Q. Which of the following is the correct representation of the decimal number 15 in base 8?
Solution
In base 8, the decimal number 15 is represented as '17'.
Correct Answer:
A
— 17
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Q. Which of the following is the correct representation of the decimal number 15 in base-2?
-
A.
1111
-
B.
1011
-
C.
1101
-
D.
1001
Solution
The decimal number 15 is represented as '1111' in binary.
Correct Answer:
A
— 1111
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Q. Which of the following is the correct representation of the decimal number 15 in hexadecimal?
Solution
In hexadecimal, the decimal number 15 is represented as F.
Correct Answer:
A
— F
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Q. Which of the following is the correct representation of the decimal number 15 in base-4?
Solution
15 in decimal is represented as '33' in base-4 (3*4^1 + 3*4^0 = 12 + 3 = 15).
Correct Answer:
A
— 33
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Q. Which of the following is the correct representation of the decimal number 25 in base-8?
Solution
25 in decimal is represented as '31' in base-8 (3*8^1 + 1*8^0 = 24 + 1 = 25).
Correct Answer:
A
— 31
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Q. Which of the following is the correct simplification of (x^2y^3)/(xy^2)?
-
A.
x^(2-1)y^(3-2)
-
B.
x^1y^1
-
C.
x^2y^5
-
D.
x^3y^1
Solution
Using the quotient rule for exponents, (x^2y^3)/(xy^2) simplifies to x^(2-1)y^(3-2) = xy.
Correct Answer:
A
— x^(2-1)y^(3-2)
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Q. Which of the following is the correct simplification of (x^3 * y^2) / (x^2 * y)?
-
A.
x^(3-2) * y^(2-1)
-
B.
x^(5) * y^(1)
-
C.
x^(1) * y^(1)
-
D.
x^(1) * y^(3)
Solution
Using the property of exponents for division, we subtract the exponents: x^(3-2) * y^(2-1) = x^1 * y^1.
Correct Answer:
A
— x^(3-2) * y^(2-1)
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Q. Which of the following is the correct simplification of (x^3 * y^2)^(2)?
-
A.
x^6 * y^4
-
B.
x^5 * y^2
-
C.
x^3 * y^2
-
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
-
B.
x^5y^2
-
C.
x^3y^2
-
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(0.01)?
Solution
log_10(1000) = 3 and log_10(0.01) = -2, thus 3 + (-2) = 1.
Correct Answer:
B
— 0
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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)?
-
A.
log_2(32)
-
B.
log_2(12)
-
C.
log_2(16)
-
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)?
-
A.
2 log_a(b)
-
B.
log_a(2b)
-
C.
log_a(b) + 2
-
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 simplification of log_a(b^c)?
-
A.
c * log_a(b)
-
B.
log_a(c) * log_a(b)
-
C.
log_a(b) / c
-
D.
log_a(c^b)
Solution
The property of logarithms states that log_a(b^c) = c * log_a(b).
Correct Answer:
A
— c * 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?
-
A.
y = (x - 2)² - 1
-
B.
y = (x + 2)² - 1
-
C.
y = (x - 2)² + 1
-
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
-
B.
72
-
C.
108
-
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 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 odd one out in the context of historical events? (2023)
-
A.
World War I
-
B.
World War II
-
C.
The Great Depression
-
D.
The Renaissance
Solution
The Renaissance was a cultural movement, while the others are significant historical events related to conflict and economic turmoil.
Correct Answer:
D
— The Renaissance
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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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