Yes C) 55 D) No E) NOTA 29) How many factors does have? A) 6 B) 8 C) 10 D) 15 E) NOTA 30) What is the sum of the first, second, and fourth largest zeros of the polynomial 2n 4 + 5n 3 5n 2 20n 12 = 0? A) 1 B) 5/2 C) 9/2 D) 3/2 E) NOTA

6 31) Base negative two consists of two digits, 1 and 0, repeated as many times as necessary to express an integer. Expressing an integer in base negative two is similar to expressing the same integer in base two, but the first integer left of the decimal takes on a value of 1, and the next a value of negative two and so on. This means that 10-2 = Express as a base negative two number: A) B) C) D) E) NOTA 32) A group of eight pirates have two boats and want to attack an oil ship and take the ransom money. The probability that they follow through on this is 1/13. The pirate s attack on an oil ship consists of all eight pirates and two boats. It takes two pirates to steer a boat (with less than two it will sink), and one boat to hold the money picked up as a ransom after a successful attack. The probability pirates are arrested before they reach the oil ship is 1/10, the probability of a successful attack given that the pirates reach the oil ship is 1/4, the probability of each individual pirate dying in the attack is 1/2, the probability that each individual pirate boat sinks in the attack is 1/4, and the probability that each individual pirate boat sinks on the way to shore after a successful attack is 1/4. What is the probability that the pirate attack will result in exactly one pirate boat successfully reaching shore with money? (If both pirate boats and at least four pirates survive the attack, the ransom will be transported in equal sums on both boats.) A) 171/10240 B) 2223/10240 C) 171/5120 D) 513/10240 E) NOTA 33) What is 11 7? A) B) C) 77 D) E) NOTA 34) What is (4-3i) 9? A) 32 32i B) 32i 32 C) i D) 0 E) NOTA 35) How many zeros does 100! end in? A) 2 B) C) 20 D) 24 E) NOTA 36) How many zeros does ! end in? A) 653 B) 53 C) 47 D) 49 E) NOTA

7 37) Two days ago I was 15. Next year I will turn 17. If today is the x th day of the a th month, and my birthday is the y th day of the b th month, what is the sum of a + b + x + y? A) 44 B) 46 C) This scenario is impossible D) 56 E) NOT 38) What is the sum of the fourth trigonal number and the third pentagonal number? A) 22 B) 32 C) 7 D) 42 E) NOTA 39) If all of the external surface of a 9x9x9 cube, composed of 729 unit cubes, is painted and then the cube is deconstructed into 1x1x1 unit cubes and two cubes are picked without replacement, what is the probability that only one is painted? A) 931/2106 B) 3968/9477 C) 9457/18954 D) 1421/6318 E) NOTA 40) Evaluate n + 2 n 1 5 A) 13/16 B) diverges C) 3/4 D) 5/8 E) NOTA