Download Test - Mu Alpha Theta

Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 1
Round 1 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
1.
Evaluate: lim(2 x2  x3  16)
2.
What is the greatest common factor of 189 and 588?
3.
Using two subdivisions, approximate the value of
4.
If 2 x  8 y1 and 9 y  3x9 , find the value of x + y.
x 2
Round 2 of 20

4
0
x 2 dx using the Trapezoid Rule.
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 2
Round 3 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
5. Find the value of
6.
What is the remainder when 252005 is divided by 13?
5
7.
8.
dy
at the point (9, –6) if x 2  xy  y 2  63 .
dx
9
0
Evaluate: 2 3 6
5 6 7
If lim  f ( x)  g ( x)   11 and lim  ( f ( x))2  ( g ( x)) 2   61 , evaluate lim  f ( x) g ( x)  .
x c
x c
x c
Round 4 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 3
Round 5 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
9. What is the sum of the squares of the roots of P( x)  x5  2 x 4  6 x3  3x 2  4 x  4 ?
10. Find the width of the largest x-interval where the graph of y  4 x 2  x 4 is concave up.
11. A circle’s radius is increasing at a rate of 4 meters per second. How fast, in square meters
per second, is the area of this circle increasing at the moment the circumference of the circle
in meters is numerically twice its area in square meters? Express your answer in terms of
.
12. A sequence an is called a root-mean sequence if (an ) 2 forms an arithmetic sequence. If bn
is a root-mean sequence with b1  1 and b2  2 , what is the smallest m > 2 such that bm is
an integer?
Round 6 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 4
Round 7 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
13. In a circle, AB and CD are perpendicular chords that intersect at E. If AE = 2, EB = 12, and
CE = 4, find the distance from E to the center of the circle in simplest radical form.
14. Two joggers are running around a circular track in opposite directions at constant speeds.
One jogger runs around the track in 56 seconds, meeting the other jogger every 24 seconds.
How many seconds does it take for the other jogger to run around the track?
 x 3  4k
15. What value of k will make f ( x)  
 kx  10
x2
x2
16. If F ( x)  x 2  x  3 and G( x)  4 x 2 , evaluate
continuous for all real x?
d
 F (G ( x))  at x = 1.
dx
Round 8 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 5
Round 9 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
17. If y  (2 x  3) 4 , find
d4 y
.
dx 4
18. A deck of ten cards consists of two 1s, two 2s, two 3s, two 4s, and two 5s. A pair of cards
with the same number is removed from the deck without replacement. If these cards are not
returned to the deck, what is the probability that the next two randomly selected cards also
have the same number in them?
 x
19. What is the area between the curve y  2sin x  3cos   and the x-axis on the interval
2
0  x  ?
5 
 1 3 1 0 
 0  , and ABC  6  , how many elements does B have?
C

20. If A  
,

 2
 
 3 0 2 3
 
1 
Round 10 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 6
Round 11 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
21. Right triangle ABC has AB as its hypotenuse. If CB = 12, AC = 6, and D is a point on AB
such that CD bisects C, find the length of CD in simplest radical form.
22. The equation of the line tangent to the graph of y  f ( x) at x  a is y  16ax  15a 2 .
Evaluate: lim
x a
f ( x)  a 2
x2  a2
23. A right triangle has perimeter 56 while the sum of the squares of the side lengths equals
1250. What is the area of the triangle?
24. Let f be continuous on 2  x  8 . If
what is the value of

6
4

8
2
f ( x) dx  14 ,
f ( x ) dx ?
Round 12 of 20

6
2
f ( x) dx  5 , and

8
4
f ( x) dx  2 ,
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 7
Round 13 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
25. Solve for x:
x  10 x  1  24  9
26. An integer N has units digit 5, tens digit 2, and hundreds digit x. What is the sum of all
possible values of x if N is a perfect square?
27. Let f be continuous on 4  x  2 with f (4)  3 and f (2)  2 . For how many numbers
c from the set {–, 0, e,
f ( x)  c ?
2 , , –1} can we be guaranteed to find at least one solution to
28. The position of a particle moving along the x-axis at time t > 0 is given by x(t )  (ln t ) 2 .
For what time value is the velocity of the particle a maximum?
Round 14 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 8
Round 15 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
29. How many positive two-digit primes are there such that when the digits of the two-digit
prime are reversed the resulting number is also prime?
30. Suppose that f is an odd function with continuous third derivative. What is the sum of the
missing entries in the following table?
x
1
0
–1
f ( x)
18
5
f ( x)
f ( x)
84
–6
31. Let f ( x)  x 4  6 x  3 and g ( x)  f 1 ( x) for x  1.5 . What is the slope of the normal line
to the graph of y  g ( x) at (7, 2)?
32. The vertices of a three-dimensional solid are the centers of the faces of a cube with edge
length 6. What is the volume of this solid?
Round 16 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 9
Round 17 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
33. If
dy
 x 2 y 2 with y = 3 when x = 2, find the value of y when x = 0.
dx
34. If five lines are drawn in the plane, what is the maximum number of acute triangles that can
be formed whose sides are on the lines?
35. Let R be the set of all (x, y) where 0  xy  1 and

12
solid produced when R is revolved about the x-axis?
x

4
. What is the volume of the
36. What is the harmonic mean of the positive integral factors of 28? The harmonic mean of a
set of numbers is the reciprocal of the average of the reciprocals of the numbers.
Round 18 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
Round 10
Round 19 of 20
Mu Alpha Theta National Convention: Hawaii, 2005
School Bowl – Mu Division
37. Two numbers are chosen uniformly at random between 0 and 1. What is the probability the
square roots of the numbers sum to less than 1?
38. An ellipse’s shape changes with time t such that for t  0 , its equation is given by
x2
y2

 1 . What is the closest the center of this ellipse will ever be to one of its
t 2  30 8t  5
foci?
39. How many integers between 1000 and 8000 have exactly four distinct digits?
40. If cos  
10
3 10
and sin  
, find the value of sin(4 ) .
10
10
Round 20 of 20