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1.

1

2 3.75

31

4+ 13

8

= (A) 32

(B) 23

(C) 148195

(D) 1318

(E) 3439

2. Let H = 3qN 5, where N is a constant. When H = 10, q = 2. Find q when H = 25.

(A) 4 (B) 6 (C) 12 (D) 1 (E) 8 3

3. i = 1

5

(1)i (i + 2) = (A) 5 (B) 5 (C) 3 (D) 3 (E) 25

4. John travels at a constant speed of 50 mph for 3 hours and then travels at a constant speed of 62 mphfor 5 hours. What is his average speed over the course of the trip?

(A) 56 mph (B) 57 mph (C) 57 12

mph (D) 58 mph (E) 58 12

mph

5. If P(3, 5) lies on the graph of line l given by ax + 7y = 12, determine the slope of a lineperpendicular to l.

(A) 473

(B) 2147

(C) 4721

(D) 2123

(E) 79

6. Briannas mother is 20 years older than Brianna. Ten years ago, Briannas mother was twice as old asBrianna. What is the mothers present age?

(A) 30 (B) 40 (C) 25 (D) 51 (E) 50

7. A 12 foot square garden has been planted to produce corn. Stalks are planted 18 inches apart in rowsand stalks may be planted along the boundary. If the rows are 12 inches apart and each stalk produces three

ears of corn, what is the largest yield for this plot?(A) 288 ears (B) 312 ears (C) 333 ears (D) 351 ears (E) 432 ears

8. In a group of 50 students, 28 are taking a math course, 36 are taking an English course, and 22 aretaking both a math and an English course. How many of the 50 students are taking neither a math nor anEnglish course?

(A) 8 (B) 6 (C) 14 (D) 28 (E) 30

9. Two high school classes took the same exam. One class of 35 students had a mean grade of 70 whilethe other class of 25 had a mean grade of 85. What is the mean grade for all students in both classes?

(A) 77.5 (B) 72.43 (C) 74.75 (D) 76.25 (E) 78

10. How many functions can be defined from a domain D = {1, 2, 3} onto a range R = {4, 5} ?

(A) 5 (B) 6 (C) 7 (D) 8 (E) 9

11. Ten liters of a 30% acid solution are obtained by mixing a 25% solution and a 50% solution. Howmany liters of the 25% solution must be used?

(A) 1 (B) 1.5 (C) 2 (D) 5 (E) 8

12.1

x y+

1

y + x=

(A)1

x(B)

2x

x2 y(C)

2x

x2 y(D)

2x

x2 + y(E)

1

x2 y

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13. A local long distance company charges its customers $.24 per minute for a long distance call. Thiscompany does not charge a fraction of this rate for parts of a minute. Instead, it rounds the length ofthe call up to the next full minute. Which of the following graphs shows the pricep of a phone calllasting tminutes with this long distance carrier?

(A) (B) (C) (D) (E)

14. If the base of a triangle is increased by 10% and the altitude is decreased by 10%, what is the percentdecrease in the area of the triangle?

(A) 1% (B) 0.1% (C) 10% (D) 11% (E) 0%

15. Suppose that log3x = logy5 = 2. Then (xy)2 =

(A) 75 (B) 45 (C) 225 (D) 405 (E) 1875

16. Suppose that you are standing at the top of a lighthouse which is built on the edge of the sea. You arelooking at a buoy and your eye level is 100 feet above sea level. If the angle of depression from the

horizontal to your line of sight is 30o, how far off shore is the buoy?

(A) 100 3 feet (B) 100 33 feet (C) 50 2 feet (D) 50 32 feet (E) 100 32 feet

17. Find the exact value of sec(arctan 2), also denoted sec[tan1 (2)].

(A) 5 (B) 3 (C) 1 (D)1

5(E)

1

3

18. The graph at right shows the graphs of two functions f and g.Which of the following describes the relationship between the twofunctions?

(A) g(x) = f(x) (B) g(x) = f(x) + 1 (C) g(x) = f(x) + 1(D) g(x) = f(x 1) (E) g(x) = f(x + 1) 1

19. What is the largest power of 5 that divides 125! ?

(A) 55 (B) 511 (C) 525 (D) 531 (E) 535

20. The base 5 representation of an integer N is A023 where A is a digit from the set {0, 1, 2, 3, 4}.

The base 7 representation of N is 106A. What is the digit A?(A) 0 (B) 1 (C) 2 (D) 3 (E) 4

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21. A 3 inch by 5 inch rectangle is to be enlarged to a similar rectangle with a width of 4 inches.Determine the length of the diagonal of the enlarged rectangle.

(A) 34 inches (B) 3 344

inches (C) 4 343

inches (D) 4 34 inches (E) 3 34 inches

22. Find the length of segment BC if segment AD haslength 5 inches and the perimeter of triangle BCD is28 inches.

(A) 8 3 inches(B) 28 3 inches(C) 17 3 inches(D) 20 3 inches(E) 53 6 inches

23. A vendor sells two sizes of pizza by the slice. Each slice has the shape of a circular sector.The small pizza has a 10 inch diameter and the large has a 20 inch diameter. Sue buys a slice from

a small pizza with a central angle of 24o and Kathy buys a slice from the large pizza with a central

angle of7

radians. Kathys slice is how many times larger than Sues slice?

(A) 1514

(B) 40 7

(C) 307

(D) 154

(E) 157

24. The zeros of a polynomial function p(x) = 2x3 x2 13x 6 are 2, r1 and r2. Find r1 + r2.

(A) 12

(B) 52

(C) 1 (D) 2 (E) 32

25. For how many of the years between 1987 and 1995 inclusive, was the budget for new books less thanfifty percent of the total library budget?

$150,000

$140,000

$130,000

$120,000

$110,000

$100,000

$90,000

$80,000

$70,000

$60,000

$50,000

$40,000

1987

1987

1990

1990

1993

1993

1995

1995

TOTAL

BUDGET

FOR

PUBLIC

LIBRARY

SYSTEM

1987-1995

BUDGET

FOR

NEW

BOOKS

IN

PUBLIC

LIBRARY

SYSTEM1987-1995

(A) three (B) four (C) five (D) six (E) seven

26. What is the probability that the sum of the numbers rolled on a pair of fair sixsided dice is a primenumber?

(A) 12

(B) 718

(C) 736

(D) 29

(E) 512

27. A function, f, defined for all real numbers, satisfies: f(x2) = (x2 + 1)f(x) and f(2) = 3. What is f(256)?

(A) 384 (B) 1280 (C) 3072 (D) 65,535 (E) 196,605

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28. An isosceles trapezoid ABCD (shown) has bases AB = 10, CD = 6. If___ ___ ___

the diagonals AC and BD intersect in point F and the altitude GE, of length 8,

___

passes through F, then what is the length of EF?

(A) 5 2 (B) 5 (C) 3 (D) 4 (E) None of these

29. A new professional tiddlywinks league will have teams from exactly four cities, to be chosen from:New York, Chicago, Los Angeles, Boston, Philadelphia, San Diego, and Atlanta. At least two of the citiesmust be among the largest three---New York, Chicago, and Los Angeles. New York and San Diego willnot be in the league together. Los Angeles and Boston will not be in the league together. Philadelphia andSan Diego will not be in the league together. Which of the following statements must be true?I. San Diego and Boston will not be in the league together.II. If either Chicago or San Diego is in the league, both will be.III. If Boston is in the league, then Chicago is in the league.

(A) I only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III

30. In the figure below lineAB is tangent to the circle at pointX, lineACis tangent to the circle at point Y,

and lineBCis tangent to the circle at pointZ. If the length of segmentAX= 12 = the length of

segmentAY, then what is the perimeter of triangle ABC?

(A) 24 (B) 12 2 (C) 12 (D) 18 (E) 8 3

31. A drawer has 8 red socks, 6 blue socks, and 4 white socks. A blindfolded person takes socksout of the drawer, one at a time. What is the minimum number of socks which must be taken from thedrawer until she is guaranteed to have three matching pairs?

(A) 6 (B) 7 (C) 8 (D) 9 (E) 10

32. Two adjacent vertices of a square are on the line y = 4and the remaining two vertices are on the parabola

y = x2 (see figure.) The exact area of the square is

(A) 6 2 5(B) 5 1(C) 7 + 2 5

(D) 1 + 5(E) 24 8 5

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33. How many triangles can be formed so that each vertex is a point on aregular 3 3 grid (see diagram at right)?

(A) 48 (B) 36 (C) 38 (D) 76 (E) 84

34. How many integers between 1 and 1000 inclusive are divisible by 6 but are notdivisible by either 9 or 15?

(A) 66 (B) 11 (C) 78 (D) 89 (E) 166

35. Find all real solutions of 9x 9x 1 = 24 .

(A) 3 (B) 2 (C) 3 2 (D) 1 2 (E) 1 3

36. Find the greatest value of the function f(x) =2x2 + 4x + 11x2 + 2x + 5

.

(A) 9 4 (B) 2 (C) 11 5 (D) 0 (E) 17 8

37. Find the sum of all of the digits of all twodigit positive integers.

(A) 45 (B) 99 (C) 450 (D) 855 (E) 900

38. Equilateral triangle ABC is cut by line l such that l

||BC.

If AC = 1 and the area DAE is equal to the area of trapezoidBDEC, find the height ofDAE. (Drawing not to scale)

(A) 38

(B) 22

(C) 64

(D) 32

(E) 24

39. Papa Domonique has three sizes of pizza, 12 inch diameter at $8.00, 16 inch diameter at $10.00 and20 inch diameter at $12.00. He only sells whole pizzas (Dont even ask for pizza by the slice!!). What isthe difference between the maximum and minimum price you can pay for exactly 236 in2 of PapaDomoniques pizza?

(A) $6.00 (B) $12.00 (C) $10.00 (D) $4.00 (E) $8.00

40. The numbers a, b, c, and d are consecutive terms in a strictly increasing geometric sequence (withcommon ratio between consecutive terms). In terms of a and b, find the yintercept of the line all of whosepoints are equidistant from the points (a, b) and (c, d).

(A)b

a(B)

a

b(C)

(a2 + b2)2

2a2b(D)

b3 + a2b2a2

(E)a2

b2

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98 entries

1. What is the sum of the prime factors of 1998?

(A) 222 (B) 42 (C) 43 (D) 122 (E) 48

2. How many even numbers, k, in the interval 100 k 300 do not have a digit 6 in their decimalrepresentation?

(A) 72 (B) 73 (c) 74 (D) 75 (E) 76

3. What is the minimum number of lines determined by 5 points in a plane with no four collinear?

(A) 4 (B) 5 (C) 6 (D) 7 (E) 84. Find the least value of the polynomial p(x) = x4 4x3 + 6x2 4x + 6.

(A) 0 (B) 6 (C) 5 (D) 1 (E) 3

5. Isosceles triangle with AB = AC and BC of length 2 isinscribed in circle with center O. Point P is the midpointof radius OA and the area ofBPC is 2 3 the area ofBAC.What is the radius of circle ?

(A)1

3 2 (B)2

3 2 (C)1

3 3 (D)2

3 3 (E) 3

6. If cot = 52

where2

, then cos =

(A) 5 2929

(B) 5 2929

(C) 2 2929

(D) 2 2929

(E) 5

7. Evaluate : 70 272 3 + 42

(A) 12716

(B) 14 316

(C) 6 (D) 8 (E) 24

8. If 125x 1 = 6253 x, then the value of x is

(A) 157

(B) 2 (C) 117

(D) 114

(E) 97

9. Given a circle whose equation is x2 + y2 8y 12 = 0 and a parabola given by y = (x 2)2 + 3,compute the distance between the center of the circle and the vertex of the parabola.

(A) 5 (B) 53 (C) 29 (D) 2 (E) 3

10. Mrs. Smiths fifth period Algebra II class contains 12 boys and 15 girls. If3 4 of the boys and 2 3 of

the girls are present today, how many student in the class are absent?

(A) 19 (B) 6 (C) 9 (D) 10 (E) 8

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11. Points A and C lie on a number line with coordinates 13 and 17, respectively. Find the coordinateof the point 3 5 of the way from C to A.

(A) 11 (B) 1 (C) 5 (D) 6 (E) 7

12. If P(3, 7) lies on the graph of y = f(x) then which point must lie on the graph of y = 2f(x) + 4?

(A) (3, 3) (B) (6, 3) (C) (3, 10) (D) (6, 11) (E) (10, 7)

13.3 + 2i4 i = (A)

3

4 2i (B) 10

17+ 11

17i (C) 10

15+ 11

15i (D) 3

4 1

2i (E) 14

17+ 11

17i

14. If A = {1, 2, 3, 4} and B = {2, 3, 5}, how many elements are contained in set C whereC = {(x, y) (x , y) A B and y > x + 1}?

(A

)6

(B

)7

(C

)4

(D

)5

(E

)3

15. How many solutions of sin 3 = 22

lie in the interval23

< 114

?

(A) 2 (B) 3 (C) 4 (D) 5 (E) 6

16. The radius of the earth is approximately 4000 miles. Given that Miami, Florida and Erie, Pennsylvania

are on the same longitudinal line and that Erie has a latitude of 44 o N while Miami has a latitude of 14o N,how far apart are these two cities?

(A) 20, 940 miles (B) 2094 miles (C) 4188 miles (D) 488 miles (E) 2000 miles

4. An equilateral triangular picture has edges oflength 12 inches. The width of the frame is2 inches as shown in the diagram. What is theexact size in square inches of the total area of theframed picture?

(A) 72 in2 (B) 54 + 36 2 in2

(C) 108 + 72 2 in2

(D

)36 in2

(E

)200 in2

3. How many diagonals (segments joining nonadjacent vertices) does a tensided regular polygon have?(A) 10 (B) 20 (C) 30 (D) 35 (E) 70

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4. A degree 3 polynomial with rational coefficients has constant term 2 and one of the roots is 2 . Oneof the other roots must be

(A) 2 (B) 2 (C) 2 (D) 2 (E) 2 2

5. For some number b, log b 3 = 4 and log b 2 2.5. What is log b 4.5?(A) 5.5 (B)

4. The line t is a transversal cutting lines a and b such that the interior angles on the same side of t have

measures (x2)o and (3x)o. If a and b are parallel, what is the value of x?

(A) 20 (B) 15 (C) 12 (D) 30 (E) 32

5. In the diagram, what is the sum of the measures of the five angles A, B C, D, and E, at the pointsof the fivepointed star?

(A) 180o (B) 270o (C) 360o (D) 540o (E) None of these

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1 . S o l v e f o r x :

p

4 x , 3 = , 2

A n o s o l u t i o n B ,

1

4

C

i

p

2 + 3

4

D 3 E

7

4

2 . P r o f e s s o r A b s t r a c t i s a s k e d t o j u d g e a b e a u t y c o n t e s t . O v e r w h e l m e d b y t h e b e a u t y o f t h e t e n

c o n t e s t a n t s , h e m o p s h i s b r o w a n d d e c i d e s h e m u s t a s s i g n t h e r s t , s e c o n d , a n d t h i r d p r i z e

c o m p l e t e l y a t r a n d o m . I n h o w m a n y w a y s c a n t h i s b e d o n e ?

A 3 B 7 2 0 C 1 0 0 0 D 3 0 E 2 5 6

3 . F i n d t h e p r o d u c t o f t h e m e a n , m e d i a n , a n d p o p u l a t i o n s t a n d a r d d e v i a t i o n o f t h i s d a t a s e t :

3 , 5 , 6 , 1 2 , 1 4

A 8 6 4 B 1 9 2

p

2 C 4 8 D 1 4 4

p

2 E 7 3

p

1 0

4 . T w o t h o u s a n d o n e h u n d r e d A r m s t r o n g s t u d e n t s t o o k a t l e a s t o n e m a t h e m a t i c s c o u r s e l a s t

y e a r . S e v e n h u n d r e d f t y A r m s t r o n g s t u d e n t s t o o k a t l e a s t o n e p s y c h o l o g y c o u r s e l a s t y e a r .

T h r e e h u n d r e d A r m s t r o n g s t u d e n t s t o o k b o t h a m a t h e m a t i c s c o u r s e a n d a p s y c h o l o g y c o u r s e

l a s t y e a r . H o w m a n y A r m s t r o n g s t u d e n t s t o o k e i t h e r a m a t h e m a t i c s c o u r s e o r a p s y c h o l o g y

c o u r s e l a s t y e a r ?

A 2 8 5 0 B 3 1 5 0 C 2 2 5 0 D 1 6 5 0 E 2 5 5 0

5 . A n e q u i l a t e r a l t r i a n g l e i s i n s c r i b e d i n a c i r c l e . E a c h s i d e o f t h e t r i a n g l e m e a s u r e s a . W h a t i s

t h e a r e a o f t h e c i r c l e ?

A a

2

B

a

2

4

C

a

2

3

D

2 a

p

3

E

3 a

2

4

6 . H o w m a n y i n t e g e r s b e t w e e n 1 a n d 1 0 0 , i n c l u s i v e , a r e d i v i s i b l e b y e i t h e r 2 o r 3 ?

A 8 3 B 6 7 C 5 0 D 3 3 E 1 6

7 . W h i c h o f t h e e x p r e s s i o n s b e l o w d e t e r m i n e s t h e l o c u s o f p o i n t s

f x y t h e d i s t a n c e f r o m x y t o 1 1 = t h e d i s t a n c e f r o m x y t o , 1 , 1 g ?

A x

2

+ y

2

= 1 B x = y C f 0 0 g D y = x

3

E x = , y

8 . I n t h e g u r e b e l o w A B k C D , m A B = 6 a n d m C D = 8 .

A B

P

CD

I f t h e a r e a o f 4 D P C = 3 2 , t h e n t h e a r e a o f 4 A P B =

A 2 4 B 1 2 C 1 8 D 3 2 E

p

1 0 8

1

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9 . A r i g h t t r i a n g l e h a s p e r i m e t e r 5 6 a n d a r e a 8 4 . D e t e r m i n e t h e l e n g t h s o f t h e t h r e e s i d e s .

A 1 4

5 6

3

7 0

3

B 1 7 2 1 1 8 C 2 4 2 5 7

D 3 1 2 1 4 E

6 9 7

4 0

4 1 1

5

6 1 5

4 0

1 0 . I f a n i n t e g e r b e t w e e n 1 a n d 2 0 0 , i n c l u s i v e , i s r a n d o m l y s e l e c t e d , w h a t i s t h e p r o b a b i l i t y t h a t

i t i s a p e r f e c t s q u a r e ?

A 5 0 B 2 5 C 1 0 D 0 7 E 0 5

1 1 . L e t s i x p o i n t s b e p l a c e d i n t h e p l a n e s o t h a t n o t h r e e a r e c o l i n e a r . H o w m a n y t r i a n g l e s c a n

b e f o r m e d s o t h a t e v e r y v e r t e x o f t h e t r i a n g l e c o m e s f r o m t h i s s e t o f s i x p o i n t s ?

A 1 5 B 2 0 C 2 5 D 3 0 E 3 5

1 2 . F o r w h a t v a l u e s o f x d o e s t h e c i r c l e x

2

+ y

2

= 1 0 i n t e r s e c t t h e l i n e x + y = 2 ?

A , 1 B 2 C , 1 3 D 1 , 2 E 2 , 3

1 3 . W h a t i s t h e p e r i m e t e r o f t h e g u r e b e l o w ?

4

6

1

1

A 1 2 B 2 1 C 2 2 D 2 4 E 2 5

1 4 . A p a c k o f 1 0 0 c a r d s , n u m b e r e d 1 t o 1 0 0 , i s t h o r o u g h l y s h u e d , a n d v e c a r d s a r e r a n d o m l y

d r a w n f r o m i t . W h a t i s t h e p r o b a b i l i t y t h a t t h e v e c a r d s w i l l b e d r a w n i n i n c r e a s i n g o r d e r ?

A

5 ! 9 5 !

1 0 0 0 !

B

9 5 !

1 0 0 !

C

5 !

1 0 0 !

D

1

5 !

E

1

9 5 !

1 5 . L e t f x = 6 x

5

+ c x

3

+ 3 5 , w h e r e c i s a n i n t e g e r . W h i c h o f t h e f o l l o w i n g c a n p o s s i b l y b e a

z e r o o f f ?

A 2 B 3 C

7

3

D

2

5

E 4

2

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1 6 . A s s u m i n g t h e m a r k s d i v i d e t h e s q u a r e ' s s i d e s i n t o e q u a l p o r t i o n s , t h e p e r c e n t o f t h e s q u a r e

w h i c h i s s h a d e d i s

A 6 0 B 6 4 C 6 8 D 7 2 E 7 5

1 7 . W h i c h o f t h e f o l l o w i n g i s e q u a l t o s i n x c o s x t a n x c o t x s e c x c s c x ?

A s i n

3

x B s e c

2

x , t a n

2

x C s i n x , c o s x

D x E s i n

2

x , c o s

2

x

1 8 . W h i c h o f t h e f o l l o w i n g i s t h e g r a p h o f a f o u r t h d e g r e e p o l y n o m i a l ?

A B C

D E

1 9 . W h e n d r o p p e d o n a h a r d s u r f a c e , a S u p e r B a l l t a k e s a s e r i e s o f b o u n c e s , e a c h o n e

9

1 0

a s h i g h

a s t h e p r e c e d i n g o n e . I f a S u p e r B a l l i s d r o p p e d f r o m a h e i g h t o f 1 0 f e e t , t h e t o t a l d i s t a n c e i t

t r a v e l s b e f o r e c o m i n g t o r e s t i s :

A 1 0 0 f t B 2 0 0 f t C 1 9 0 f t

D 1 8 0 f t E a n i n n i t e d i s t a n c e

2 0 . T h e p r o d u c t , i n b a s e e i g h t , o f t h e t w o b a s e e i g h t n u m b e r s 7 7 7 a n d 7 7 7 7 i s :

A 7 , 7 7 7 , 7 7 7 B 7 , 7 6 6 , 7 7 7 C 7 , 7 6 7 , 0 0 1 D 1 0 , 0 0 0 , 0 0 1 E 7 , 7 7 6 , 7 0 1

2 1 . A b a s e t e n t w o - d i g i t n u m b e r i s s e l e c t e d a t r a n d o m . W h a t i s t h e p r o b a b i l i t y t h a t t h e n u m b e r

i s f o u r t i m e s t h e s u m o f i t s d i g i t s ?

A 0 B

9

1 0

C

4

9

D

1

2 5

E

2

4 5

3

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2 2 . T h e t w o s h o r t e s t a l t i t u d e s o f a r i g h t t r i a n g l e h a v e l e n g t h s 2 4 c m a n d 3 0 c m . F i n d t h e l e n g t h

o f t h e l o n g e s t a l t i t u d e .

A 3 2 c m B 3 6 c m C 4 0 c m D 4 8 c m E 5 0 c m

2 3 . I f s i n =

2

3

, t h e n c o s 2 i s

A

1

9

B

4

p

5

3

C

5

9

D

p

5

3

E ,

p

5

3

2 4 . A s e m i c i r c l e i s i n s c r i b e d i n a u n i t s q u a r e w i t h a d i a m e t e r o n o n e s i d e o f t h e s q u a r e . F i n d t h e

a r e a o f t h e s h a d e d r e g i o n b o u n d e d b y t h i s s e m i c i r c l e a n d o n e o f t h e d i a g o n a l s o f t h e s q u a r e .

A

, 1

4

B ,

1

2

C

, 1

8

D

, 2

1 6

E

, 2

4

2 5 . T h e n u m b e r o f r e a l n u m b e r s s a t i s f y i n g t h e e q u a t i o n

p

x = 2 , x

p

x i s

A 0 B 1 C 2 D 3 E i n n i t e

2 6 . F o r w h a t n o n z e r o v a l u e s o f k i s t h e p a r a b o l a y = 2 x

2

+ 2 x + k t a n g e n t t o t h e x - a x i s ?

A 2 B 4 C 8 D

1

2

E

1

4

2 7 . S u p p o s e t h a t A , B , C a n d D a r e a l l r e a l p o s i t i v e n u m b e r s . I f D = C

n

p

A B , t h e n n e q u a l s

A

l o g D , l o g C

l o g A + l o g B

B

l o g A + l o g B

l o g D , l o g C

C

l o g A l o g B

o g D

o g C

D

o g D

o g C

l o g A l o g B

E

l o g A + B

l o g D , C

2 8 . L e t f x = 2 x

2

, 3 x + 1 . I f t h e g r a p h o f a f u n c t i o n y = g x i s f o r m e d b y s h i f t i n g t h e g r a p h

o f f 2 u n i t s u p a n d 3 u n i t s t o t h e r i g h t , t h e n g i s g i v e n b y

A g x = 3 x

2

+ 9 x + 1 2 B g x = 2 x

2

, 1 1 x + 1 8 C g x = 2 x

2

, 1 5 x + 2 1

D g x = 2 x

2

, 1 5 x + 3 0 E g x = 2 x

2

+ 9 x + 1 2

4

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2 9 . I n t h e g i v e n g u r e , l e t 4 A B C b e e q u i l a t e r a l , a n d l e t D a n d E b e m i d p o i n t s o f A B a n d A C ,

r e s p e c t i v e l y . T h e r a t i o o f t h e a r e a o f t h e q u a d r i l a t e r a l D E F G t o t h e a r e a o f 4 A B C i s

A

BC

DE

F G

A

5

8

B

1

2

C

3

8

D

3

4

E

1

3

3 0 . I n t h e g i v e n g u r e , t h e l e n g t h o f A E i s 2 u n i t s , t h e m e a s u r e o f

6

D E A = 6 0

, a n d t h e t o t a l

a r e a o f t h e g u r e i s 6 s q u a r e u n i t s . T h e l e n g t h o f A B i s

A

BC

D

E

A

p

3 B

p

3

2

C

p

3 , 1 D 2 E

p

2

3 1 . A j u m b o r e c t a n g u l a r c h o c o l a t e b a r m e a s u r e s 1 0 c m i n l e n g t h , 5 c m i n w i d t h , a n d 2 c m i n

t h i c k n e s s . D u e t o t h e e s c a l a t i n g c o s t s o f c o c o a , m a n a g e m e n t d e c i d e s t o r e d u c e t h e v o l u m e o f

t h e c a n d y b a r b y 2 8 . T h e y w o u l d l i k e t o k e e p t h e s a m e t h i c k n e s s a n d r e d u c e t h e l e n g t h a n d

w i d t h b y t h e s a m e n u m b e r o f c e n t i m e t e r s . W h a t i s t h e l e n g t h o f t h e r e d u c e d c a n d y b a r ?

A 7 . 2 c m B 9 c m C 4 c m D 1 c m E 9 c m

2

3 2 . F o u r s u s p e c t s o f a c r i m e m a d e t h e f o l l o w i n g s t a t e m e n t s t o t h e p o l i c e .

A n d i : C a r l a d i d i t .

B o b : I d i d n o t d o i t .

C a r l a : D a v e d i d i t .

D a v e : C a r l a l i e d w h e n s h e s a i d I d i d i t .

I f t h e c r i m e w a s c o m m i t t e d b y o n l y o n e p e r s o n , a n d e x a c t l y o n e o f t h e f o u r s u s p e c t s t o l d t h e

t r u t h , w h o c o m m i t t e d t h e c r i m e ?

A A n d i B B o b C C a r l a

D D a v e E N o n e o f t h e f o u r s u s p e c t s

5

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3 3 . P h o n e n u m b e r s i n a c e r t a i n t o w n a r e o f t h e f o r m 2 1 4 - . I f y o u l i v e i n t h i s t o w n , w h a t

i s t h e p r o b a b i l i t y t h a t y o u r p h o n e n u m b e r c o n t a i n s v e i d e n t i c a l d i g i t s ?

A 3 B 0 0 0 0 3 C 0 0 0 1 D 0 0 0 3 E 0 1

3 4 . A r e c t a n g u l a r c l a s s r o o m s e a t s s e v e n t y - t w o s t u d e n t s . I f t h e s e a t s w e r e a r r a n g e d w i t h t h r e e

m o r e s e a t s i n e a c h r o w , t h e r e w o u l d b e t w o f e w e r r o w s . F i n d t h e o r i g i n a l n u m b e r o f r o w s .

A 7 B 6 C 9 D 8 E 1 2

3 5 . A f a m i l y i s t r a v e l i n g d u e E a s t a t a c o n s t a n t r a t e o f 6 0 m i l e s p e r h o u r o n a r o a d t h a t p a s s e s

t o t h e n o r t h o f a f a m o u s l a n d m a r k . A t n o o n , t h e f a m i l y ' s p o s i t i o n r e l a t i v e t o t h e l a n d m a r k

i s 3 0

W e s t o f N o r t h . O n e h o u r l a t e r , t h e i r p o s i t i o n i s 3 0

E a s t o f N o r t h . H o w f a r f r o m t h e

l a n d m a r k w e r e t h e y a t t h e m o m e n t w h e n t h e y w e r e c l o s e s t t o i t ?

A 3 0

p

2 m i B 3 0

p

3 m i C 6 0 m i D 3 0 m i E 1 0

p

3 m i

3 6 . A m a t h p r o f e s s o r n e e d s t o b r i n g f o u r i t e m s t o t h e c l a s s r o o m : t h e t e x t b o o k , h i s n o t e s , a

g r a p h i n g c a l c u l a t o r , a n d a p e n . I t i s e q u a l l y l i k e l y t h a t t h e p r o f e s s o r w i l l f o r g e t o r r e m e m b e r

e a c h o f t h e s e i t e m s . A s s u m i n g t h a t t h e p r o f e s s o r h a s f o r g o t t e n a t l e a s t o n e i t e m , w h a t i s t h e

p r o b a b i l i t y t h a t h e h a s f o r g o t t e n e x a c t l y t w o i t e m s ?

A

2

1 3

B

1

4

C

1

5

D

3

8

E

2

5

3 7 . T w o m e n a n d t w o b o y s n e e d t o c r o s s a l a k e . T h e i r b o a t w i l l c a r r y e i t h e r o n e m a n o r t w o

b o y s . W h a t i s t h e f e w e s t n u m b e r o f o n e - w a y t r i p s i t w i l l t a k e t o g e t e v e r y b o d y t o t h e o t h e r

s i d e o f t h e l a k e ?

A 4 B 5 C 8 D 9 E 1 1

3 8 . T h e g r a p h o f t h e f u n c t i o n f i s g i v e n b e l o w . W h i c h o f t h e g r a p h s c o r r e s p o n d s t o y = f , 2 x ?

-2 -1 1 2

-2 -1 1 2

A

1 2 3

B

-2-1 1 2 3 4

C

-1 1 2

D

-1 1 2

E

6

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3 9 . T h e f o l l o w i n g h i s t o g r a m s h o w s t h e g r a d e d i s t r i b u t i o n s f o r a l a r g e u n i v e r s i t y m a t h c l a s s . I f a

p i e c h a r t w e r e t o b e c o n s t r u c t e d f o r t h i s s a m e d a t a , w h a t w o u l d b e t h e c e n t r a l a n g l e , m e a s u r e d

i n d e g r e e s , f o r t h e s e c t o r c o r r e s p o n d i n g t o t h e n u m b e r o f A s t u d e n t s ?

F D C B Agrades

12

16

25

3235

number of students

A 1 2

B 2 4

C 1 0

D 3 6

E 2 0

4 0 . A d d t h e r e c i p r o c a l s o f t h e t w o

R o o t s o f x

2

+ p x + q

S e t e q u a l t o z e r o .

T h e n y o u ' l l b e a h e r o

B y n d i n g t h e a n s w e r t h a t ' s t r u e :

A ,

p

q

B

q

p

C

p

q

D ,

q

p

E p q

7

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1. All of the following are true for a given angle A, 0 < A < 2

, except:

a) sin2 A = 1 cos2 A b) 1 = sec2 A tan2 A c) sec2 A 1 = tan2 A

d) 1 = csc A sin A e) cos2 A = sin2 A

2. A New York deli advertises Best Submarine Sandwiches in Town. Each sandwich consists of one meat choice,one cheese choice and one bread choice. They offer 10 types of meat, 4 types of cheese, and 6 types of bread.How many submarine sandwiches can be made from these selections?

a) 20 b) 240 c) 120 d) 12 e) none of these

3. 4% of 300% of x is equal to

a) .12x b) .07x c) 3.04x d) .304x e) 1200x

4. How many 4 digit numbers are divisible by 5?

a) 1000 b) 1800 c) 2400 d) 4800 e) 5000

5. In a certain state lottery, you pick any 3 numbers in the range 1 to 30. The three distinct numbers are randomlydrawn from an urn, and you win the big jackpot if your three numbers are drawn (in any order). What arethe chances that you will win the big jackpot?

a) 1 in 10 b) 3 in 29 c) 1 in 4060 d) 3 in 8120 e) 1 in 24,360

6. Let a, b, and c represent the sides of a right triangle, with c being the hypotenuse length. Which of the followingstatements is true?

a) a3 + b3 = c3 b) a3 b3 > c3 c) a3 + b3 > c3 d) a3 + b3 < c3 e) a3 b3 = c3

7. Bill and Bob are two oarsmen who row at the same rate. Bill is rowing on a river going x miles with thecurrent, and then x miles against the current. Bob rows 2x miles on a lake with no current. If you know therowing rate is greater than the rivers current, then which of the following is true?

a) Bill takes longer. b) Bob takes longer. c) Bill and Bob take the same amount of time.

d) Cannot determine without knowing the value of the currents rate. e) Rate doesnt matter. No Solution

8. An auto dealer has three brands of car to choose from and the following 9 options which you may or maynot wish to to include: CD player upgrade, 4 wheel drive, glow-in-the-dark hubcaps, moon roof, trailer hitch,digital compass, roof rack, rear spoiler, and leather seats. If you can have as many or as few options added toyour car as you want, how many different cars are possible?

a) 1088640 b) 362880 c) 4096 d) 1536 e) 512

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9. Determine whether the series 12 + 8 + 163

+ 329

+ converges or diverges. If it converges, find the sum.

a) converges, sum = 26 b) converges, sum = 36 c) converges, sum = 40

d) converges, sum = 63 e) diverges

10. If p

q > 0, which of the following must be true?

a) If q = 0, then p < 0 b) If q < 0, then p < 0 c) If p > 0, then q > 0

d) If q < 1, then p > 1 e) None of the Above

11. If y = 10log x

x2, for x > 0, which of the following is true?

a) y varies directly with x b) y varies inversely with x2 c) y varies as the square of x

d) y varies directly with log x e) y varies inversely with x

12. Suppose that 0 < < 2

and sec = x3

. Which of the following must be true?

a) tan =x29x

b) sin = xx29 c) csc =

xx29 d) cot =

x293

e) tan = 3x29

13. A circle with center at point O has radius 1 inch. A point B is 3 inches away from O. The tangents to thecircle from B touch the circle at points A and C. What is the area of the quadrilateral, OABC?

a) 2.25 b) 3 c) 2

2 d)

10 e) 2

3

14. The logically equivalent statement to Not all dogs have fleas. is

a) Some dogs have fleas. b) Some dogs do not have fleas. c) There exists a dog with fleas.

d) All pets with fleas are not dogs. e) Every dog does not have fleas.

15. A box of chocolates has 10 pieces of chocolate that look identical, but four of them have caramel in the middle,

four have an orange cream in the middle and two are solid chocolate. If two are randomly selected, what is theprobability at least one contains caramel?

a) 2/3 b) 4/15 c) 8/45 d) 21/25 e) 1/2

16. A restaurant manager must get 6 of 9 waiters to work catering a banquet. Two of the waiters (Tom and Joe)refuse to work together. How many ways can the manager assemble the team where Tom and Joe are not bothon it?

a) 56 b) 84 c) 7 d) 35 e) 49

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17. Define a function as f(x) = ex

x!, x = 0, 1, 2, ...; where is positive and real-valued. The maximum value of

f(x) occurs at which value of x?

a) the greatest integer less than b) the greatest integer less then 1/ c) 0

d) 0 if 1 and 1 if > 1 e) none of the above

18. Suppose that triangle ABC is equilateral and define the following functions on the position of ABC:

s reflects ABC about the altitude to that base (ie., if A is the top vertex, then s switches B and C),

r rotates ABC 120 sending A B, B C, and C A,t rotates ABC 240 sending A C, B A, and C B.

Which of the following compositions return ABC back to its original position?

a) s t b) s r s r c) t s d) r r e) s r s t

19. Let x be an odd natural number. If x is divided by 6, it leaves a remainder of y. If y2

is divided by 4, it leavesa remainder of z. Which of the following must be true of z?

a) z = 3 b) z = 5 c) z = 1 d) z is even e) none of the above

20. If a regular hexagon has all of its diagonals (line segments connecting non-consecutive vertices), then how manynon-overlapping regions are there in the interior of the hexagon?

a) 12 b) 18 c) 20 d) 24 e) none of the above

21. What is the largest perfect square that divides 15! ?

a) 28345272 b) 216365272 c) 210365272 d) 216365272 11 13 e) 211365372

22. The two circles are tangent to each other, and are each tangent to the square surrounding them. If each circlehas radius 1, then the area of the entire square is

a) 10 +

2 b) 8 + 2

2 c) 4 + 6

2 d) 2 + 8

2 e) 6 + 4

2

"!

#

"!#23. The sum of the solutions of a quadratic equation is 11

12and the product is 1

6. A possible quadratic equation

that has these solutions is

a) 12x2 + 11x 2 = 0 b) 12x2 11x + 2 = 0 c) 6x2 2x 11 = 0

d) 6x2 11x + 1 = 0 e) 6x2 2x + 11 = 0

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24. The number of positive integers less than 30 that can be written as the product of two distinct primes is

a) 13 b) 11 c) 10 d) 8 e) 7

25. Triangle P QR has angle R as a right angle, QT = QV, and P S = P V. The measure of angle SV T is

a) 55 b) 35 c) 75 d) 45 e) 90

rrrr

rrrr

rr

P Q

R

rSrVr

Tr

26. If the roots of the equations x2 px + q = 0 are r1 and r2, then r21 + r22 =a) p2 + q2 b) p2 2q c) p2 q2 d) p2 e) q2

27. If N is a positive integer greater than 1, then N(N-1)(N+1) is always

a) divisible by 2N b) greater than 6 c) divisible by 3

d) greater than N3 e) divisible by N2

28. cos(2 sin17

3) =

a) 59

b) 32

c) 223

d) 2149

e) 12

29. A fair coin is tossed until it lands heads up. What is the probability that heads first appears on an odd-numberedtoss?

a) 15

b) 12

c) 23

d) 34

e) 79

30. In how many distinct ways can the letters in the word SAVANNAH be arranged?a) 560 b) 13400 c) 6720 d) 3360 e) 20160

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31. We come upon a woman on the bank of a river who possesses only 5 rocks that she intends to toss in front ofherself within leaping distance to enable her to cross the river and yet keep her feet dry. Each rock has theprobability 2/3 that it will adhere to the river bottom and serve as a stepping stone (and probability 1/3 itwill float down river). She needs 4 rocks to accomplish her task. What is the probability she is able to crossthe river on her rocks?

a) 112/243 b) 16/81 c) 16/243 d) 80/243 e) 32/243

32. How many zeros does the number 100! end in?

a) 10 b) 11 c) 21 d) 20 e) 24

33. What is the length of the altitude of the 3-4-5 right triangle from the vertex at the right angle?

a) 2.4 b) 1.8 c) 2 d)

3 e)

5

34. An isosceles triangle has two sides of length 10 inches. If the length of the third side is a whole number ofinches, and the area of the triangle is a whole number of square inches, what is the area of the triangle?

a) 37 in2 b) 40 in2 c) 45 in2 d) 48 in2 e) 50 in2

35. Solve for x. log3(x + 1) log9 x = 1.a) 1

8b) 73

5

2c) 1

2d) 1+

37

2e) 7+3

5

2

36. In a group of 10 people, the mean age is 20.5 years. One more person joins this group. How old is this 11thperson if the mean age of the group increased to 21 years?

a) 21.5 b) 26 c) 31 d) 25 e) none of these

37. Find the degree measure of the smallest positive angle that satisfies the equation sin + cos =32

.

a) 6 b) 10 c) 12 d) 15 e) 22.5

38. Quadrilateral ABCD has CD AB, the measure of angle D = 45, the measure of angle C = 120, AD = 122and CD = 27.

The area of the quadrilateral is

a) 252 + 24

3 b) 342 c) 324 d) 252 + 72

3 e) 234 + 6

2

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39. A triangular arrangement of six numbers is called magic if the sums of the numbers along the three edges ofthe triangle are equal, as in the diagram below. If the numbers 1, 2, 3, 4, 5, and 6 are positioned randomly inthe six squares below, what is the probability that a magic triangle will result?

1

2 34

56

d

d

dd

a) 1/6 b) 1/30 c) 1/120 d) 1/180 e) 1/720

40. Greg heads off to disc-golf practice on his bike. He takes a low sloping hill down to a traffic light. While waitingfor the light to turn green, he realizes he forgot his discs, and pedals back up the hill to his house. After pickingup discs, he takes a shorter (more dangerous) more steep descent to the practice field. What function belowbest represents his distance from his house as a function of time?

a) b)

e

ee

ee

e

c) d)

e)7

77

77

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1. The year 2002 is a numerical palindrome because it reads the same forwards and backwards. Howmany four digit numbers are palindromic?

(A) 90 (B) 900 (C) 1000 (D) 1111 (E) 9000

2. Simplify: 27 < 11 27 + 11

(A 2 (B) 4 38 (C) 16 (D) 6 (E) 4

3. If a2 = 74,701,449 and (a + 12 = 74,718,736, then a =

(A 8,633 (B 8,643 (C 8,742 (D 8,743 (E 8,843

4. A shoe store has a sale offering 25% off the retail price. As an employee, Tracy gets a 40% discount

off the sale price. If Tracy pays $9.99 for a pair of shoes, how much money does she save off theretail price?

(A $12.21 (B $6.50 (C $2.22 (D $22.20 (E $18.58

5. In which of these is y a function of x?

I. y = 2/ + e II. y4 = x III. y = x2 IV. y =

1 if x is rational

0 if x is irrational

(A III only (B I, III, and IV (C II, III, and IV (D II, III (E all of the above

6. If f(x) = 3-x and g(x) = 3x < 1, then (fo g)(1) equals

(A)

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11. When the electricity is interrupted, an alarm clock restarts at 12:00 am. Suppose the electricitygoes out for an hour in the night. When you get up, the clock reads 4:30 am, but the radio DJannounces the time as 8:45 am. At what time did the electricity go out ?

(A 4:15 am (B 3:45 am (C 3:30 am (D 3:15 am (E 3:00 am

12. For one root of ax2 + bx + c = 0 to be triple the other, the coefficients a, b, c must be related asfollows:

(A 3b2 = 16c (B b2 = 16c (C 6b2 = 8c (D b2 = 12ac (E 3b2 = 16ac

13. Lines l1 and l2 are parallel. Which triangle listed below has greatest area?

(A 6ABC (B 6ABD (C 6ABE (D 6ABF (E These areas are equal.

14. Suppose the operation * is defined as a * b =a + b

a. For integers k and j, simplify th e following:

[(k+ j * (k. j] * ( < 1. (The dot, . , is for multiplication.

(Ak. j

k+j + k. j(B

1

k+ j(C

k.j + kk. j

(D k (Ej(k+ 1

k+j + k. j

15. A lake contains 5,000,000 gallons of water. Suppose the water is to be drained by a pump at

a constant daily rate. If9

10of the volume of the lake remains after the first days pumping, how

many gallons are left in the lake after the sixth day?

(A 0.5 (B 5 (C 50 (D 2,000,000 (E 3,000,000

16. If f(x) = f(x + 1) < x and f(0) = 5, find f(3).

(A) 6 (B) 8 (C) 4 (D) 3 (E) 2

17. Find the area of the region determined by the system of inequalities:

x * 0

x < y ) 0

x + y ) 4(A 2 (B 3 (C 4 (D 6 (E 8

18. Two regular 6-sided dice are rolled simultaneously. What is the maximum number of times they

could be rolled without the same pair of numbers appearing twice?

(A) 6 (B) 11 (C) 21 (D) 30 (E) 36

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19. In a Toyota Celica, your speed in miles per hour is calculated by the number of revolutions made bythe tires assuming you keep the standard size P195 60R15 tires on your car. These tires whenmounted have a total diameter of 24 inches. Suppose you hook up your Celica with new wheelsand Z

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26. The city of Savannah decides to dye the water in the Forsyth Park fountain green for next monthsSt. Patricks Day. The 2,000 gallons of water in the fountain already contain 1% green dye left overfrom last year. The city planning committee determines that perfect St. Patricks Day Green will occurif the concentration of dye is 45%. Which of the following is closest to the number of gallons ofwater which will have to be removed and replaced with pure dye in order to reach perfect St. PatricksDay Green?

(A 880 (B 8800 (C 900 (D 90 (E 1000

27. In the accompanying figure AAv AC, BBv AB , CCv CB.Then area of6 AvBvCv =

(A 4 . area 6ABC (B 6 . area 6ABC(C 7 . area 6ABC (D 8 . area 6ABC(E 10 . area 6ABC

28. Find the shortest distance from a point on the graph of y = x to the point (1,0.

(A 1 (B1

2(C

2

2(D

3

2(E 2

29. At a Gymboree class, Judah watches a spherical soap bubble of radius r land on a flat mat andform a hemisphere. Assuming that the volume remains the same, what is the radius of thehemisphere?

(A) 2r (B) r 2 (C) 2 33

r (D) r 23 (E) r

30. The graph of the function at rightdepicts the amount of money y indollars in an automatic teller machine(ATM after x minutes. If x = 0corresponds to 3:00 pm, which of thefollowing is (are correct.I. A withdrawal of $600 was made at

3:20 pm.

II. A deposit of $400 was made at4:00 pm.III. A deposit of $300 was made at

4:10 pm.IV. The largest single transaction is valued

at $400.V. From 3:00 pm until 4:40 pm the overall

monetary difference in the machine was$100

A. I and IIB. IV only

C. III and VD. IV and VE. II, III, and IV

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31. If the probability that a randomly thrown dart lands on red is one m, then an > am.

(A 4 (B 151,200 (C 5,040 (D 210 (E 24

34. All four children in the Jones family have the same birthday. The twins were born two yearsafter their older sister and six years before their younger brother. The standard deviation of theirfour ages (in years) is:(A) 3 (B) 4 (C) 6 (D) 9 (E) 10

35. If 8 = log 3 r1 + log 3 r

12 + log 3 r

14 + . . . + log 3 r

1

2n + . . . then r =

(A 38 (B 8 (C 9 (D 64 (E 81

36. A ship sails from Seaport 1 to Seaport 2 along the equator. Upon reaching Seaport 2, the

navigator determines that the angle the sun makes with the horizontal is 60o at the same timewhen the sun is directly overhead at Seaport 1. Approximating the earth as a sphere with radius4002 miles and assuming that all rays from the sun to the earth are parallel, find the distance theship sailed.

(A 2001 miles (B 2001 2 miles (C 2001 3 miles(D 1334/ miles (E 667/ miles

37. How many distinct arrangements are there using all six letters in LESSER?

(A) 6! (B)6!

4(C)

6!

2(D) 4! (E) none of these

38. ABC is a triangle with mB = 90o and mA = 30o. Points P, Q, and R are on AB, BC and CA,

respectively, and 6PQR is an equilateral triangle. If the length of BC is 4, and Q is the midpoint of

BC, determine the length of PR.

(A 2 2 (B 52 (C 4 (D 7 (E 32

39. The function y = 2csc/x 0, e > 1, making f(x) a strictly decreasing function.Therefore, the maximum value occurs at x = 0.

18. (B) We can think of our functions as permuting the vertices in the following way:

s : (A,B,C )

(A,C,B) r : (A,B,C )

(B,C ,A) t : (A,B,C )

(C,A,B)

Then the correct answer is (b) since(s r s r)(A,B,C ) = s(r(s(B,C ,A))) = s(r(B,A,C )) = s(A,C,B) = (A,B,C ).Note that none of the other functions bring us back to the original position of our triangle:(s t)(A,B,C ) = s(C,A,B) = (C ,B,A)(t s)(A,B,C ) = t(A,C,B) = (B,A,C )(s r s t)(A,B,C ) = s(r(s(C,A,B))) = s(r(C ,B,A)) = s(B,A,C ) = (B,C ,A)(r r)(A,B,C ) = r(B,C ,A) = (C,A,B)

19. (C) Since x is an odd natural number, when divided by six, the only possible remainders are y = 1, 3, or 5.

Therefore y2 = 1, 9, or 25. When each of those values for y2 is divided by 4, the remainder, z, equals 1.

20. (D) Note that the 3 diagonals that connect opposite vertices together all meetin exactly one point. They also divide our hexagon into 6 regions. Each ofthe 6 regions is divided into 4 non-overlapping regions by the remaining diagonals,leaving a total of 24 regions.

21. (C) 15! = 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 = 13 11 72 53 36 211.Therefore, the largest perfect square is 72 52 36 210.

22. (E) From the diagram, the side of the square has length = 1 +

2 + 1 = 2 +

2.

Therefore, the area = (2 + 2)2 = 4 + 42 + 2 = 6 + 42.

"!#

"!#r rdddddd

11

1 1 "!#

"!#r r@@@@@@

1

1

2

2

2

23. (B) Given two solutions a, b we have 0 = (xa)(x b) = x2 (a + b)x + ab. Since the sum of the two solutionsis 11

12and the product of the two solutions is 1

6, one possible quadratic equation would be: x2 11

12x + 1

6= 0

Multiplying by 12 yields another possible equation: 12x2 11x + 2 = 0.

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24. (E) Primes less than 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.Numbers under 30 which are a product of two distinct primes:2 3 = 6, 2 5 = 10, 2 7 = 14, 2 11 = 22, 2 13 = 26,3 5 = 15, 3 7 = 21.There are 7 such numbers.

25. (D) With QT= QV, and P S= P V, we can label the triangle with angles x,y,z,a, and b asThis lead to the following system of equations:

a + b + 90 = 180a + 2y + b + 2x = 360

x + y + z = 180

rrrr

rrrr

rr

P Q

R

rSrVr

T

ra bx

x

y

yz

Solving the first equations yields a + b = 90. Substitute into the second equation to get 90 + 2y + 2x = 360 =2y + 2x = 270 = x + y = 135. Finally, substituting into the last equation yields 135 + z = 180 = z = 45.

26. (B) Since the two roots are r1 and r2, we have (x r1)(x r2) = x2 (r1 + r2) + r1r2 = x2 px + q. So,p = r1 + r2 and q = r1r2.

p2 = (r1 + r2)2 = r21 + 2r1r2 + r

22, then r

21 + r

22 = p

2

2r1r2 = p

2

2q.

27. (C) Since N1, N, and N+ 1 are consecutive positive integers, one of them must be divisible by 3, and hencetheir product is divisible by 3. If N = 2 then the product is 6 and the other answers are false.

28. (A) cos(2 sin173

) = cos2 sin2 where = sin173

. From the right triangle below, we have cos =23

and sin =73

. This gives

cos(2 sin173

) = cos2 sin2 = 29 7

9= 5

9

23

7

29. (C) A fair coin tossed until a head appears on an odd-numbered tossed can happen in the following ways:

H, TTH, TTTTH, TTTTTTH, ...

The probability of getting one of those scenarios is 12

+ 18

+ 132

+ ... This is a geometric series with a = 12

and

r = 14

whose sum = 1/211/4 =

1/23/4

= 23

.

30. (D) The number of ways to arrange 8 letters = 8! but, since some letters are indistinguishable (two Ns andthree As), not all arrangements are distinct. For any given arrangement, there are 2! = 2 ways to arrange thetwo Ns and 3! = 6 ways to arrange the three As, resulting in 2 6 = 12 indistinguishable arrangements.Overall, the total number of distinct arrangement must be 8!

2!3!

= 8

7

5

4

3 = 3360

31. (A) P(able to cross) = P(only 4 rocks are needed) + P(all 5 rocks are needed)

P(only 4 rocks are needed) = P(4 successes in a row)P(all 5 rocks are needed) = P(1 failure and 4 successes) 4Multiply by 4 to represent the possible locations for the single failure.

P(only 4 rocks are needed) + P(all 5 rocks are needed) = ( 23

)4 + 4(23

)4( 13

) = 1681

+ 64243

= 48243

+ 64243

= 112243

.

32. (E) 100! has 20 multiples of 5 and 4 multiples of 25. Hence, the exponent on 5 in the prime factorization is20+4 = 24 and the exponent on 2 is > 50 since there are 50 even numbers. Each 25 product results in a zeroand there will be 24 such products, so 100! will end in 24 zeros.

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33. (A) Let a equal the length of the altitude from the vertex at the right angle.From the picture, Area = 1

2(4)(3) but area also equals 1

2(5)(a).

So, 12 = 5a and a = 125

= 2.4 is the length.

&&&&&&&&

a3

5

434. (D) Draw an isosceles triangle with height b and base 2a. The third side(the base)

must be a whole number strictly between 0 and 20 making a = 12

, 1, 32

, . . . , 9, or 192

.However, there are two further restriction involving a.

a2 + b2 = 100 and Area = 12

(2a)(b) = ab must be a whole number. But with the first equation, b =

100 a2,and therefore the area = a

100 a2. Only two assignments of a will satisfy the area is a whole number

condition: a = 6 or a = 8. Regardless, the area is (6)(8) = 48 square inches.

eeeeee

a a

b10 10

35. (B) log3(x + 1) log9 x = 1. Then, 9log3(x+1)log9 x = 91 = 9log3(x+1)

9log9 x= 9 = (32)log3(x+1)x = 9 =

32log3(x+1) = 9x = 3log3(x+1)2 = 9x = (x + 1)2 = 9x = x2 + 2x + 1 = 9x = x2 7x + 1 = 0 =x = 7

4942

= x = 745

2. Both values are positive so both satisfy the original equation.

36. (B) Let x = age of the 11th person, The sum of the ages for the first 10 people is 10(20 .5) = 205. The

sum of the ages for the 11 people is given by 205 + x and the average is205+x

11 = 21. Multiply by 11 to get205 + x = 231 = x = 26

37. (D) sin + cos =32

= (sin + cos )2 = (32

)2 = sin2 + 2 sin cos + cos2 = 32

=1 + 2 sin cos = 3

2= 2sin cos = 1

2= sin2 = 1

2= 2 = 30, 150, 390,... = = 15, 75, 195,...

The smallest positive angle is 15.

38. (A) Quadrilateral ABCD with CD AB, measure of angle D = 45, measure of angle C = 120, AD = 122and CD = 27 looks like(not drawn to scale)

A B

CD

122

27

45 120

=

A B

CD12

12

15

12

15 4

3

12230t

t

Therefore, AB = 15 + 4

3 and Area = 12h(b1 + b2) =12(12)(27 + 15 + 4

3) = 6(42 + 4

3) = 252 + 24

3.

39. (B) There are 6! = 720 ways to position the 6 numbers. 24 of these arrangements result in a magic triangle,as we show below, for a probability of 24

720= 1

30. Since some two numbers must be on the same edge as 6, the

smallest possible edge sum for a magic triangle is 1 + 2 + 6 = 9, as in the given triangle. Similarly, since sometwo numbers must be on the same edge as 1, the largest possible edge sum for a magic triangle is 1+5+6 = 12.Thus, if E represents the edge sum for a magic triangle, then 9 E 12. Notice that if we add the three edgesums in a magic triangle, each vertex number is added twice, so that 3E = 21 + V, where V is the sum of the

vertex numbers. Thus, if E = 9, then V = 6 and the three vertex numbers must be 1, 2, and 3, as in the giventriangle. There are a total of 3! = 6 magic triangles with E = 9, obtained by rotating or reflecting the giventriangle. If E = 10 then V = 9 and the vertex numbers must be 1, 3, and 5 in order to get a magic triangle.Once again there are 6 arrangements of these vertex numbers; each arrangement determines a unique magictriangle with E = 10. If E = 11 then V = 12 and the vertex numbers must be 2, 4, and 6 in order to get amagic triangle; this leads to 6 more magic triangles. Finally, if E = 12 then V = 15 and the vertex numbersmust be 4, 5, and 6; this yields 6 more magic triangles, for a total of 24.

40. (C) When Greg first leaves his home, he starts from a stopped position and gradually increases his speed downthe hill. So, the distance from his house starts out increasing very slowly. Upon coming to a stop at the trafficlight, he must reduce his speed slowly, so that his distance is again increasing very slowly. ( c) is the only graphthat represents this scenario.