Q. If two angles are supplementary and one angle measures 3 times the other, what is the measure of the smaller angle? (2023)
A.
30 degrees
B.
45 degrees
C.
60 degrees
D.
75 degrees
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Solution
Let the smaller angle be x. Then the larger angle is 3x. Since they are supplementary, x + 3x = 180. Solving gives x = 45 degrees.
Correct Answer:
A
— 30 degrees
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Q. If two circles intersect at two points, what can be said about their centers?
A.
They are the same.
B.
They are equidistant from the intersection points.
C.
They lie on the same line.
D.
They are at a fixed distance apart.
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Solution
The centers of two intersecting circles are equidistant from the points of intersection.
Correct Answer:
B
— They are equidistant from the intersection points.
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Q. If two circles intersect at two points, what can be said about their relationship?
A.
They are concentric.
B.
They are tangent to each other.
C.
They are secant to each other.
D.
They do not intersect.
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Solution
Circles that intersect at two points are said to be secant to each other.
Correct Answer:
C
— They are secant to each other.
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Q. If two circles intersect at two points, which of the following statements is true? (2021)
A.
The centers of the circles are equidistant from the intersection points.
B.
The line joining the centers of the circles passes through the intersection points.
C.
The circles are tangent to each other.
D.
The area of intersection is always a triangle.
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Solution
The centers of the circles are equidistant from the intersection points, making option 0 true.
Correct Answer:
A
— The centers of the circles are equidistant from the intersection points.
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Q. If two dice are rolled, what is the probability that the sum is greater than 10?
A.
1/12
B.
1/6
C.
1/8
D.
1/4
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Solution
The combinations for a sum greater than 10 are (5,6), (6,5), (6,6), giving a probability of 3/36 = 1/12.
Correct Answer:
B
— 1/6
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Q. If two dice are rolled, what is the probability that the sum of the numbers on the dice is 7?
A.
1/6
B.
1/12
C.
1/36
D.
5/36
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Solution
The combinations that give a sum of 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1), totaling 6 combinations. The total outcomes are 36, so the probability is 6/36 = 1/6.
Correct Answer:
A
— 1/6
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Q. If two linear equations are represented as ax + by = c and dx + ey = f, under what condition will they be parallel?
A.
If a/e = b/d
B.
If a/d = b/e
C.
If a/b = c/f
D.
If c/f = d/e
Show solution
Solution
Two lines are parallel if their slopes are equal, which occurs when a/d = b/e.
Correct Answer:
B
— If a/d = b/e
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Q. If two linear equations are represented by the lines y = 2x + 1 and y = 2x - 3, what can be inferred about their relationship?
A.
They intersect at one point.
B.
They are parallel.
C.
They coincide.
D.
They are perpendicular.
Show solution
Solution
Both lines have the same slope (2) but different y-intercepts, indicating they are parallel.
Correct Answer:
B
— They are parallel.
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Q. If two linear equations are represented by the lines y = 2x + 3 and y = 2x - 1, what can be inferred about their relationship?
A.
They intersect at one point.
B.
They are parallel lines.
C.
They are the same line.
D.
They intersect at infinitely many points.
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Solution
Both lines have the same slope (2) but different y-intercepts, indicating they are parallel.
Correct Answer:
B
— They are parallel lines.
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Q. If two linear equations have the same slope but different y-intercepts, what can be inferred about their graphs?
A.
They are identical lines.
B.
They are parallel lines.
C.
They intersect at one point.
D.
They intersect at infinitely many points.
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Solution
Lines with the same slope but different y-intercepts are parallel and will never intersect.
Correct Answer:
B
— They are parallel lines.
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Q. If two liquids are mixed in the ratio 1:4, what is the ratio of the second liquid to the total mixture?
A.
4:5
B.
4:1
C.
1:5
D.
1:4
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Solution
Total parts = 1 + 4 = 5. Ratio of second liquid to total = 4:5.
Correct Answer:
A
— 4:5
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Q. If two numbers are 18 and 24, what is their HCF? (2023)
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Solution
The HCF of 18 and 24 is 6, as it is the largest number that divides both.
Correct Answer:
B
— 12
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Q. If two numbers are in the ratio 2:3 and their LCM is 60, what is the sum of the two numbers?
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Solution
Let the numbers be 2x and 3x. Then, LCM(2x, 3x) = 6x = 60, which gives x = 10. The numbers are 20 and 30, and their sum is 50.
Correct Answer:
B
— 40
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Q. If two parallel lines are cut by a transversal, which of the following pairs of angles are always equal?
A.
Alternate interior angles
B.
Corresponding angles
C.
Consecutive interior angles
D.
All of the above
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Solution
When two parallel lines are cut by a transversal, alternate interior angles and corresponding angles are equal, while consecutive interior angles are supplementary. Therefore, the correct answer is 'All of the above'.
Correct Answer:
D
— All of the above
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Q. If two parallel lines are cut by a transversal, which of the following pairs of angles are equal?
A.
Alternate interior angles
B.
Corresponding angles
C.
Consecutive interior angles
D.
Both A and B
Show solution
Solution
Both alternate interior angles and corresponding angles are equal when two parallel lines are cut by a transversal.
Correct Answer:
D
— Both A and B
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Q. If two parallel lines are cut by a transversal, which of the following statements is true? (2023)
A.
Alternate interior angles are equal
B.
Corresponding angles are supplementary
C.
Consecutive interior angles are equal
D.
All angles are acute
Show solution
Solution
When two parallel lines are cut by a transversal, alternate interior angles are equal.
Correct Answer:
A
— Alternate interior angles are equal
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Q. If two partners decide to dissolve their partnership, what is the first step they should take?
A.
Notify their clients
B.
Liquidate the assets
C.
Draft a dissolution agreement
D.
File for bankruptcy
Show solution
Solution
The first step in dissolving a partnership is to draft a dissolution agreement that outlines the terms of the dissolution.
Correct Answer:
C
— Draft a dissolution agreement
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Q. If two partners decide to share profits equally but one partner contributes more capital, what is the likely outcome?
A.
The partner with more capital will receive a larger share of profits
B.
The partners will renegotiate their profit-sharing agreement
C.
The partnership will dissolve
D.
The partner with less capital will receive a larger share of profits
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Solution
Typically, partners who contribute more capital may negotiate for a larger share of profits.
Correct Answer:
B
— The partners will renegotiate their profit-sharing agreement
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Q. If two partners in a business decide to split their profits equally, which of the following statements is true? (2023)
A.
They must have equal investments in the business.
B.
They can have different levels of involvement in the business.
C.
They are legally required to file taxes jointly.
D.
They cannot change the profit-sharing ratio later.
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Solution
Partners can agree to share profits equally regardless of their investment or involvement levels.
Correct Answer:
B
— They can have different levels of involvement in the business.
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Q. If x + 2 = 5, what is the value of x?
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Solution
To find x, subtract 2 from both sides of the equation: x = 5 - 2, which gives x = 3.
Correct Answer:
B
— 3
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Q. If x + 3 = 7, what is the value of x?
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Solution
To find x, subtract 3 from both sides: x = 7 - 3, which gives x = 4.
Correct Answer:
A
— 4
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Q. If x = 2 and y = 3, what is the value of 2^(x+y)?
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Solution
Substituting x and y gives us 2^(2+3) = 2^5 = 32.
Correct Answer:
B
— 16
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Q. If x = 2 and y = 3, what is the value of x^y + y^x?
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Solution
Calculating, we find 2^3 + 3^2 = 8 + 9 = 17.
Correct Answer:
B
— 17
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Q. If x = 2^3 and y = 2^2, what is the value of x/y? (2023)
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Solution
We have x = 8 and y = 4. Thus, x/y = 8/4 = 2.
Correct Answer:
A
— 2
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Q. If x ≡ 2 (mod 3) and y ≡ 1 (mod 3), what is the value of (x + y) mod 3?
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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?
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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 5), which of the following values of x is NOT possible?
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Solution
x can be expressed as 4 + 5k, where k is an integer. 19 is not congruent to 4 modulo 5.
Correct Answer:
D
— 19
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Q. If x ≡ 4 (mod 6), which of the following could be a possible value of x?
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Solution
10 mod 6 = 4, so x could be 10.
Correct Answer:
A
— 10
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Q. If x ≡ 4 (mod 7) and x ≡ 5 (mod 11), what is the smallest positive integer solution for x?
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Solution
Using the method of successive substitutions or the Chinese Remainder Theorem, we find that the smallest positive integer solution is x = 25.
Correct Answer:
B
— 25
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Q. If x ≡ 4 (mod 7) and y ≡ 3 (mod 7), what is the value of (x + y) mod 7?
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Solution
x + y ≡ 4 + 3 ≡ 7 (mod 7), which is equivalent to 0.
Correct Answer:
A
— 0
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