MARQUETTE UNIVERSITY

1 9 9 7

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 four 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. A plumber has to replace a large 12 inch diameter pipe with smaller 2 inch diameter pipes. In order to carry the same amount of water how many smaller pipes must be used?

    (A)

    (B)

    (C)

    (D)

    (E)

    6 pi (Greek pi)

    6

    12

    36

    36 pi (Greek pi)

  2. What is the value of

    1 - 2 + 3 - 4 + 5 - 6 + . . . - 96 + 97 - 98 + 99 - 100?
    (A)

    (B)

    (C)

    (D)

    (E)

    0

    -100

    -50

    1

    50

  3. A used car dealer has a car valued at $10,000. He sells it to a customer at a 10% profit based on the worth of the car. The customer later sells the car back to the dealer at a 10% loss. Then:

    (A)

    (B)

    (C)

    (D)

    (E)

    Dealer comes out even

    Dealer makes $1100 on the deal

    Dealer make $1000 on the deal

    Dealer loses $900 on the deal

    Dealer loss $1000 on the deal

  4. A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed three times, what is the probability of getting 2 tails and 1 head.

    (A)

    (B)

    (C)

    (D)

    (E)

    2 / 27

    1 / 9

    1 / 3

    4 / 15

    2 / 9

  5. What is the value of

    19972 - 19962
    (A)

    (B)

    (C)

    (D)

    (E)

    1

    1996

    1997

    3993

    3994

  6. From a group of dogs and cats, 15 cats leave. There are then left two dogs for each cat. After this 45 dogs leave, there are then 5 cats for each dog. The number of cats in the beginning was:

    (A)

    (B)

    (C)

    (D)

    (E)

    40

    43

    29

    50

    None of the above

  7. The limit of x2 - 1 / x - 1 as x approaches 1 as a limit is:

    (A)

    (B)

    (C)

    (D)

    (E)

    0

    indeterminate

    x - 1

    2

    1

  8. Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in 3 hours and the other in 4 hours. At what time p.m. should the candles be lighted so that, at 4 p.m., one candle stub is twice the length of the other?

    (A)

    (B)

    (C)

    (D)

    (E)

    1 : 24

    1 : 28

    1 : 36

    1 : 40

    1 : 48

  9. Triangles ABC and ADC are right triangles. Find the value of h if AB=40 and AC=30.

    (A)

    (B)

    (C)

    (D)

    (E)

    18

    20

    21

    24

    25

    Diagram of the square

  10. Simplify
    log2 256
    -------
    log3 81
     - log8 (4 3)
    -------
    log5 25
    (A)

    (B)

    (C)

    (D)

    (E)

    0

    1

    2

    -1

    -2

  11. Assume that the following three statements are true:

    I. All fresshmen are human II. All students are human III. Some students think

    Given the following 4 statements:

    (1)
    (3)
    All freshmen are students
    No freshmen think
    (2)
    (4)
    Some freshmen think
    Some humans who think are not students

    Those which are logical consequences of I, II, and III are:

    (A)

    (B)

    (C)

    (D)

    (E)

    2

    4

    2, 3

    2, 4

    None

  12. If (x + y)5 is expanded to the form A x5 + B x4 y + C x3 y2 + D x2 y3 + E x y4 + F y5, what is the value of the coefficient C?

    (A)

    (B)

    (C)

    (D)

    (E)

    0

    1

    5

    3

    10

  13. Let f (x) = a x2 + b x + c. If it happens that c=b2 / 4a, then the graph of y = f (x) will certainly:

    (A)

    (B)

    (C)

    (D)

    (E)

    have a maximum

    have a minimum

    be tangent to the x-axis

    be tangent to the y-axis

    lie in one quadrant only

  14. If a x = c q = b and c y = a z = d, then:

    (A)

    (B)

    (C)

    (D)

    (E)

    xy = qz

    x/y = q/z

    x + y = q + z

    x - y = q - z

    x y = q z

  15. The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as:

    (A)

    (B)

    (C)

    (D)

    (E)

    1 : 2

    2 : 3

    2 : 5

    3 : 4

    3 : 5

  16. Given that 0.5 < X < 1.0 and a = x, b = x x, c = x ( x x ), which of the following is true?

    (A)

    (B)

    (C)

    (D)

    (E)

    a > b > c

    b > c > a

    c > b > a

    a > c > b

    b > a > c


    PART II

    1. (13 points)

    1. Mom makes two pancakes, one with both sides brown, the other with one side brown and one side yellow. You have not seen the pancakes yet, except that mom just gives you one of the pancakes with a brown side up. The pancake was arbitrarily selected, and so was the choice of which side is up.

      1. What is the probability that your pancake's other side is also brown? Explain your answer.

      The story continues. Now mom comes in with the other pancake, on her own plate. The choice of which side is up is again arbitrarily, but her pancake's top side is also brown.

      1. Now what is the probability that your pancake's other side is also brown? Explain your answer.

    2. (13 points)
    Two people agree to meet at a given place between noon and 1 p.m. Each person will arrive at a random time between noon and 1 p.m. and wait exactly 15 minutes before leaving again. What is the probability that the meeting actually takes place?

    3. (13 points)
    Find all solutions X1, X2, X3, Y of the system

    X3 + X2 = Y X1

    X1 + X3 = Y X2

    X2 + X1 = Y X3

    4. (13 points)
    Given a right angle triangle ABC with legs BC = 3, AC = 4. Find the length of the angle bisector from C to the hypotenuse.