CALIFORNIA STATE UNIVERSITY, BAKERSFIELD
MATHEMATICS FIELD DAY 2004
Individual Medley, Junior – Senior Level
For each of the following questions, blacken the appropriate circle on the answer
sheet. Each correct answer is worth four points. One point is deducted for each
incorrect answer. An unanswered question is given zero points. Note that
random guessing may adversely affect your score.
You have 50 minutes to complete the Examination. When the exam is over, give
your answer sheet to the proctor.
Junior-Senior Individual Exam – March 2004
1.
The sum of the reciprocals of the solutions to the equation ax 2  bx  c  0 are:
1 1

a b
b
d)
c
a)
2.
3.
4.
If f  x  
b
c
x3
and f 1  x  denotes the inverse of f  x  then f  f 1  4   is:
x2
c) 11
b) 2
d) 4
e) none of the above
3
What is the value of C such that the line 3 y  5 x  C  0 has a y-intercept of 4,
a) -20
b) 4
d) 4
e) 12
c) 4
5
3
A card is drawn at random from a standard deck of 52 cards. What is the probability that the
card is a King or a Heart?
1
52
d) 17
52
b) 1
13
e) 3
4
c) 4
13
The angle between the hour hand and minute hand on a standard wall clock at 7:30 is:
a) 30
d) 52.5
6.
c) 
a) -4
a)
5.
c
b
a
e) 
b
b) 
b) 37.5
e) 60
c) 45
What must b be so that when x3  bx 2  4 is divided by x  2 the remainder is zero?
a) -2
d) 2
b) -1
e) none of the above
c) 1
Junior-Senior Individual Exam – March 2004
7.
How many distinct real roots does the equation x 4  3x3  2 x 2  0 have?
a) 0
d) 3
8.
b) 1
e) 4
For points  x, y  on the curve y  1  x 2 , the maximum value of the sum of x and y is
b) 5
a) 1
4
e) not enough information
d) 
9.
c) 2
c) 3
2
A point P is randomly selected from the rectangular region with vertices  0, 0  ,  3, 0  ,
 3,1 ,  0,1 .
 5, 0 ?
1
a)
d) 5
2
6
What is the probability that P is closer to the origin than it is to the point
b) 3
c) 2
5
3
e) 1
10. The area of the region enclosed by the graphs of y  3  x and y  x  3 is:
a) 9
2
d) 18
b) 9
e) 36
1
11. The domain of the function f  x  
a)
d)
 ,  
 0,  
c) 9 2
b)
x 2  x3
is
 1,1
c)
 0,1
e) none of the above
12. Two candies are drawn from a bag that contains 4 blue, 5 green, and 6 red candies. The
probability that at least one of the candies is not red is:
a) 12
d) 3
35
5
b) 13
e) 6
35
c) 2
5
7
Junior-Senior Individual Exam – March 2004
13. Let f  x   x2 , g  x   1  x , and h  x   3x . The function s  x   3  6 x  3x 2 can be
written as the composition
a) s  f g h
d) s  h f g
b) s  g f h
e) s  h g f
14. The function f  x  
a)
d)
ln 1  x 
1 x
 , 1
 0,  
c) s  g h f
has domain
b)
e)
 ,1
1, 
c)
 0,1
15. Ben and Jerry’s Homemade, Inc. sells 20 different flavors of ice cream. How many 3-dip
cones are possible if order of flavors is to be considered and no flavor is repeated:
20!
17!3!
d) 3! 203
20!
17!
e) 320
a)
b)
c) 203
16. Each of the following lists has an average of 12. Which list has the largest standard
deviation?
a)
b)
c)
d)
e)
0
0
0
0
0
0
1
3
6
12
20
12
12
12
12
20
23
21
18
12
20
24
24
24
24
17. The center of the circle x 2  y 2  8 y after it is rolled one revolution in the positive direction
down the x-axis is at:
a)
d)
 4, 8
8 , 8
b)
e)
 4, 8 
16 , 8
c)
8 , 4
Junior-Senior Individual Exam – March 2004
F  n  1 F  n  2   1
18. The function F  n  is defined so that F  n  
F  n  3
for n  4 . If
F 1  F  2  F  3  1 then F  6  is equal to:
a) 2
d) 8
b) 3
e) 11
c) 7
19. Niki is four years older than Nathan, and Nathan is half of Nancy’s age. The total of their
ages is 60 years. How old is Niki?
a) 14
d) 23
b) 18
e) 28
c) 19
1
5
20. Solve for x: log 2 4  log 2  x  
2
2
b) 1
a) -2
2
e) none of the above
d) 2
c) 1
2
21. The domain of the function ln  6  x  2 x 2  is:
a) x  2 or
d) 2  x  3
x3
2
2
b) x  3
x2
c) 3  x  2
2
1
1
22. Simplify the complex fraction:
1
x
y
1
1
1 y
1 y
x
d)
2y  x
or
e) 0  x
1
a)
2
x
y
yx
yx
x
e)
2y  x
b)
c)
yx
yx
Junior-Senior Individual Exam – March 2004
23. The graph of the function x  y  1 is symmetric about:
a)
b)
c)
d)
e)
The x-axis
The y-axis
The origin
Both x-axis and y-axis
The x-axis, y-axis, and the origin
24. The area of the triangle formed by the lines x  y  0 , x  0 , and 2 x  3 y  30 is:
a) 15
d) 75
b) 30
e) 90
c) 45

 7 
25. The sin  tan 1    is
 24  

a) 24
d) 24
25
25
b) 7
c) 7
25
25
e) can not be determined
Junior-Senior Individual Exam – March 2004
CALIFORNIA STATE UNIVERSITY, BAKERSFIELD
MATHEMATICS FIELD DAY 2004
Team Medley, Junior-Senior Level
Each correct answer is worth ten points. Partial credit may be given. An
unanswered question is given zero points.
No calculators are allowed. You have 50 minutes to complete the Exam. When
the exam is over, give only one set of answers per team to the proctor.
Elegance of solutions may affect score and may be used to break ties.
Junior-Senior Team Exam – March 2004
1.
If the solution set of the inequality x3  px 2  qx  r  1 is x : x  1, 2  x  3 . Find p,
q, and r.
2.
The sum of an infinite geometric series with common ratio r is 2
the geometric series is 4
3.
27
and the second term in
3
. What could the first term in this series be?
The parabola y  x 2  m and the line with positive slope y  mx intersect only at one point.
What are the coordinates of the point of intersection?
Junior-Senior Team Exam – March 2004
4.
Find the dimensions of the rectangle with maximum area that can be inscribed (see Figure
below) between the curve y  9  x 2 and the x-axis?
5.
The square ABCD is shown in the figure below. The points E and F bisect the line segments
AB and BC, respectively. What is the area of the triangle AGD in terms of the length AB?
Junior-Senior Team Exam – March 2004
6.
What is the difference 10106  1010102 in base 4?
7.
The figure below shows a circle with radius r. Find the circumference of the circle given
that the chord AB (see Figure) has length 6.
8.
Solve for x.
1
1
2
3
4
5
x
1

2
1
3
4
5
6
x
Junior-Senior Team Exam – March 2004
9.
A lighthouse is on an island (A), 4 miles off shore from the nearest point O of a straight
beach; a store is at point B, 4 miles down the beach from O. If the lighthouse keeper can
row 4 miles per hour and walk 5 miles per hour, he should proceed to some point C on the
beach between O and A to get from the lighthouse to the Store in the least possible time?
[See Figure below]. Find C.
10. The sequence a  n, k  is defined by the relationship a  n  1, k   a  n, k 1  a  n, k  for all
integers n and integers k  1, 2, 3, ... n , where a  n, 0  1 for every value of n and a 1,1  1 .
What is the value of
a  6, 3
6
 a  6, k 
?
k 0
Junior-Senior Team Exam – March 2004