2007 Marywood Mathematics Contest

Transcription

1 007 Marywood Mathematics Contest Level II Sponsored by SEMI-GROUP The Student Mathematics Club of Marywood University February 4, 007 Directions:. This exam consists of 40 questions on 7 pages. Please check to make sure that you have all the pages.. Allot your time accordingly. This is a 60-minute test. Do not spend too much time on any one problem. If a question seems to be too difficult, make your best possible guess. 3. Do not worry if you do not finish the test. Your score will be the number of correct responses. 4. On the scantron form provided for you, darken in the space corresponding to the correct answer. Please mark all answers carefully and erase completely when changing an answer. Mark only one answer for each question. Only those answers on the answer sheet will be counted. 5. There is a sheet of blank paper on the last page which you can (carefully) tear off and use as scratch paper. 6. NOTE: In order to ensure uniformity, proctors are NOT allowed to answer any questions pertaining to specific problem content. Please do NOT open the test until you are told to do so.

2 . A linear function f(x) satisfies f( 3) = and f() = 3. What is the y-intercept of its graph? A. 5 B. 5 C. 7 D. E. None of these.. Find the values of x, y, z if the degree measures of the angles are as shown. A. x = 0, y = 6, z = 9 0x 0y + 7 B. x =, y = 0, z = 7 3x + 5 z x 7 C. x = 7, y = 8, z = 0 D. x = 0, y =, z = 5 E. x = 6, y = 9, z = 0 3. How many four-letter sequences (without repetition) can you make from the following set? {a, b, c, d, e, f, g} A. 8 B. 840 C. 7 4 D. 49 E Given f(x) = x + 3 and g(x) = 5 x, (f g)(x) = A. 5x + 5 B. 5x + 3 C. 5 x + 3 D. 5x + 3 E. None of these. 5. A woman has $.5 in change in her purse, comprised entirely of dimes and quarters. Given that there are more quarters than dimes in her purse, what is the total number of coins? A. 9 B. 0 C. D. E The circle with center C has radius. The sector ABC has area A A. π B. π 6 6 E. None of these. C. π 3 D. π 9 D 30 C B

3 7. Find the sum of the digits of the number A B C D E. None of these 8. If x + y = 40 and xy =, then x + y = A. 7. B. 8. C. 7 or 7. D. 8 or 8. E. None of these. 9. Any point (x, y) on a particular circle in the Cartesian plane has the property that the sum of the squares of its distances from (4, ) and (, 5) is 45. What is the radius of the circle? A. B. 5 3 C. D. 35 E. None of these. 0. The remainder when x 3 3x + 6 is divided by x + is A. 8. B. 0. C. 8. D. 4. E. 6.. If xyz = 4 and y z = 5, what is the value of x y? A. 0 B. 0 C..5 D. E If the operation is defined by the equation x y = x + y, what is the value of a in the equation a = a 3? A. 0 B. C. D..5 E A fair coin is tossed 3 times. Find the probability of obtaining at least one head. A. 7 8 B. 4 C. D. 6 E The square root of 3 is A B.. 3 C. ( ) 3. D E. None of these.

4 5. When tuning a piano, a technician strikes a tuning fork for the A above middle C and sets up a wave motion that can be approximated by y = 0.00 sin(880πt) where t is the time in seconds. What is the period of this function? A. 000 B. 880 C. 880 D. 440 E. π If two coins are removed at random from a purse containing three nickels and eight dimes, what is the probability that both coins will be dimes? A B C D. 64 E Compute the value of A. 0 7 B. 3 ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) C. 7 0 D. 8 E In the figure shown, OAB and ODC are sectors of two concentric circles centered at O. The length of OD is one third the length of OA. What is the ratio of the area of the region ABCD to the area of the sector ODC? B A. 7. B. 8. C. 0. D.. C E.. O D A 9. Solve for x: log(x 3) = log x + log(x ). A. x = 5 B. x = 4 C. x = 3 D. x = E. x = 0. One angle of an isosceles triangle is twice another angle of the triangle. Which of the following could NOT be an angle of this triangle? A. 8 B. 36 C. 45 D. 7 E. 90 3

5 . Find the intersection points of the circle x + y + 6x 6y + 5 = 0 with the x-axis. A. ( 5, 0) and (, 0) B. (0, 5) and (0, ) C. (5, 0) and (, 0) D. (0, 5) and (0, ) E. None of these.. If a divided by 4 leaves a remainder of and b divided by 4 leaves a remainder of 3, then when a + b is divided by 4, the remainder is A. 0 B. C. D. 3 E. 4 In the next two problems, select the equation whose graph most closely resembles the one which is given A. y = 0x B. y = x C. y = 0x D. y = 0 x E. y = x A. y = x 6 C. y = 6 x E. y = 6 x 3 B. y = 6 x D. y = 6 x 5. What are all the x-intercepts of the graph of y = cosx sin x for 0 x π? A. 0, π, π B. π, 3π C. 3π 4, 7π 4 D. π 6, π, 5π 6, 3π E. None of these. 4

6 6. What is the value of log 3 log 4 5 log 6 7 log 4 3 log 6 5 log 8 7? A. 4. B.. C. D. 3 E A group containing boys and girls took a test. If exactly /3 of the boys and exactly 3/4 of the girls passed the test, and if an equal number of boys and girls passed the test, then what fraction of the entire group passed the test? A. 6. B. 7. C D.. E The circles in the figure shown are concentric. The chord shown is tangent to the inner circle and has length. What is the area of the shaded region? A. 4π. B. 3π. C. 36π. D. 40π. E. The area cannot be determined with the information given. 9. On this contest you are taking now, there are 40 multiple choice problems and each problem has five associated answers with exactly one of them being correct. If a student chooses his/her answer for each problem randomly and independently (i.e., whatever he/she chooses on one problem does not affect his/her choice on any other problem), which of the following is the most likely score this student will receive? (Each problem is worth point and there is no penalty for wrong answers.) A. 3 B. 8 C. 3 D. 8 E The GCD (greatest common divisor) of and 4873 is A. 4. B. 6. C.. D. 4. E. None of these. 5

7 3. Given ABC with B = 90, AB = and BC = 5, find sin A. A. 9 B. 5 9 C. 9 D. 5 9 E. None of these. 3. One diagonal of a rhombus is of length 0 cm and each side of the rhombus is of length 3 cm. What is the area of the rhombus? A. 30 cm. B. 60 cm. C. 65 cm. D. 0 cm. E. 30 cm. ( a 33. Let f(x) = f = a + b for every positive rational number x = b) a, where a, b are ( ) b 3 relatively prime positive integers. For example f(3) = f = 3 + = 4. Find the product of all the positive rational numbers that satisfy f(x) = 007. A.. B. 006!. C. 006!. D E. None of these. 34. Randomly pick a point from the inside of a circle of radius 3 and assume that all the points inside this circle have equal chance of being picked. There is a smaller circle of radius which lies completely inside the previous circle. What is the probability that the point picked is outside the smaller circle? A. 9 B. 3 C. 3 D. 8 9 E. It depends on where the smaller circle s center is. 35. If P(x) = x kx 007, find k such that x + is a factor of P(x). A. 008 B. 007 C. 006 D. 007 E

8 36. Suppose f is a function such that 3f(x) + f( x) = x + 9 for every real number x. What is the value of f()? A.. B.. C. 3. D. 4. E In ABC, the lengths of the sides are x a, x and x respectively where x > and a > 0. What is the smallest value of x so that such a triangle does NOT exist for any a > 0? A.. B.. C. 3. D. 5. E How many three digit integers (numbers between 00 and 999 inclusive) have exactly two of the digits the same (for example 3 counts but doesn t)? A. 7. B. 6. C. 43. D. 70. E. None of these. 39. How many of those three digit integers with exactly two of the digits the same in the previous problem are divisible by 5? A. 43. B. 54. C. 45. D. 48. E. None of these. 40. One adult and two children are trying to cross a river in a small boat. The boat can carry only one adult or two children. What is the smallest number of times the boat must revisit the initial side they are on to get all of them to the opposite side? Assume at least one person must be in the boat each time it crosses the river and both children are old enough to be in the boat alone. A.. B. 3. C. 4. D. 5. E. It s impossible for them to cross the river. 7