Q. Which of the following is the correct simplification of (x^3 * y^2)^(2)?
-
A.
x^6 * y^4
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B.
x^5 * y^2
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C.
x^3 * y^2
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D.
x^2 * y^3
Solution
Using the power of a product property, (a*b)^n = a^n * b^n, we get (x^3)^2 * (y^2)^2 = x^6 * y^4.
Correct Answer: A — x^6 * y^4
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Q. Which of the following is the correct simplification of (x^3y^2)^2?
-
A.
x^6y^4
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B.
x^5y^2
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C.
x^3y^2
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D.
x^2y^3
Solution
Using the power of a product property, (x^3y^2)^2 = x^(3*2)y^(2*2) = x^6y^4.
Correct Answer: A — x^6y^4
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Q. Which of the following is the correct simplification of log_10(1000) - log_10(10)?
Solution
Using the property of logarithms, log_10(1000) = 3 and log_10(10) = 1. Therefore, log_10(1000) - log_10(10) = 3 - 1 = 2.
Correct Answer: C — 3
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Q. Which of the following is the correct simplification of log_10(1000) using properties of logarithms?
Solution
log_10(1000) = log_10(10^3) = 3.
Correct Answer: A — 3
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Q. Which of the following is the correct simplification of log_2(8) + log_2(4)?
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A.
log_2(32)
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B.
log_2(12)
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C.
log_2(16)
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D.
log_2(6)
Solution
Using the property of logarithms, log_2(8) + log_2(4) = log_2(8*4) = log_2(32) = log_2(16).
Correct Answer: C — log_2(16)
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Q. Which of the following is the correct simplification of log_5(25) - log_5(5)?
Solution
log_5(25) = 2 and log_5(5) = 1, thus log_5(25) - log_5(5) = 2 - 1 = 1.
Correct Answer: A — 1
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Q. Which of the following is the correct simplification of log_a(b^2)?
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A.
2 log_a(b)
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B.
log_a(2b)
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C.
log_a(b) + 2
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D.
log_a(b) - 2
Solution
Using the power rule of logarithms, log_a(b^2) simplifies to 2 log_a(b).
Correct Answer: A — 2 log_a(b)
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Q. Which of the following is the correct vertex form of the quadratic equation y = x² - 4x + 3?
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A.
y = (x - 2)² - 1
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B.
y = (x + 2)² - 1
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C.
y = (x - 2)² + 1
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D.
y = (x + 2)² + 1
Solution
Completing the square for the equation y = x² - 4x + 3 results in y = (x - 2)² - 1.
Correct Answer: A — y = (x - 2)² - 1
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Q. Which of the following is the greatest common divisor (GCD) of 48 and 180?
Solution
The GCD of 48 and 180 is 12, as it is the largest number that divides both without a remainder.
Correct Answer: A — 12
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Q. Which of the following is the greatest common factor (GCF) of 48 and 180?
Solution
The GCF of 48 and 180 is 12.
Correct Answer: A — 12
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Q. Which of the following is the least common multiple (LCM) of 9 and 12?
-
A.
36
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B.
72
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C.
108
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D.
144
Solution
The LCM of 9 and 12 is 36, as it is the smallest number that both can divide evenly.
Correct Answer: A — 36
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Q. Which of the following is the result of simplifying (2^3)^2?
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A.
2^5
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B.
2^6
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C.
2^7
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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)?
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A.
It can be expressed as 2^x + 2^y.
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B.
It can be expressed as 2^x * 2^y.
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C.
It is always greater than 2.
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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?
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A.
It can have at most two real roots.
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B.
It can have at most three real roots.
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C.
It can have no real roots.
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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.
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B.
It has at least one real root.
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C.
It has no real roots.
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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?
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A.
It has two distinct real roots.
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B.
It has one real root with multiplicity 2.
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C.
It has no real roots.
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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)
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A.
2^2
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B.
2^0
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C.
2^1
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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)
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A.
x = 1
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B.
x = 2
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C.
x = 3
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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
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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_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?
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A.
log_5(0)
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B.
log_5(1)
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C.
log_5(5)
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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)
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B.
log_a(b/c) = log_a(b) - log_a(c)
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C.
log_a(b^c) = c * log_a(b)
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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 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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