MARQUETTE UNIVERSITY

1 9 9 6

COMPETITIVE SCHOLARSHIP EXAMINATION
IN
MATHEMATICS

Do not open this booklet until you are directed to do so.
  1. Fill out completely the following information about yourself.

    PRINT  __________________________________________________________________________________________
    Last name First name Initial Phone No.

    ADDRESS  ________________________________________________________________________________________
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    Your hight school: Name ____________________________________ City ____________________________

    High School Counselor or Advisor: ______________________________________________________________

  2. This examination consists of two parts. The time allowed for each will be approximately 60 minutes. Should you finish Part I early, you may proceed to Part II.

  3. Part I consists of 16 objective-type questions. Each question has five possible answers marked: A., B., C., D., E. Only one answer is correct. You are to circle the letter corresponding to the correct response for as many problems as you can.

    Example: If x=5 and y=-2, then x+4y is

            (A) -3    (B) -2    (C) -1    (D) 0    (E) +1.

  4. Part II consists of 4 subjective-type questions. Show a summary of your work in this booklet for each question you attempt, whether or not you obtain a complete solution. Scratch paper is provided but be sure to show the essential steps of your work concisely in the space provided for each question. Only the work appearing in this booklet will be scored. You will be scored on your method of attack, ingenuity, insight, inventiveness, and logical developments as well as your solutions.

  5. Pencils and scratch paper will be provided. No tables, rulers, compasses, protractors, slide rules, calculators, or other aids are permitted.

    1. The scoring of questions in Part I has been devised to discourage random guessing and will be computed as follows:

      (three times number correct) - (number wrong).

    2. The scoring for the three questions in Part II will be 13 points per question. Partial credit will be given so it will be to your advantage to do as much as you are able to do on each question.

  6. For the scoring committee. Do not write in this space.

    Part I:

    No. Correct: __________

    No. Wrong: __________

    Part II:

    Score in 1: __________

    Score in 2: __________

    Score in 3: __________

    Score in 4: __________

    Score in Part I: __________

    Score in Part II: __________


    T O T A L : __________












PART I

  1. Let S be the set of integers 0, 1, 2, . . . , 24, 25. The number of members of S that leave a remainder of 0 when divided by 6 is:

    (A)

    (B)

    (C)

    (D)

    (E)

    4

    5

    6

    7

    8

  2. The vertex of the parabola y=3x2+12x+b will be a point on the x axis if the value of b is
    (A)

    (B)

    (C)

    (D)

    (E)

    36

    12

    -16

    -12

    -36

  3. Let a,b > 0. When simplified, square root of -(a2 -b2)2+(a2+b2)2 equals

    (A)

    (B)

    (C)

    (D)

    (E)

    0

    ab

    Square rot of a2+b2

    2ab

    Square root of a2-b2

  4. The average of a set of 50 numbers is 38. If two numbers of the set, namely 45 and 55, are discarded, the average of the remaining numbers is

    (A)

    (B)

    (C)

    (D)

    (E)

    38.5

    37.5

    37

    36.5

    36

  5. When the base of a triangle is increased 20% and the altitude to this base is decreased 20% the change in area is

    (A)

    (B)

    (C)

    (D)

    (E)

    2% increase

    4% increase

    0%

    2% decrease

    4% decrease

  6. 2-(2k+1)-2-(2k-1)+2-2k is equal to
    (A)

    (B)

    (C)

    (D)

    (E)

    2-2k

    2-(2k-1)

    -2-(2k+1)

    0

    2

  7. If the line y=mx+1 intersects the ellipse x2+4y2=1 exactly once, then the value of m2 is
    (A)

    (B)

    (C)

    (D)

    (E)

    1
    2
    2
    3
    3
    4
    4
    5
    5
    6

  8. If a and b are real numbers the equation 3x-5+a=bx+1 has a unique solution x:

    (A)

    (B)

    (C)

    (D)

    (E)

    for all a and b

    if a not= 2b

    if anot= 6

    if b not= 0

    if b not= 3

  9. A triangle has angles of 30o and 45o . If the side opposite the 45o angle has length 8, then the side opposite the 30o angle has length

    (A)

    (B)

    (C)

    (D)

    (E)

    4

    4(square of 2)

    4(square of 3)

    4(square of 6)

    6

  10. The number of cubic feet in the volume of a cube is the same as the number of square inches in its surface area. The length of the edge, expressed in feet, is

    (A)

    (B)

    (C)

    (D)

    (E)

    6

    864

    1728

    6 times 1728

    2304

  11. The square root of -64 is

    (A)

    (B)

    (C)

    (D)

    (E)

    64i

    -64i

    -8

    8i

    8

  12. The ratio of females to males in a math club is 7 to 4. If three females and twelve males are absent from a meeting, the ratio of females to males is 5 to 2. How many members of the math club attend the meeting?
    (A)

    (B)

    (C)

    (D)

    (E)

    24

    60

    63

    84

    99

  13. Two cylists, 30 miles apart and starting at the same time, would be together in 6 hours if they traveled in the same direction, but would pass each other in 2 hours if they traveled towards each other in opposite directions. The ratio of speed of the faster cylist to that of the slower is:

    (A)

    (B)

    (C)

    (D)

    (E)

    2
    1
    3
    2
    2
    3
    4
    3
    3
    1

  14. In a general triangle ADE (as shown) lines EB and EC are drawn. Which of the following angle relations is true?

    (A)

    (B)

    (C)

    (D)

    (E)

    x+z=a+b

    y+z=a+b

    m+x=w+n

    x+z+n=w+c+n

    x+y+n=a+b+m

    Diagram of the triangle

  15. Let S be the sum of the first nine terms of the sequence x+a, x2+2a, x3+3a, . . . Then S equals:

    (A)

    50 a+x+x8
    x+1
    (B)

    50 a-x+x16
    x-1  
    (C)

    x4-1
           
    x+1

    + 45a

    (D)

    x 10-x
               
     x-1

    + 45a
    (E)

    x11-x
               
    x-1

    + 45a

  16. The derivative of x3+x2 is

    (A)

    (B)

    (C)

    (D)

    (E)

    x3

    3x2+2x

    3x2

    2x

    3x2+2/x


    PART II

    1. (13 points)

    1. Student A eats lunch at noon in the cafeteria once a week (including weekends) on a random schedule. Student B also eats there at noon, once a week (including weekends), on a random schedule. On average, how many times will A see B in a week?

    2. If A eats lunch in the cafeteria 3 times a week instead, and B eats there twice a week, how many times a week will they meet, on average?

    3. If A eats lunch in the cafeteria m times a week, and B eats there n times a week, how many times a week will they meet, on average?

    2. (13 points)

    1. Consider the following game: We are given a pile of n jelly beans, where n is a positive integer. Two players alternately must eat 1 or 2 jelly beans off the pile. The winner is the person who eats the last jelly bean.

      For which values of n is there a winning strategy for the beginner? Describe the strategy.

    2. Consider a game involving a pile of n jelly beans as above, except that the players alternately must eat 1, 3, or 5 jelly beans.

      For which values of n is there a winning strategy for the beginner? Describe the strategy.

    3. Consider a game involving a pile of n jelly beans as above, except that the players alternately must eat 1, 2 or 4 jelly beans.

      For which values of n is there a winning strategy for the beginner? Describe the strategy.

    3. (13 points)
    A 20 sided die is labeled with the integer values 1, 2, 3, . . . , 20. The die is ``loaded'', so that it is twice as likely to roll any given odd number as any given even number.

    1. Find the probability of rolling a number greater than 10.

    2. Find the probability of rolling a number greater than 11.

    4. (13 points)

    Inside square ABCD (see figure) with sides of length 20 inches, segment AE is drawn, where E is the point on DC which is 6 inches from D. The perpendicular bisector of AE is drawn and intersects AE, AD, and BC at points M, P, and Q respectively. Find the ratio of segment PMto segment MQ.

    Diagram of the square