Q. Which of the following is the correct simplification of log_2(8) + log_2(4)?
-
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 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 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 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 4^(x+1) = 16? (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 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 undefined?
-
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 pairs of equations represents parallel lines?
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A.
2x + 3y = 6 and 4x + 6y = 12
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B.
x - y = 1 and x + y = 1
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C.
3x + 2y = 5 and 3x - 2y = 5
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D.
x + 2y = 3 and 2x + 4y = 6
Solution
The first pair has the same slope (2/3) and thus represents parallel lines.
Correct Answer: A — 2x + 3y = 6 and 4x + 6y = 12
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Q. Which of the following pairs of linear equations has no solution?
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A.
x + y = 2 and x + y = 4
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B.
2x - y = 1 and 4x - 2y = 2
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C.
3x + 2y = 6 and 6x + 4y = 12
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D.
x - 2y = 3 and 2x - 4y = 6
Solution
The first pair represents parallel lines, which means they will never intersect, hence no solution.
Correct Answer: A — x + y = 2 and x + y = 4
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Q. Which of the following polynomials is a perfect square?
-
A.
x^2 + 4x + 4
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B.
x^2 - 4
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C.
x^2 + 2x + 3
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D.
x^2 - 2x + 1
Solution
The polynomial x^2 + 4x + 4 can be factored as (x + 2)^2, making it a perfect square.
Correct Answer: A — x^2 + 4x + 4
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Q. Which of the following polynomials is a quadratic polynomial?
-
A.
x^3 - 2x + 1
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B.
2x^2 + 3x - 5
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C.
4x + 7
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D.
x^4 - x^2 + 1
Solution
A quadratic polynomial is defined as a polynomial of degree 2, which is 2x^2 + 3x - 5.
Correct Answer: B — 2x^2 + 3x - 5
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Q. Which of the following represents the slope of the line represented by the equation y = mx + b?
Solution
In the equation y = mx + b, 'm' represents the slope of the line.
Correct Answer: A — m
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Q. Which of the following sequences cannot be a harmonic progression?
-
A.
1, 1/2, 1/3
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B.
2, 4, 8
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C.
3, 1, 1/3
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D.
5, 10, 15
Solution
The sequence 2, 4, 8 does not have reciprocals that form an arithmetic progression, hence it cannot be a harmonic progression.
Correct Answer: B — 2, 4, 8
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Q. Which of the following sequences is a geometric progression?
-
A.
1, 2, 4, 8
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B.
1, 3, 6, 10
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C.
2, 4, 8, 16
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D.
1, 1, 1, 1
Solution
The sequence 2, 4, 8, 16 has a constant ratio of 2, making it a geometric progression.
Correct Answer: C — 2, 4, 8, 16
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Q. Which of the following sequences is a harmonic progression?
-
A.
1, 2, 3
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B.
1, 1/2, 1/3
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C.
2, 4, 6
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D.
3, 6, 9
Solution
The sequence 1, 1/2, 1/3 has reciprocals 1, 2, 3 which are in arithmetic progression, thus it is a harmonic progression.
Correct Answer: B — 1, 1/2, 1/3
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Q. Which of the following sequences is an arithmetic progression?
-
A.
2, 4, 8, 16
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B.
1, 3, 5, 7
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C.
5, 10, 15, 25
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D.
3, 6, 9, 12
Solution
An arithmetic progression has a constant difference between consecutive terms. The sequence 1, 3, 5, 7 has a common difference of 2.
Correct Answer: B — 1, 3, 5, 7
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Q. Which of the following statements about exponents is incorrect?
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A.
a^(m+n) = a^m * a^n
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B.
a^(m-n) = a^m / a^n
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C.
a^m * b^m = (ab)^m
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D.
a^m + a^n = a^(m+n)
Solution
The statement a^m + a^n = a^(m+n) is incorrect; addition of exponents does not apply in this manner.
Correct Answer: D — a^m + a^n = a^(m+n)
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Q. Which of the following statements about geometric progressions is true?
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A.
The ratio of consecutive terms is constant.
-
B.
The sum of terms is always positive.
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C.
The first term must be greater than the second.
-
D.
The common ratio can only be an integer.
Solution
In a geometric progression, the ratio of consecutive terms is indeed constant, which defines the progression.
Correct Answer: A — The ratio of consecutive terms is constant.
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Q. Which of the following statements about harmonic progression is true?
-
A.
The sum of the terms is always positive.
-
B.
The terms can be negative.
-
C.
The terms are always integers.
-
D.
The common difference is always positive.
Solution
In a harmonic progression, the terms can be negative as long as their reciprocals form an arithmetic progression.
Correct Answer: B — The terms can be negative.
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Q. Which of the following statements about the graph of a quadratic function is true?
-
A.
It is always a parabola that opens upwards.
-
B.
It can be a straight line.
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C.
It can intersect the x-axis at three points.
-
D.
It is symmetric about its vertex.
Solution
The graph of a quadratic function is symmetric about its vertex.
Correct Answer: D — It is symmetric about its vertex.
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Q. Which of the following statements is true about the graph of a function that is periodic?
-
A.
It repeats its values at regular intervals.
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B.
It is always increasing.
-
C.
It has no maximum or minimum values.
-
D.
It is a straight line.
Solution
A periodic function is characterized by repeating values at regular intervals, such as sine and cosine functions.
Correct Answer: A — It repeats its values at regular intervals.
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Q. Which of the following statements is true about the inverse of a function?
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A.
The inverse of a function is always a function.
-
B.
The inverse of a function is not necessarily a function.
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C.
The inverse of a function is always linear.
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D.
The inverse of a function cannot be graphed.
Solution
The inverse of a function is a function only if the original function is one-to-one.
Correct Answer: B — The inverse of a function is not necessarily a function.
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Q. Which of the following statements is true regarding an arithmetic progression?
-
A.
The sum of any two terms is constant.
-
B.
The difference between consecutive terms is constant.
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C.
The product of any two terms is constant.
-
D.
The ratio of any two terms is constant.
Solution
In an arithmetic progression, the difference between consecutive terms is constant, which is the defining property of an AP.
Correct Answer: B — The difference between consecutive terms is constant.
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Q. Which of the following statements is true regarding harmonic progressions?
-
A.
The sum of the terms is always constant.
-
B.
The product of the terms is always constant.
-
C.
The reciprocals of the terms form an arithmetic progression.
-
D.
The terms are always integers.
Solution
In a harmonic progression, the reciprocals of the terms indeed form an arithmetic progression.
Correct Answer: C — The reciprocals of the terms form an arithmetic progression.
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Q. Which of the following statements is true regarding the graph of a linear equation?
-
A.
It can be a curve.
-
B.
It is always a straight line.
-
C.
It can have multiple slopes.
-
D.
It can intersect the x-axis at multiple points.
Solution
The graph of a linear equation is always a straight line.
Correct Answer: B — It is always a straight line.
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Q. Which of the following statements is true regarding the graph of the equation y = mx + b?
-
A.
The graph is always a circle.
-
B.
The slope m indicates the steepness of the line.
-
C.
The y-intercept b is always negative.
-
D.
The graph can never be horizontal.
Solution
In the equation y = mx + b, m represents the slope, which indicates how steep the line is.
Correct Answer: B — The slope m indicates the steepness of the line.
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Q. Which of the following words best captures the author's attitude towards the current state of inequality?
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A.
Pessimistic.
-
B.
Indifferent.
-
C.
Concerned.
-
D.
Skeptical.
Solution
The author expresses concern about the persistence and impact of inequality in society.
Correct Answer: C — Concerned.
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Q. Which of the following words best captures the author's view on the urgency of addressing inequalities?
-
A.
Critical.
-
B.
Optional.
-
C.
Negligible.
-
D.
Controversial.
Solution
The author emphasizes the critical nature of addressing inequalities, suggesting it is an urgent issue.
Correct Answer: A — Critical.
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Q. Which of the following words from the passage is closest in meaning to 'perpetuate'?
-
A.
End
-
B.
Continue
-
C.
Diminish
-
D.
Challenge
Solution
In the context of the passage, 'perpetuate' means to continue, as it refers to the ongoing nature of inequalities.
Correct Answer: B — Continue
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