IN

MATHEMATICS

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PRINT __________________________________________________________________________________________

Last name First name Initial Phone No.ADDRESS ________________________________________________________________________________________

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Your hight school: Name ____________________________________ City ____________________________

High School Counselor or Advisor: ______________________________________________________________ - 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.
- Part I consists of 16 objective-type questions. Each question has
five possible answers marked: A., B., C., D., E. Only
__one__Example: If

`x=5`and`y=-2`, then`x+4y`is(A)

`-3`(B)`-2`(C)`-1`(D)`0`(E)`+1`. - 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
be sure to show the essential steps of your work concisely in the space provided for each question.*but*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.*Only* - Pencils and scratch paper will be provided. No tables, rulers,
compasses, protractors, slide rules, calculators, or other aids are permitted.
- 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)`. - 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.

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

- 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)

- 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

- 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

- 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

- What is the value of
1997 ^{2}- 1996^{2}(A) (B)

(C)

(D)

(E)

1 1996

1997

3993

3994

- 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

- The limit of
`x`as^{2}- 1 / x - 1`x`approaches 1 as a limit is:(A) (B)

(C)

(D)

(E)

0 indeterminate

`x`- 12

1

- 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

- 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

- Simplify
log _{2}256

-------

log_{3}81- log _{8}(4^{3})

-------

log_{5}25(A) (B)

(C)

(D)

(E)

0 1

2

-1

-2

- 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 studentsThose which are logical consequences of I, II, and III are:

(A) (B)

(C)

(D)

(E)

2 4

2, 3

2, 4

None

- If
`(x + y)`is expanded to the form^{5}`A x`, what is the value of the coefficient^{5}+ B x^{4}y + C x^{3}y^{2}+ D x^{2}y^{3}+ E x y^{4}+ F y^{5}`C`?(A) (B)

(C)

(D)

(E)

0 1

5

3

10

- Let
`f (x) = a x`. If it happens that^{2}+ b x + c`c=b`, then the graph of^{2}/ 4a`y = f (x)`will certainly:(A) (B)

(C)

(D)

(E)

have a maximum have a minimum

be tangent to the

`x`-axisbe tangent to the

`y`-axislie in one quadrant only

- If
`a`and^{x}= c^{q}= b`c`, then:^{y}= a^{z}= d(A) (B)

(C)

(D)

(E)

`xy = qz``x/y = q/z``x + y = q + z``x - y = q - z``x`^{y}= q^{z} - 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

- Given that
`0.5 < X < 1.0`and`a = x, b = x`, which of the following is true?^{x}, c = x^{ ( x x )}(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)**- 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.
- 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.

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

2. **(13 points)**3. **(13 points)**`X`of the system_{1}, X_{2}, X_{3}, Y`X`_{3}+ X_{2}= Y X_{1}X

_{1}+ X_{3}= Y X_{2}X

_{2}+ X_{1}= Y X_{3}4. **(13 points)**`ABC`with legs`BC = 3, AC = 4`. Find the length of the angle bisector from`C`to the hypotenuse. - 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.