Download Sophomore Olympiad

Senior Olympiad
1. If x  0 , then
3 4
x equals:
A) x
B) 7 x
C) 12 x
D) 64 x
E) none of these
2. If x  0 , then
A)
B)
C)
D)
E)
 3x   2x
2 3
2
 6x 
2
equals:
 18x 7
 18x 6
 6x 9
6x 9
none of these
3. A factorization of 8 x 3  27 is:
A)
B)
C)
D)
E)
2 x  33
2 x  34 x2  12 x  9
2 x  34 x 2  6 x  9
2 x  34 x2  6 x  9
none of these
4. If the area of a circle is  square inches, then the length in inches of the radius of the
circle is:
A)
B)
C)
D)
E)

2


2
none of these
x
3
4
5
5. If      , then x equals:
5
4
3
2
2

3
2
3
3
2
none of these
A) 
B)
C)
D)
E)
6. An equation for the line that passes through the point 2, 3 and is parallel to the line
x  4 is:
A)
B)
C)
D)
E)
x2
x3
y2
y3
none of these
7. If 3, 2 is the midpoint of the line segment joining x, y  and 5, 1 , then y equals:
A)
B)
C)
D)
E)
1
3
4
5
none of these
1  2 x for x  1
8. If f ( x)   2
, then f (5) equals:
for x  1
 x
A)
B)
C)
D)
E)
 34
 17
9
25
none of these
9. The distance between the points  1, 4, 2 and 3, 1, 4 is:
A)
B)
C)
D)
E)
29
9
29
3
none of these
10. If f ( x)  3x  1  5 , then f ( x  1)  f ( x) equals:
A)  5
B) 3
C) 3x  1  3x  4  10
D) 3x  4  3x  1
E) none of these
11. If x  a  bi is a complex number such that x 3  11  2i and x 2  3  4i where
i   1 , then a  b equals:
A)  3
B)  1
C) 3
D) 4
E) none of these
12. If x  1 and log 3 x  1  log 3 2 x  1  2 , then x equals:
A)
B)
C)
D)
E)
2
5
1
2
5
2
none of these
13. The smallest x  0 for which y  sin (2 x) attains its minimum value is:
A)
B)
C)
D)
E)

4

2
3
4
3
2
none of these
14. If x and y are real numbers such that xy  0 , then the point x, y  must be in
quadrant:
A)
B)
C)
D)
E)
II or IV
II or III
I or III
I or IV
none of these
15. If sin x 
A)
B)
C)
D)
E)
12
12
and tan x  , then cos x equals:
13
5
5
13
64
144
144
65
13
5
none of these
16. The area of the region enclosed by the graph of x  y  2 is:
A)
B)
C)
D)
E)
2
4
6
8
none of these
17. An equation for the graph obtained by shifting the graph of y  2 x  5 three units to
the left is:
A)
B)
C)
D)
E)
y  2x  8
y  2x  1
y  2 x  11
y  2x  2
none of these
18. The period of y  sin x is:
A)
B)
C)
D)
E)
0

2

2
none of these
19. If f ( x)  4  5 cos(  x) for all real numbers x , then the range of f is:
A)
B)
C)
D)
E)
 1, 1
 9, 9
 9, 1
1, 9
none of these
20. Using interval notation, the solution set for 2 x 2  5 x  3 is:
 1 
A)   , 3 
 2 
1 
B)  ,  3   ,  
2 
1

C)   ,    3,  
2

 1

D)   ,  
 2 
E) none of these
21. If i   1 and j is a nonnegative integer, then
4j
i
n
equals:
n 0
A) 1
B)  1
C) i
D)  i
E) none of these

2 
22. The coefficient of x in the expansion of  x 

x

4
10
is:
A)  210
B)
16
C)
210
D) 3360
E) none of these

 3
23. The value of  3  
4
k 1 
A)
B)
C)
D)
E)
k 1
is:
4
7
12
7
7
4
3
none of these
24. The number of integer values of x which satisfy
A) 2
B) 5
C) 6
D) 21
E) none of these
1 2 1
 
is:
13 x 10
25. If i   1 , then 1  i  equals:
8
A)  2
B) 2
C) 2i
D) 16i
E) none of these
26. If 3x  2 y   4 and 2 x  7 y  39 , then x  y equals:
A) 2
B) 7
C) 12
D) 35
E) none of these
27. The area of the region determined by x  0 , y  0 , x  y  24 , and 2 x  y  32 is:
A)
B)
C)
D)
E)
32
192
224
384
none of these
28. If x 2  10 x  k  0 has a unique solution for x , then k equals:
A)  25
B)
0
C)
10
D)
25
E) none of these
29. A bag contains exactly 3 red marbles and x green marbles. If the probability of
1
randomly selecting a red marble from the bag is , then x equals:
4
A)
B)
C)
D)
E)
1
3
6
12
none of these
30. The number of distinct four-digit numbers that can be formed using the digits 0, 1, 2,
3, 4, 5, 6, 7, 8, and 9 if the first digit cannot be 0 and repeated digits are allowed is:
4
A)
24
B)
C) 900
D) 10000
E) none of these
31. If three fair coins are tossed, then the probability of obtaining exactly two tails is:
A)
B)
C)
D)
E)
1
3
3
8
1
2
2
3
none of these
32. If 3 x  9 x 1 , then x equals:
A)  1
1
B) 
2
C)
0
D)
2
E) none of these
1 
 1
33. An equation for the tangent line to the circle x 2  y 2  1 at the point 
,
 is:
2
 2
A) y  x  2
B) y  x  2
C) y   x  2
D) y   x  2
E) none of these
34. An equation of the slant (or oblique) asymptote of the curve defined by
2 x 2  3x  2 xy  y  6 is:
A) y  x  1
B) y   x  1
C) y  x
1
D) y  
2
E) none of these
35. If f ( x)  2 x 2  3 and h  0 , then
f ( x  h)  f ( x )
equals:
h
A) 4x  2h
B) 2h 2  3
3
C) 2h 
h
D) 4hx  2h 2
E) none of these
36. Where defined, the inverse of f ( x) 
A)
B)
C)
D)
E)
x7
2x  1
2x  1
x
 x7
2x  1
x
none of these
x7
is:
2x  1
37. When the polynomial px is divided by x  1 , the remainder is 1 . When px is
divided by x  1 , the remainder is  1 . If px is divided by x 2  1 , then the
remainder is:
A)  1
B) 1
C)  x
D) x
E) none of these
n
n
1 1  1 0
 0 6
38. If 




 , then n equals:
0 1 1 1 
 6 0 
A)  7
B)  5
C) 5
D) 7
E) none of these
39. The radius of the circle x 2  y 2  2 x  4 y  11  0 is:
A)
B)
C)
D)
E)
2
4
6
11
none of these
40. If x is the acute angle satisfying 1  sec x  tan x  3 , then sec x equals:
A)
B)
C)
D)
E)
2 3
3
3
2
2
2 3
none of these
41. If p and q are statements and the symbol ‘  ’ denotes and, ‘  ’ denotes or, ‘~’
denotes negation, and ‘  ’ denotes implication, then ~  p  q  is equivalent to:
A)
B)
C)
D)
E)
p  ~ q
pq
~ p  ~ q
~ p  q
none of these
42. The area of the triangle with vertices 0, 0 , 6, 2 , and 8, 10 is:
A)
B)
C)
D)
E)
12
16
22
25
none of these
43. The number of sets A which satisfy 2, 3  A  1, 2, 3, 4, 5 is:
A)
B)
C)
D)
E)
4
5
30
32
none of these
44. If x 2  x  3 , then x 4  x equals:
A)
B)
C)
D)
E)
9
x2  3
12  4 x
12  6 x
none of these
45. If x  y   xy
A)
B)
C)
D)
E)
x y
for all positive real numbers x and y , then 11  2 equals:
1
2
8
9
none of these
46. The coordinates of the vertex of the parabola y  4 x 2  8 x  1 are:
A)
B)
C)
D)
E)
 2, 1
1, 13
0, 1
 1,  3
none of these
47. If each of the three integers 264 , 376 , and 642 have the same remainder when
divided by the integer n , then the largest possible value of n equals:
A) 2
B) 5
C) 7
D) 12
E) none of these
48. The total cost of a pencil, eraser, and notebook is $1.00. If the notebook costs more
than two pencils, three pencils cost more than four erasers, and three erasers cost
more than a notebook, then the cost (in cents) of an eraser is:
A)
B)
C)
D)
E)
19
21
26
55
none of these
49. If  3  x  1 , then
A)
B)
C)
D)
E)
4 x 2  4 x  1  x 2  6 x  9  x 2  2 x  1 equals:
 2x  1
5
 2x  3
2x  1
none of these
50. If r, s , r  s , is the solution set of the quadratic equation ux 2  u  v x  v  0
where uv  0 , then the solution set of the quadratic equation
u 2 x 2  u 2  v 2 x  v 2  0 is:


A) r, s
B)
C)
D)
E)
r
2

, s2
 r,  s
2r, 2s
none of these