Q. Which of the following is the least common multiple of 8 and 12?
Solution
The least common multiple of 8 and 12 is 24, but the next multiple is 48.
Correct Answer:
B
— 48
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Q. Which of the following is the result of (2^3)^2?
-
A.
2^5
-
B.
2^6
-
C.
2^7
-
D.
2^8
Solution
Using the power of a power property, (a^m)^n = a^(m*n), we get (2^3)^2 = 2^(3*2) = 2^6.
Correct Answer:
B
— 2^6
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Q. Which of the following is the result of (x^2y^3)^2?
-
A.
x^4y^6
-
B.
x^2y^3
-
C.
x^2y^6
-
D.
x^4y^3
Solution
Using the power of a power property, we multiply the exponents: (x^2)^2 = x^4 and (y^3)^2 = y^6.
Correct Answer:
A
— x^4y^6
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Q. Which of the following is the result of simplifying (2^3)^2?
-
A.
2^5
-
B.
2^6
-
C.
2^7
-
D.
2^8
Solution
Using the power of a power property, (a^m)^n = a^(m*n), we get (2^3)^2 = 2^(3*2) = 2^6.
Correct Answer:
B
— 2^6
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Q. Which of the following is the smallest multiple of 7 that is greater than 50?
Solution
The multiples of 7 are 7, 14, 21, 28, 35, 42, 49, 56, ... The smallest multiple greater than 50 is 56.
Correct Answer:
A
— 56
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Q. Which of the following is true about the expression 2^(x+y)?
-
A.
It can be expressed as 2^x + 2^y.
-
B.
It can be expressed as 2^x * 2^y.
-
C.
It is always greater than 2.
-
D.
It is equal to 2 when x and y are both 0.
Solution
Using the property of exponents, 2^(x+y) = 2^x * 2^y.
Correct Answer:
B
— It can be expressed as 2^x * 2^y.
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Q. Which of the following is true about the roots of a cubic function?
-
A.
It can have at most two real roots.
-
B.
It can have at most three real roots.
-
C.
It can have no real roots.
-
D.
It must have at least one real root.
Solution
A cubic function can have at most three real roots, and it is guaranteed to have at least one real root due to the Intermediate Value Theorem.
Correct Answer:
B
— It can have at most three real roots.
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Q. Which of the following is true about the roots of a polynomial of odd degree?
-
A.
It has an even number of roots.
-
B.
It has at least one real root.
-
C.
It has no real roots.
-
D.
It has exactly two real roots.
Solution
A polynomial of odd degree must have at least one real root due to the Intermediate Value Theorem.
Correct Answer:
B
— It has at least one real root.
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Q. Which of the following is true about the roots of the polynomial P(x) = x^2 + 4x + 4?
-
A.
It has two distinct real roots.
-
B.
It has one real root with multiplicity 2.
-
C.
It has no real roots.
-
D.
It has two complex roots.
Solution
The polynomial can be factored as (x + 2)^2, indicating it has one real root with multiplicity 2.
Correct Answer:
B
— It has one real root with multiplicity 2.
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Q. Which of the following is true about the roots of the polynomial x^2 + 4x + 4?
-
A.
It has two distinct real roots.
-
B.
It has one real root with multiplicity 2.
-
C.
It has no real roots.
-
D.
It has two complex roots.
Solution
The polynomial can be factored as (x + 2)(x + 2), indicating it has one real root with multiplicity 2.
Correct Answer:
B
— It has one real root with multiplicity 2.
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Q. Which of the following is true for the expression 2^(x+1) / 2^(x-1)? (2023)
-
A.
2^2
-
B.
2^0
-
C.
2^1
-
D.
2^3
Solution
Using the property of exponents, we have 2^(x+1 - (x-1)) = 2^(x+1-x+1) = 2^2.
Correct Answer:
C
— 2^1
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Q. Which of the following is true for the expression 2^(x+3) = 8? (2023)
-
A.
x = 1
-
B.
x = 2
-
C.
x = 3
-
D.
x = 0
Solution
Since 8 can be expressed as 2^3, we have 2^(x+3) = 2^3, thus x + 3 = 3, leading to x = 0.
Correct Answer:
A
— x = 1
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Q. Which of the following is true for the expression 4^(x+1) = 16? (2023)
-
A.
x = 1
-
B.
x = 2
-
C.
x = 3
-
D.
x = 0
Solution
Since 16 can be expressed as 4^2, we have 4^(x+1) = 4^2, leading to x + 1 = 2, thus x = 1.
Correct Answer:
A
— x = 1
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Q. Which of the following logarithmic expressions is equivalent to log_10(0.01)?
Solution
Since 0.01 is 10^-2, log_10(0.01) = -2.
Correct Answer:
A
— -2
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Q. Which of the following logarithmic expressions is equivalent to log_2(8) - log_2(4)?
-
A.
log_2(2)
-
B.
log_2(1)
-
C.
log_2(0)
-
D.
log_2(3)
Solution
Using the property of logarithms that states log_a(b) - log_a(c) = log_a(b/c), we find log_2(8) - log_2(4) = log_2(8/4) = log_2(2).
Correct Answer:
A
— log_2(2)
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Q. Which of the following logarithmic expressions is equivalent to log_3(81)?
Solution
Since 81 is 3^4, log_3(81) equals 4.
Correct Answer:
A
— 4
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Q. Which of the following logarithmic expressions is undefined?
-
A.
log_5(0)
-
B.
log_5(1)
-
C.
log_5(5)
-
D.
log_5(25)
Solution
Logarithm of zero is undefined, hence log_5(0) is the correct answer.
Correct Answer:
A
— log_5(0)
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Q. Which of the following logarithmic identities is incorrect?
-
A.
log_a(b) + log_a(c) = log_a(bc)
-
B.
log_a(b/c) = log_a(b) - log_a(c)
-
C.
log_a(b^c) = c * log_a(b)
-
D.
log_a(b) * log_a(c) = log_a(bc)
Solution
The last identity is incorrect; it should be log_a(b) + log_a(c) = log_a(bc).
Correct Answer:
D
— log_a(b) * log_a(c) = log_a(bc)
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Q. Which of the following logarithmic properties is used to simplify log_a(b^c)?
-
A.
Power Rule
-
B.
Product Rule
-
C.
Quotient Rule
-
D.
Change of Base Formula
Solution
The Power Rule states that log_a(b^c) = c * log_a(b).
Correct Answer:
A
— Power Rule
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Q. Which of the following numbers is a common multiple of 3 and 5?
Solution
15 is the smallest common multiple of 3 and 5, and 30 is also a common multiple. However, 15 is the first one listed.
Correct Answer:
A
— 15
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Q. Which of the following numbers is a common multiple of 4 and 6?
Solution
12 is the smallest common multiple of 4 and 6, and 24 is also a common multiple, but 12 is the smallest.
Correct Answer:
A
— 12
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Q. Which of the following numbers is a factor of 144?
Solution
12 is a factor of 144 because 144 ÷ 12 = 12, which is an integer.
Correct Answer:
B
— 12
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Q. Which of the following numbers is a factor of 48?
Solution
12 is a factor of 48, as 48 divided by 12 equals 4, which is an integer.
Correct Answer:
B
— 12
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Q. Which of the following numbers is a factor of 60?
Solution
12 is a factor of 60 as 60 divided by 12 equals 5 with no remainder.
Correct Answer:
B
— 12
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Q. Which of the following numbers is a multiple of both 4 and 6?
Solution
12 is the smallest number that is a multiple of both 4 and 6.
Correct Answer:
A
— 12
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Q. Which of the following numbers is a multiple of both 8 and 12? (2023)
Solution
The LCM of 8 and 12 is 24, which is a multiple of both.
Correct Answer:
A
— 24
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Q. Which of the following numbers is divisible by 11?
-
A.
121
-
B.
123
-
C.
124
-
D.
125
Solution
121 is divisible by 11, as 121 = 11 x 11.
Correct Answer:
A
— 121
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Q. Which of the following numbers is divisible by 15?
Solution
A number is divisible by 15 if it is divisible by both 3 and 5. 30 meets both criteria.
Correct Answer:
A
— 30
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Q. Which of the following numbers is divisible by 7?
Solution
28 is divisible by 7 (28 ÷ 7 = 4). The other numbers are not.
Correct Answer:
A
— 28
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Q. Which of the following numbers is divisible by 8?
Solution
A number is divisible by 8 if the last three digits form a number that is divisible by 8. 64 is divisible by 8.
Correct Answer:
A
— 64
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