The College Scholastic Ability Test(csat) of Korea Mathematics I
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100 tests on mathematics inha university in tashkent 2015
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The College Scholastic Ability Test(CSAT) of Korea Mathematics I (A type, B type) 1. When two matrices
are
, find the sum of all entries of the matrix
. 2. Compute the following limit lim →∞
. 3. Let
be a sequence and
th
term,
⋯
. When
, find . 4. Find the sum of all natural numbers satisfying the exponential inequality ≤
.
5. Let and
. Find the sum of all constants
satisfying
. 6. Let and two sequences satisfying
∞ and
∞ . Then find
∞
. 7. Let be a geometric sequence with
. Find ∞
. 8. Let be a constant with and
be the intersection point between two curves
and
. We denote the -coordinate of by
. Then, compute lim →∞
. 9. Let be an arithmetic sequence with
. Find . 10.
Let
and
be two
× matrices satisfying
, where is an identity matrix. Then, which one consists of true statements among the following statements? (a) The inverse matrix of exists. (b) . (c)
. ① (a) ② (c) ③ (a), (b) ④ (b), (c) ⑤ (a), (b), (c) 11. When is defined by lim
→ for each natural number , find
.
12. Compute ×
. 13. When the common ratio geometric sequence
is
and
, find
. 14. Find the solution of the logarithmic equation log
. 15. Let
be an arithmetic sequence with
. When , find . 16. Let
. When the sum of all entries of
is , find . 17. Compute ×
. 18. When two matrices are
, find the sum of all entries of the matrix
. 19. Compute lim →∞
×
. 20. When a sequence
satisfies the relations
≥ , find . 21. Find the sum of all real numbers which make the following system of linear equations
have solutions except .
22. We assume thar for a natural number , is defined by log odd log even Find the number of ordered pairs such
that and ≤ ≤ . 23. Let
be an arithmetic sequence with
and the common difference . If is the sum from the -st term to -th term,
⋯
, and
, find the common difference . 24. 9. For each natural number , is defined by
log
odd log even .
When a sequence is defined by
, find
. 25. For a real number , we write log , where
is a
natural number and ≤ . We list numbers ⋯
⋯ such that for each , is a
multiple of . Compute log . 26. For a natural number with
≥ , is defined by the solution of tan , . Compute lim →∞ . 27.
For a positive real number , we write log , where
is an integer and ≤ . For a natural number , we define by the product of all 's satisfying . Compute lim →∞
log
. 28. Let
. Find the sum of all entries of .
29. Compute lim → ∞
. 30. Compute log
. 31. Let be a geometric sequence consisting of positive real numbers. When satisfies
, find . 32. Let
be an arithmetic sequence with
. Find such that . 33. When and
satisfy the following equations
,
simultaneously, find the constant .
34. Let
be a sequence satisfying
∞
. Compute lim
→∞
. 35. Let be a sequence with
, ⋅
≥
. We find
From the given information ⋅ ≥
. Since
for each natural number with ≥ , we have
. If we put
, we have ≥
and , which leads to ≥ . Therefore, × ≥ When we write , find . 36. For each natural number , we define a point on the -plane as follows. (i) The coordinates of first three point are
, (ii) the middle point of the line segment
is
equal to the middle point of the line segment
. For an example, the point
is . If the coordinate of
, find
. 37. Let be the set of all integers and ∈ . For each natural number , we define ≤ ≤ log and define to be the number of elements in ∩ . For example,
. Compute
. 38. When
, find the sum of all entries of the inverse matrix
. 39. Compute lim →∞
. 40. When a sequence satisfies
,
for each natural number
, find
. 41. When is an arithmetic sequence with the and the common difference is 2, find
. 42. Find the solution of the equation log
. 43. When three numbers becomes a positive arithmetic sequence and three numbers becomes a positive geometric sequence, find
. 44. Let be a
× matrix,
be the
× identity matrix and be the
× zero matrix. We assume that satisfies and
. When and
are real numbers satisfying
, find
. 45. For a positive real number , we write log , where
is an integer and ≤ . Find the number of natural numbers
≤ and
≤ . 46. Let and
be two
× matrices satisfying
, where is the identity matrix. Then, which one consists of true statements among the following statements? (a)
. (b)
. (c)
. ① (a) ② (b) ③ (a), (b) ④ (a), (c) ⑤ (a), (b), (c) 47. Let
. Find the sum of all entries of . 48. Compute × log
. 49. When lim →∞
×
, find . 50. Find the sum of all natural numbers satisfying the exponential inequality
. 51. Let be a sequence satisfying
for each natural number
, find ⋯ . 52. Solve the following log equation log log . 53. Let
be a sequence satisfying
log for each natural number
, find
. 54. For a natural number , we write log , where
is an integer and ≤ . Find the number of natural numbers satisfying ≤ . (You may use the following inequality .) Mathematics II 55. lim →
ln
. 56. Find the maximal value of function sin cos .
57. Compute the integral
. 58. Compute ′ for the function cos . 59. Let be the product of all real roots of the equation . Find
. 60. When , compute the limit lim →∞
. 61. Find the number of all ordered triples such that (i) are
natural numbers with ≤ ≤ ≤ and (ii) × × is an odd integer. 62. When tan , find cos . 63. When the maximal value of the function cos sin is , find a positive number .
′ for the function
. 65. When the straight line passing through the two points is perpendicular to the line , find . 66. When is the product of all roots of the equation , find . 67. When sin with
sin
68. Find the product of all real roots of the equation
. 69. Compute ′
for the function ln . 70. Let be positive real numbers and be a point on the hyperbola
. When the tangent line to at the point is perpendicular to one of the asymptotic lines of
, find
. 71. If the maximal value of the function
cos
sin is
, find . 72. When a continuous function satisfies
, find
. 73. Compute lim → . 74. Let
≤ and
≤ .
∪ ≤ , find the constant .
with
∠ ∠ . When
we choose a point satisfying
, find
. 76. When is the solution of the equation cos cos , find
tan . 77. Find the product of all real roots of the equation
. 78. When tan
with , find sec
. 79. When the minimal value of function is , find the constant . 80. Find the slope of the tangent line to the curve ln on the point .
81. When the coefficient of in the expansion of the polynomial is
, find the positive number .
, find the constant .
defined by ≥
is continuous in the set of all real numbers, find the constant .
be a polynomial function and let ′
. When the graph of passes through the point , find
. 85. Compute lim → .
86. When the function
satisfies lim →
, find
the constant . 87. Compute lim →
. 88. Find area of the region enclosed by the curve
and straight line
. 89. When a real number satisfies
, find . 90. When the function satisfies lim →
, find the constant . 91. We assume that the function
is defined by
≤
. Find a real number such that the function is continuous at .
92. Compute lim →
. 93. Let and
be real numbers. A function defined by
≥ is differentiable at
, find
. 94. When the tangent line to the function
at the point is , find
. 95. When , find lim →
. 96. Let . Find a real number satisfying
. 97. Compute
lim → . 98. When , find lim →
. 99. When
, find ′
. 100. Let be a real number. When the coefficient of in the expansion of is , find the coefficient of
101. Compute .
The College Scholastic Ability Test(CSAT) of Korea answers
1. 9 2. 4
3. 4. 15
5. 3 6. 54
7. 8. 3
9. 25 10. ⑤
11. 33 12. 10
13. 12 14. 26
15. 10 16. 4
17. 12 18. 10
19. 6 20. 256
21. 11 22. 220
23. -1 24.
log 25. 16 26.
27. 28. 7
29. 30. 3
31. 16 32. 10
33. 2 34.
35. 105 36. 23
37. 573 38. 4
39. 5 40. 2
41. 240 42. 19
43. 10 44. 21
45. 50 46. ⑤
47. 4 48. 4
49. 50. 9
51. 95 52. 12
53. 21 54. 77
55. 56.
57. 2 58. 8 59. 25
60. In3 61. 220
62. 63. 3
64. 15 65. 3
66. 64 67.
68. -4 69. 14
70. 68 71. 28
72. e-1 73.
74. 15 75. 7
76. 35 77. -
78. 3 79. 14
80. - 81. 2
82. 3 83. 11
84. 12 85. 7
86. 2 87. 3
88. 89. 25 90. 12
91. 13 92. 5
93. 7 94. 1
95. 12 96.
97. 11 98. 2
99. 7 100. 84
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