ALGEBRA TEST
(SECOND COURSE)
IUN
2002 STATE HIGH SCHOOL MATHEMATICS CONTEST
1)
If | x +  | < - 2, then
(A) x < - 2 + 
(B) x > - 2 + 
(D) x is not a real number
2)
3)
4)
(C) x < - 2 - 
(E) x is any real number
The point of intersection of the lines x - y = a and x + y = 2a is
 a 3a 
(A)  , 
2 2 
a a
(B)  , 
 2 2
 3a a 
(D)  , 
 2 2
(E) none of these
 3a a 
(C)   , 
 2 2
The minimum number of points of intersection of a cubic polynomial in x and a
quadratic polynomial in x is
(A) 0
(B) 1
(C) 2
(D) infinitely many
(E) can not be determined
If m and n are integers such that 2m - n = 3, then what will m - 2n equal?
(A) -3 only
(B) 0 only
(D) any integer
(E) none of these
(C) an integer multiple of 3
5)
Six identical cardboard pieces are piled on top of one another, and the result is
shown in the diagram.
The third piece to be placed is:
6)
(A) A
(B) B
(D) D
(E) E
AB is a diameter of a circle of radius 1 unit. CD is a chord perpendicular to AB
that cuts AB at E. If the arc CAD is 2/3 of the circumference of the circle, what is
the length of the segment AE?
(A) 2
(B) 3
3
(D) 3
7)
(C) C
2
(C) 
2
2
(E) none of these
It takes Chris 1 12 times as long to perform Task X as Allison. It takes Allison 1 23
times as long to perform Task X as Barry. If Chris, Allison, and Barry work
together, they can perform Task X in 3 hours. How long does it take Chris to
perform Task X working alone?
(A) 43.2 minutes
(B) 6 hours
(D) 10 hours
(E) 15 hours
(C) 25 hours
8)
9)
10)
11)
How many real values of x satisfy the following equation:
200210  2002 20  2002 30  2002 40  (2002 x ) x
(A) 0
(B) 1
(D) can not be determined
(E) none of these
(C) 2
The perimeter of the triangle that is bounded by the coordinate axes and the line
3
3
y  x  is:
4
4
(A) 2
(B) 1
(D) 3
(E) none of these
(C)
5
4
The year 2002 is special because it is a palindrome—it reads the same forwards
and backwards. How many four digit numbers are palindromes?
(A) 8100
(B) 200
(D) 100
(E) 90
(C) 180
How many prime numbers p satisfy 114  p  126 ?
(A) 0
(B) 1
(D) 3
(E) 4
(C) 2
12)
The third term of an arithmetic sequence is a and the seventh term of an
arithmetic sequence is b . Find the first term of the sequence.
(A)
ba
8
(D) b  2a
13)
14)
15)
(B)
3a  b
2
(E)
5a  b
4
(C)
b  2a
3
Each month after he bought a new computer John used 20% of the hard disk
space still available. After three months only 640KB remained. How many KB
did his computer have when it was brand new?
(A) 1400
(B) 1500
(D) 1200
(E) 1250
(C) 1150
Bill sets out on a journey. For the first one third of the distance he drives at
30mph; he drives the second third at 40mph; he drives the last third of the
distance at 50mph. What is his average speed?
(A) 35mph
(B) 38.3mph
(D) 42.4mph
(E) 43mph
(C) 40mph
Find integer b so that one of the solutions of the equation
x 3  (b 2  4b  3) x 2  bx  3b  0 will be x  2 .
(A) b =  4
(B) b  2
(D) b  3
(E) b  4
(C) b  2
16)
17)
18)
Solve for x :
1 5 
(A) 

 2 
1  5 
(B) 

 2 
1 5 1 5 
,
(D) 

2 
 2
1  5 1  5 
,
(E) 

2 
 2
1 5 1 5 
,
(C) 

2 
 2
How much water must we evaporate from 40 liters of a 30% salt solution to
obtain a 50% salt solution?
(A) 16 liters
(B) 25 liters
(D) 22 liters
(E) 24 liters
Solve for x :
(A) 
(D)
19)
x2  5  x 1  0
3
(C) 15 liters
20  10 x  5  15
(B)
0
(E)
5
(C)
 2
The ratio of men to women in an algebra class is 3 to 5. If there are a total of 40
students in the class, how many are women?
(A) 15
(B) 16
(D) 25
(E) 35
(C) 24
20)
A function f (x) is defined recursively by f ( x  1)  6  f ( x) . If f (1) 
1
, find
2
the value of f ( f (2)) .
(A) 108
(D)
21)
3
2
(B) 18
(C) 3
(E) 36
One of the bases of triangle ABC has length x-1, and the length of the hypotenuse
is x 2  2 x  10 . Find x so that the area of the triangle ABC is 6.
22)
(A) 1
(B) 2
(D) 4
( E) 5
(C) 3
Compute the following sum: log 2
1
2
3
15
 log 2  log 2    log 2
2
3
4
16
(A)  4
(B)
log 2 15
2
(D)  1
(E)
(C)  2
23)
24)
25)
26)
The quarter circle shown below has center C and radius 10. If the perimeter of
rectangle CPQR is 28, what is the perimeter of the shaded region?
(A) 16+5 
(B) 18+20 
(D) 16 + 20 
(E) none of the above
(C) 16+10 
What is the units' digit of
1 + 9 + 9 2    9 2000  9 2001  9 2002 ?
(A)0
(B) 1
(D) 9
(E) none of these
(C) 3
Let log b x 3  4   log b 2 x 2  1  1 . Find b so that x = 2 is a solution.
(A) 4
(B) 5
(D)13
(E) 36
What is
(C) 9
1 1 1 1 1 1 1
      ?
i i 2 i3 i 4 i5 i6 i7
(A)-1
(B)  i
(D) i
(E)1
(C)0
27)
28)
29)
30)
A store raises the price of an item by 15 % , then lowers the price by 15 % .
Compared to the original price, the new price has
(A) decreased by 2.25 %
(B) decreased by 5 %
(D) increased by 2.25 %
(E) increased by 5 %
(C) stayed the same
Given a square with diagonal d , area A , and perimeter P , find
P 2  16 A  16d 2 .
(A) 2A2
(B) 2A2
(D) 2P 2
(E) 2d 2
(C) 2P2
How many times will the graph of y  x  3  1 intersect the graph of y  3 ?
(A) 0
(B)1
(D) 3
(E) 4
(C) 2
What is the vertex of the parabola y  5 x 2  8 x  3 ?
 8 31 
(A)  , 
5 5 
 8 31 
(B)   ,  
 5 5
 4 31 
(D)   ,  
5
 5
 4

(E)   , 6 
 5

 8

(C)   , 3 
 5

31)
32)
A small circle is inscribed into the large one so that it is passing through the
center of the large one. Find proportion of the area of the small circle to the are
shaded on the picture:
A) 1/4
(B)1/3
(D) 2/7
(E)3/(2)

What is 1   2
(A)17
(D)
1
9

2 2
(C)1/
?
16
25
9
(E)
4
(B)
(C)
9
25