Q. If x = sin^(-1)(3/5), what is cos(x)?
-
A.
4/5
-
B.
3/5
-
C.
5/4
-
D.
1/5
Solution
Using the identity cos^2(x) + sin^2(x) = 1, we find cos(x) = √(1 - (3/5)^2) = 4/5.
Correct Answer: A — 4/5
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Q. If x = sin^(-1)(3/5), what is the value of cos(x)?
-
A.
4/5
-
B.
3/5
-
C.
2/5
-
D.
1/5
Solution
Using the identity cos(x) = sqrt(1 - sin^2(x)), we have cos(x) = sqrt(1 - (3/5)^2) = sqrt(16/25) = 4/5.
Correct Answer: A — 4/5
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Q. If x = tan^(-1)(1), then the value of x is:
Solution
tan^(-1)(1) = π/4.
Correct Answer: A — π/4
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Q. If x = tan^(-1)(1), what is the value of x?
-
A.
π/4
-
B.
π/3
-
C.
π/6
-
D.
0
Solution
tan^(-1)(1) = π/4, since tan(π/4) = 1.
Correct Answer: A — π/4
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Q. If x = tan^(-1)(1/√3), what is the value of x?
-
A.
π/6
-
B.
π/4
-
C.
π/3
-
D.
0
Solution
tan^(-1)(1/√3) = π/6, since tan(π/6) = 1/√3.
Correct Answer: A — π/6
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Q. If x = tan^(-1)(√3), then what is the value of sin^(-1)(x)?
-
A.
π/3
-
B.
π/4
-
C.
π/2
-
D.
π/6
Solution
x = tan^(-1)(√3) = π/3, thus sin^(-1)(x) = sin^(-1)(√3/2) = π/3.
Correct Answer: A — π/3
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Q. If x = tan^(-1)(√3), what is the value of sin(2x)?
-
A.
√3/2
-
B.
1
-
C.
√2/2
-
D.
0
Solution
Since tan^(-1)(√3) = π/3, then 2x = 2π/3 and sin(2x) = sin(2π/3) = √3/2.
Correct Answer: B — 1
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Q. If X is facing south and turns 90 degrees to the right, then 180 degrees to the left, which direction is he facing now?
-
A.
North
-
B.
South
-
C.
East
-
D.
West
Solution
X turns right to face west and then turns left to face south.
Correct Answer: D — West
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Q. If X is taller than Y and Y is shorter than Z, who is the shortest? (2023)
-
A.
X
-
B.
Y
-
C.
Z
-
D.
None of the above
Solution
Y is the shortest as X is taller than Y and Y is shorter than Z.
Correct Answer: B — Y
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Q. If X is the father of Y and Y is the mother of Z, what is the relationship between X and Z? (2020)
-
A.
Grandfather
-
B.
Father
-
C.
Uncle
-
D.
Brother
Solution
X is the father of Y, and Y is the mother of Z, making X the grandfather of Z.
Correct Answer: A — Grandfather
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Q. If X is the husband of Y and Z is the daughter of Y, how is Z related to X?
-
A.
Daughter
-
B.
Sister
-
C.
Niece
-
D.
Cousin
Solution
Z is the daughter of X as Y (Z's mother) is married to X.
Correct Answer: A — Daughter
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Q. If X is the sister of Y and Y is the brother of Z, what is the relationship of X to Z? (2020)
-
A.
Sister
-
B.
Cousin
-
C.
Aunt
-
D.
Brother
Solution
X is the sister of Y, and since Y is the brother of Z, X is also the sister of Z.
Correct Answer: A — Sister
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Q. If x ≡ 2 (mod 3) and y ≡ 1 (mod 3), what is the value of (x + y) mod 3?
Solution
x + y ≡ 2 + 1 ≡ 3 (mod 3), which is equivalent to 0, but since we are looking for the value, it is 0 mod 3.
Correct Answer: B — 1
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Q. If x ≡ 3 (mod 7) and x ≡ 5 (mod 11), what is the smallest positive integer solution for x?
Solution
Using the method of successive substitutions or the Chinese Remainder Theorem, the smallest solution is x = 26.
Correct Answer: B — 26
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Q. If x ≡ 4 (mod 7) and y ≡ 3 (mod 7), what is the value of (x + y) mod 7?
Solution
x + y ≡ 4 + 3 ≡ 7 (mod 7), which is equivalent to 0.
Correct Answer: A — 0
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Q. If x^2 + 2x + 1 = 0, what is the value of x?
Solution
This is a perfect square: (x + 1)^2 = 0 => x = -1.
Correct Answer: A — -1
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Q. If x^2 - 6x + 9 = 0, what is the value of x?
Solution
This is a perfect square: (x - 3)^2 = 0, so x = 3.
Correct Answer: A — 3
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Q. If x^3 - 8 = 0, what is the value of x?
Solution
x^3 = 8 => x = 2
Correct Answer: A — 2
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Q. If x^3 = 8, then the value of x is?
Solution
Taking the cube root of both sides, x = 8^(1/3) = 2.
Correct Answer: A — 2
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Q. If x^4 = 81, what is the value of x?
Solution
Taking the fourth root, x = 81^(1/4) = 3.
Correct Answer: A — 3
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Q. If y = cos^(-1)(1/2), what is the value of y?
-
A.
π/3
-
B.
π/4
-
C.
π/6
-
D.
π/2
Solution
cos^(-1)(1/2) = π/3, since cos(π/3) = 1/2.
Correct Answer: A — π/3
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Q. If y = cos^(-1)(x), then dy/dx is:
-
A.
-1/√(1-x^2)
-
B.
1/√(1-x^2)
-
C.
0
-
D.
1
Solution
The derivative of cos^(-1)(x) is dy/dx = -1/√(1-x^2).
Correct Answer: A — -1/√(1-x^2)
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Q. If y = cos^(-1)(x), then what is dy/dx?
-
A.
-1/√(1-x^2)
-
B.
1/√(1-x^2)
-
C.
1/x
-
D.
-1/x
Solution
The derivative of y = cos^(-1)(x) is dy/dx = -1/√(1-x^2).
Correct Answer: A — -1/√(1-x^2)
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Q. If y = cos^(-1)(x), what is dy/dx?
-
A.
-1/√(1-x^2)
-
B.
1/√(1-x^2)
-
C.
0
-
D.
1
Solution
The derivative of cos^(-1)(x) is dy/dx = -1/√(1-x^2).
Correct Answer: A — -1/√(1-x^2)
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Q. If y = sin^(-1)(x), then what is the derivative dy/dx?
-
A.
1/√(1-x^2)
-
B.
1/(1-x^2)
-
C.
√(1-x^2)
-
D.
1/x
Solution
The derivative of y = sin^(-1)(x) is dy/dx = 1/√(1-x^2).
Correct Answer: A — 1/√(1-x^2)
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Q. If y = sin^(-1)(x), then x = sin(y) implies:
-
A.
y = x
-
B.
y = -x
-
C.
y = 1-x
-
D.
y = 1+x
Solution
By definition, if y = sin^(-1)(x), then x = sin(y).
Correct Answer: A — y = x
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Q. If y = sin^(-1)(x), what is dy/dx?
-
A.
1/sqrt(1-x^2)
-
B.
1/(1-x^2)
-
C.
sqrt(1-x^2)
-
D.
1/x
Solution
The derivative of y = sin^(-1)(x) is dy/dx = 1/sqrt(1-x^2).
Correct Answer: A — 1/sqrt(1-x^2)
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Q. If y = sin^(-1)(x), what is the second derivative d^2y/dx^2?
-
A.
0
-
B.
1/√(1-x^2)^3
-
C.
-1/√(1-x^2)^3
-
D.
undefined
Solution
The second derivative d^2y/dx^2 = -1/√(1-x^2)^3.
Correct Answer: C — -1/√(1-x^2)^3
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Q. If y = tan^(-1)(x), then the range of y is:
-
A.
(-π/2, π/2)
-
B.
(0, π)
-
C.
(-π, π)
-
D.
[0, 1]
Solution
The range of y = tan^(-1)(x) is (-π/2, π/2).
Correct Answer: A — (-π/2, π/2)
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Q. If y = tan^(-1)(x), then what is the second derivative d^2y/dx^2?
-
A.
0
-
B.
-2/(1+x^2)^2
-
C.
2/(1+x^2)^2
-
D.
1/(1+x^2)
Solution
The first derivative dy/dx = 1/(1+x^2). The second derivative d^2y/dx^2 = -2/(1+x^2)^2.
Correct Answer: B — -2/(1+x^2)^2
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