State Mathematics Algebra II Contest

May 3, 2001

1. If the length and width of a rectangle were increased by 1, the area would be 84. The area
would be 48 if the length and width were diminished by 1. Find the perimeter P of the original
rectangle.

a. 10 < P < 20 b. 20 < P < 30 c. 30 < P < 40

d. 40 < P < 50 e. none of the above

2. In the sequence K , w , x , y , z , 0 , 1 , 1, 2 , 3 , 5 , 8 , K , each term is the sum of the two terms to

its left. Find the value of 2 ⋅ (w + x + y + z ).

a. 2 b. 1 c. 0
d. −1 e. −2

2 x + 4 − 2 ( 2 x+ 1 )
3. Simplify:
2 (2 x + 3 )
3 3
a. b. c. 2x
8 4
x 3 3
d. e.
4 2

4. A coin is biased so that the probability of obtaining a head is 0.25 . Another coin is biased
where the probability of obtaining a head is 0.6 . If both coins are tossed, find the probability
of obtaining at least one head.

11 7 3
a. b. c.
20 10 10
9
d. e. none of the above
10

5. If a polynomial, F(x), has real coefficients with zeros at 2, 1 + i , 3 − i , then this polynomial
must have a degree of :

a. at least 5 b. exactly 6 c. exactly 3

d. at least 6 e. none of the above

1
6. The sum of the coefficients of the quadratic equation whose graph passes through ( 0, − 1 ),
( 1, 4), and ( 2,13) is:

a. 0 b. 2 c. 3
d. 4 e. not unique

+ 4x − 9x + 6
= 4 9x
2 2
7. The sum of the solutions for the equation 86 x is:

a. 1 b. −2 c. 0
2 16
d. e.
5 15

8. Mary has d liters of punch that is d% grape juice. How many liters of grape juice must she add
to make the punch 3d% grape juice?

d2 2d 2 d
a. b. c.
100 − 3d 100 − 3d 100 + 3d
3d 2
d. e. none of the above
100 + d

9. The equation x + 3 − 2 = p, where p is a constant integer has exactly three distinct

solutions. Find the value of p.

a. 0 b. 1 c. 2
d. 3 e. 4

10. A particular question on a multiple-choice test that had 4 possible choices was omitted by 1/3
of the students taking the test. Of those answering the question, 7 selected choice A, 9 selected
B, 5 selected C, and 3 selected D. How many students took the test?

a. 33 b 27 c. 24
d. 42 e. 36

3 h (x ) + 4 2
11. Let h ( x + 1) = for x = 1, 2 , 3 , L and h (1) = − , find h (3) .
3 3

a. 1 b. 0 c. 5
d. 2 e. 6

2
12. If x + y = 5 and x 2 + 3xy + 2 y 2 = 40 , find the value of 2 x + 4y .

a. 16 b. 17 c. 19
d. 18 e. none of the above

13. For what value of m will the triangle formed by the lines y = 5 , y = mx − 6 , and y = − mx − 6
be equilateral?

a. 2 b. 3 c. 5
3 5
d. e.
2 3

14. If y ∗ = y 2 − 1, find ( y ∗ )∗ .

a. y 4 − 2y 2 b. y4 + y2 c. y4 − y2
d. y 2∗ − 1 e. none of the above

~
15. Define N( A ) as the number of elements in set A and let A be the complement of A.
Let A, B and C be sets for which the following are true: A ⊂ B , 2 N ( A) = N ( B ) ,
~ ~
2 N ( B ∩ C ) = N (C ) , N(A) = 30, N(C) = 40, and N ( A ∩ C ) = 0 . Find N ( A ∩ B ∩ C ) .

a. 10 b. 15 c. 20
d. 25 e. 5

16. Let > x < denote the largest prime number less than x and 〈〈x〉〉 denote the smallest prime
number greater than x. Find the value of the following expression:
41 + > 33 < − 〈〈33〉〉 + 〈〈 > 23 < 〉〉

a. 28 b. 38 c. 35
d. 25 e. 58

17. Let g ( x) = ax 7 + bx 3 + cx − 7 , where a , b , and c are constants. If g (− 5) = 5 , then g (5) is

a. −7 b. −5 c. −15
d. −19 e. not enough information

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18. The altitudes of a triangle are three distinct integers, the larger two of which are 12 and 66.
Find the length of the shortest altitude. Which of the following intervals contains the integer
solution?

a. (0,3) b. [3,6) c. [6,9)

d. [9,12) e. none of the above

19. M is a two digit number whose value is N when the digits are reversed. The difference M − N
is a positive perfect cube. Which of the following statements is true.

a. The number N can end in any digit other than 5.

b. The number N does not exist.
c. There are exactly seven values of N.
d. There are exactly ten values of N.
e. None of these

20. Given A (1, − 2) , B (5, 1) , and C (−2 , 2) , find the equation of the angle bisector at A.

a. 5x − y = 7 b. 7x − y = 2 c. 7y − x = 2
d. y=x+3 e. none of the above

3x − 5 x+ a
21. Let g ( x) = . The inverse of g can be written in the form . Find the product of
2x + 1 bx + c
a, b and c.

a. − 40 b. − 30 c. 30
d. − 35 e. − 45

22. If 1 ⋅ 10a + 2 ⋅ 10 b + 3 ⋅ 10 c + 4 ⋅ 10 d = 24,130 and a ≠ b ≠ c ≠ d , then what does

a b c d
+ + + equal?
2 4 8 16

5 5 3
a. 2 b. c. 2
16 16 16
7 3
d. 1 e. 2
16 8

4
23. Solve the following system of equations simultaneously for x where x > 0 .
y = 11 x + 3x 2
y = 11 x 2 − 3x
What is the value of x?
7
a. 1 b. 2 c. 1+
2
3 7
d. e.
11 4

24. If a + b = 3 and a 2 + b 2 = 6 , find the numerical value for a 3 + b 3.

a. 13.5 b. 18 c. 9
d. 9.5 e. 243

25. Which of the following expressions is equivalent to log 4 (8 ⋅ 2 x ) ?

x +3
a. 64x b. 2x + 6 c.
2
d. 2x e. 16x

26. Determine the approximate percentage of the interval [− 5 , 15] for which the inequality
7
x>4− is satisfied.
x+4

a. 55 % b. 65 % c. 60 %
d. 45 % e. none of the above

27. The difference between two positive numbers is 4 3 . The product of the two numbers is 8.
What is the absolute value of the difference of their reciprocals?

3 3 3
a. b. c.
2 3 4
d. 2 3 e. 3

28. If the discriminant of ax 2 + 2bx + c = 0 is zero, then which of the following statements is
true about a, b, and c?

a. they form an arithmetic progression

b. they are unequal
c. only b is negative
d. they are all negative numbers
e. they form a geometric progression

5
 ex − e −x
29. Solve = 2 for x, where x > 0.
2
a. ln ( 2 + 5 ) b. ln ( 2 ± 5 ) c. ln 2 + ln 5
d. 4 e. none of the above

30. If a, b, and c are integers such that (

3
)( )
4 + 3 2 − 2 a 3 4 + b 3 2 + c = 20 , find the value of
a + b − c.

a. 10 b. 18 c. 6
d. 23 4 e. 3
4

31. Find the sum of all proper fractions whose denominators are less than or equal to 100. (Include
unreduced fractions in the sum.)

a. 2075 b. 1050 c. 1175

d. 1275 e. 2475

c
32. Given a right triangle with sides of length a, b, and c and area, a 2 + b 2 − c 2 . Find , the
b
ratio of the legs of the right triangle.
3
a. 1 b. c. 4
2
1
d. e. none of the above
4

33. Lynn purchased two candles of equal length. One of the candles will burn up completely in 5
hours, while the other candle requires 7 hours to burn up completely. If the candles are lit at
the same time, approximately how long will they burn before one of the candles is twice the
length of the other?

a. 3.9 hr b. 3.2 hr c. 2.8 hr

d. 3 hr e. none of the above

34. A polynomial P(x) has remainder − 5 when divided by x + 1 and remainder 7 when divided by
x − 5 . What is the remainder when P(x) is divided by the product of x + 1 and x − 5 ?

a. 2x + 3 b. 2x − 3 c. 3x − 2
d. 35 e. − 35

6
35. Find the distance between the origin and the center of the circle
x 2 + y 2 − 6 x − 4 y + 11 = 0

a. 13 b. 13 c. 2 13
d. 5 2 e. 2 5

If log 2 (a 3 b) = x and log 2   = y , what is the value of log 2 a ?

3a
36.
 b 
x+y x+y
a. − log 2 4 3 b. c. log 2 4 3
4 4
x− y x−y
d. e. − log 2 4 3
4 4

37. Find the intersection of the lines 2 x + 3 y = A and x + 2 y = B .

a. (3B − 2 A, A − 2 B ) b. (3 A − 2 B , B − 2 A) c. (2 A − 3B , 2 B − A)
(
d. 12 A, 12 B ) e. (3B − 2 A, 2 B − A)

38. What is the sum of the numerical coefficients of the expression( a + b) 8 ?

a. 218 b. 128 c. 250

d. 256 e. none of the above

x +1
. Solve f −1   = 3 .
1
39. Given f ( x ) =
x −1 x 

a. x =1 b. x=2 c. −3
1 1
d. x= e. x=
2 3

40. In the xy-plane, how many lines whose x-intercept is a positive prime number and whose
y-intercept is a positive integer pass through the point (4, 3)?

a. 0 b. 1 c. 2
d. 3 e. 4