2006 Marywood Mathematics Contest

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1 006 Marywood Mathematics Contest Level II Sponsored by SEMI-GROUP The Student Mathematics Club of Marywood University March 8, 006 Directions:. This exam consists of 40 questions on 8 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.. 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 . The maximum value of the quadratic function f(x) = x + 4x is A. 3 B. 3 C. D. E.. A cylindrical can has no top. Its base circumference is 8π and its height is 3. What is the surface area of the can? A. 04π B. π C. 0π D. 36π E. 68π 3. The local lawn and garden store stocks a 7% solution of weed killer and also a % solution. How many gallons of the 7% solution should be mixed with the % solution to produce 40 gallons of % solution? A. gallons B. gallons C. 0 gallons D. 0 gallons E. None of these. 4. The combined area of two squares is 40 square centimeters. The side of one square is three times as long as the side of the other square. Find the dimensions of the larger square. A. cm cm B. 3 cm 3 cm C. cm cm D. 6 cm 6 cm E. None of these.. If x y = 3 and xy = 0, then x + y = A. 7 B. 7 C. 49 D. 9 E Melinda has a collection of nickels, dimes, and quarters totaling coins. The number of dimes she has is 3 more than four times the number of nickels, and the number of quarters she has is 9 less than the number of dimes. How many dimes does she have? A. B. 63 C. 44 D. 3 E. None of these.

3 7. π < A < π, sin A = 0.6, find sin A. A. 0.8 B. 0.8 C D E If a > 0 and a a a = 8, what is a? A. B. 4 C. 8 D. 8 E Two numbers add to 9. What is the largest possible value for their product? A. 9/ B. 8/4 C. 0 D. 8 E. None of these. 0. Determine the number of distinct real roots for the quadratic equation x + x = 0. A. 0 B. C. D. 3 E. None of these.. A satellite traveling in a circular orbit,600 km above the surface of the Earth takes hours to make an orbit. The radius of the Earth is 6,400 km. Find the linear speed of the satellite (to the nearest,000 km/hour). A. 0,000 km/hour B.,000 km/hour C. 0,000 km/hour D.,000 km/hour E. None of these.. Three points O(0, 0), A(a, a) and B(0, 3a + ) form a right triangle with OAB = 90. What is the area of OAB? A. B. C. D. 0 E. None of these.

4 3. Frank s present age is 0 years more than Mike s present age. In years, Mike s age will be three-fifths of Frank s age. How old is Frank now? A. 0 B. 0 C. D. E. None of these. 4. Find the seventh term in the geometric sequence, 6, 8,. A. 48 B. 87 C D. 367 E. None of these.. 00 n= n = = A. 303 B. 000 C. 00 D. 00 E. None of these. ( ) b 6. Given that log b 6 = B, evaluate log b 36 A. 36 B. B C. 6B D. 6 E. None of these. 7. For the angles a, b, c, d and e in the following figure, what is a + b + c + d e? a c b e d A. 90 B. 80 C. 70 D. 3 E How many solution(s) does the equation x = 6 x have? A. 0 B. C. D. 3 E. None of these. 3

5 9. Find the product and sum of the roots of the quadratic equation x + 6x 7 = 0. A. 7 and 3 B. 7 and 3 C. and 6 D. 6 and 7 E. None of these. 0. What is the radius of the circle x + y 4x + 0y = 7? A. 7 B. 6 C. D. 4 E. None of these.. log b 0.3, log b 7.89, and log b Then what is the approximate value of log b 006? (Notice that 7 9 = 006.) A B C D E A pair of fair dice are tossed. What is the probability of getting 6 as the sum of the two numbers on the top side of the dice? A. B. C. D. E. None of these. 3. If we define an operation as (a, b) (c, d) = (ac, bc + d), then (, ) (, 3) = A. (, 8) B. (, ) C. (, ) D. (, 4) E. (, ) 4. If tan x = and 80 < x < 70, what is sin x? A. B. C. D. E.. If in ABC, AC =, BC =, and A = 30, what is B? A. 4.4 B. 30 C. 30 or 0 D. 4 E. 4 or 3 4

6 6. Find the area of the equilateral triangle inscribed inside a circle with radius A. 7 3 B. 4 3 C. 36π D. 7 E. 7π 7. What is the remainder of when it s divided by? A. 0 B. C. D. 3 E Because you worked very hard and got an A in your math class, your parents decided to give you a bonus allowance each week for the next 6 weeks. You are given three options to choose from. Option begins at cent for the first week and doubles each week, while option begins at $ for the first week and increases by $ each week. Option 3, however, is a one-time cash bonus of $00 for the entire period. Rank the three options from the best to the worst (from your own financial point of view, not your parents ). A.,, 3 B., 3, C., 3, D.,, 3 E. 3,, 9. An ellipse is centered at (0, 0) with foci at ( 4, 0) and (4, 0). Its eccentricity e = 0.8. What is the equation for the ellipse? A. x 64 + y 6 = D. x 6 + y 9 = B. x + y 6 = E. None of these. C. x + y 9 =

7 30. How many five-digit numbers can be formed from the digits,, 3, 4,, 6 such that there is no repeating digit in the number and the number is divisible by 3? A. 40 B. 0 C. 70 D. 360 E. None of these. 3. Find 00 n= A. B n(n + ) = C D E A function f(x) satisfies the following property: f(x + ) = 3f(x) f(x ). We are also given that f() = 3, f(4) = 3 and f() = 63. What is f()? A. 7 B. C. 3 D. 3 E Among the 78 th grade students at Math-Is-Fun Academy, 4 students are taking AP calculus, 36 are taking AP chemistry, and 38 are taking AP physics. Which of the following statement is definitely true? A. The facts of this problem are wrong. B. There is exactly one student who is taking neither AP calculus nor AP chemistry. C. There is exactly four students who is taking neither AP chemistry nor AP physics. D. There is exactly one student who is taking both AP calculus and AP physics. E. None of these. 34. If the equation x +bx b = 0 has two distinct real roots, which of the following is true? A. b > 0 B. b 8 C. 8 < b < 0 D. b < 8 or b > 0 E. None of these. 6

8 3. Built-For-Replacement, or BFR, makes desktop computers for home users. A study finds that the probability of a randomly selected BFR computer needing repair or replacement within the first years is %. Dr. Geek at Someville Technology Institute bought 3 BFR computers yesterday. What is the probability that at least one of Dr. Geek s computer will need repair or replacement within years? A. % B. 47.% C..% D. 66% E. 78% 36. At an executive meeting at BFR, our favourite computer manufacturer, there are 8 executives and a secretary. The CEO (one of the 8 execcutives) must sit in the CEO chair at one end of the long table. The secretary always sits next to the CEO on the left or on the right. Other attendees can sit anywhere they wish. There are only 9 chairs total. How many different ways can these 9 people sit around the table? A B C D. 040 E An expression like a = is called a continued fraction. One way to find the value of a is to notice that a = + a, and solve this equation for a gives us that a = ±. However, a obviously should be positive, so a = +. Use the same technique to find the value of the following continued fraction b = A. + B. + 3 C. D. E. None of these. 7

9 38. The point (3, ) is on the hyperbola x 6 y through (3, ) and tangent to the hyperbola? A. B. C. 3 =. What is the slope of the line passing D. 3 E. None of these. 39. What is the smallest possible value of x + 7 if x is a positive real number? x A. B. 3 C. 9 D. 9 E. None of these. 40. Find the area of the shaded region in the following figure, assuming that the two circles have the same radius and the center of one circle is on the other circle A. 4π 3 B. 8π 3 C. 8π 3 D. 8π 3 3 E. None of these. 8

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