Q. If a hollow sphere and a solid sphere of the same mass and radius roll down the same incline, which one reaches the bottom first?
A.
Hollow sphere
B.
Solid sphere
C.
Both reach at the same time
D.
Depends on the angle of incline
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Solution
The solid sphere has a smaller moment of inertia compared to the hollow sphere, thus it accelerates faster and reaches the bottom first.
Correct Answer:
B
— Solid sphere
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Q. If a hollow sphere rolls down an incline, how does its acceleration compare to that of a solid sphere?
A.
Greater
B.
Less
C.
Equal
D.
Depends on mass
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Solution
The hollow sphere has a greater moment of inertia, resulting in less acceleration compared to the solid sphere.
Correct Answer:
B
— Less
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Q. If a household uses 250 liters of water per day, how much water will it use in a month (30 days)?
A.
6000 liters
B.
7500 liters
C.
8000 liters
D.
9000 liters
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Solution
250 liters/day * 30 days = 7500 liters
Correct Answer:
B
— 7500 liters
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Q. If a hybrid set contains elements from both set X and set Y, which of the following is a valid conclusion?
A.
All elements of set X are in the hybrid set.
B.
All elements of set Y are in the hybrid set.
C.
Some elements of set X may not be in the hybrid set.
D.
The hybrid set cannot contain elements from both sets.
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Solution
It is possible for some elements of set X to not be included in the hybrid set, as hybrid sets can vary in composition.
Correct Answer:
C
— Some elements of set X may not be in the hybrid set.
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Q. If a hybrid set contains elements from three different categories, which of the following is a valid inference?
A.
All elements belong to the same category.
B.
At least one element must belong to each category.
C.
No elements can belong to more than one category.
D.
The categories must be mutually exclusive.
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Solution
In a hybrid set with elements from different categories, at least one element must belong to each category.
Correct Answer:
B
— At least one element must belong to each category.
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Q. If a hybrid set contains elements from three different sets, what can be inferred about its composition?
A.
It must contain at least one element from each set.
B.
It can only contain elements from the largest set.
C.
It will have no duplicates.
D.
It can include elements that are not in any of the three sets.
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Solution
A hybrid set formed from multiple sets typically includes at least one element from each contributing set.
Correct Answer:
A
— It must contain at least one element from each set.
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Q. If a hybrid set contains elements from three different sets, what can be inferred about its size compared to the individual sets? (2023)
A.
It will always be smaller than the smallest set.
B.
It can be larger than the sum of the sizes of the individual sets.
C.
It will be equal to the largest set.
D.
It can be smaller than the sum of the sizes of the individual sets.
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Solution
The size of a hybrid set can vary and may be smaller than the sum of the sizes of the individual sets due to overlapping elements.
Correct Answer:
D
— It can be smaller than the sum of the sizes of the individual sets.
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Q. If a hybrid set contains elements from three different sets, which of the following is a possible characteristic of this hybrid set?
A.
It can only contain elements from two of the sets.
B.
It must contain all elements from each of the three sets.
C.
It may contain elements that are common to all three sets.
D.
It cannot have any elements that are unique to one set.
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Solution
A hybrid set can include elements that are common to all three sets, as well as unique elements.
Correct Answer:
C
— It may contain elements that are common to all three sets.
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Q. If a hybrid set contains elements {x, y, z} and another set contains {y, z, w}, what is the hybrid set's complement with respect to the universal set {x, y, z, w, v}?
A.
{v}
B.
{x, w}
C.
{y, z}
D.
{x, y, z, w}
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Solution
The complement of the hybrid set with respect to the universal set includes elements that are in the universal set but not in the hybrid set, which is {v}.
Correct Answer:
A
— {v}
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Q. If a hybrid set is created from set T = {1, 2} and set U = {2, 3, 4}, what is the resulting hybrid set?
A.
{1, 2, 3, 4}
B.
{2}
C.
{1, 3, 4}
D.
{1, 2, 4}
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Solution
The hybrid set formed by the union of set T and set U includes all unique elements from both sets, which are {1, 2, 3, 4}.
Correct Answer:
A
— {1, 2, 3, 4}
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Q. If a hybrid set is defined as A ∪ B, which of the following is true?
A.
It includes all elements from A and B.
B.
It includes only the common elements of A and B.
C.
It includes elements from A but not from B.
D.
It includes elements from B but not from A.
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Solution
The union of sets A and B includes all elements from both sets, regardless of whether they are common or unique.
Correct Answer:
A
— It includes all elements from A and B.
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Q. If a hybrid set is defined as the combination of set R = {x, y} and set S = {y, z}, which of the following is the correct representation of the hybrid set?
A.
{x, y, z}
B.
{y}
C.
{x, y}
D.
{y, z}
Show solution
Solution
The hybrid set formed by combining set R and set S includes all unique elements from both sets, which are {x, y, z}.
Correct Answer:
A
— {x, y, z}
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Q. If a hybrid set is defined as the combination of set R = {x, y} and set S = {y, z}, what is the resulting hybrid set?
A.
{x, y, z}
B.
{y, z}
C.
{x, y}
D.
{x, y, y, z}
Show solution
Solution
The resulting hybrid set is {x, y, z}, which includes all unique elements from both sets.
Correct Answer:
A
— {x, y, z}
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Q. If a hybrid set is defined as the combination of two sets with some overlapping elements, which of the following is a characteristic of such a set?
A.
It cannot contain duplicate elements.
B.
It must contain all elements from both sets.
C.
It can only contain elements from one of the sets.
D.
It is always larger than the individual sets.
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Solution
A hybrid set cannot contain duplicate elements, as it is defined to include unique elements from both sets.
Correct Answer:
A
— It cannot contain duplicate elements.
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Q. If a hybrid set is defined as the intersection of two sets, which of the following is true?
A.
It contains all elements from both sets.
B.
It contains only the elements that are common to both sets.
C.
It can be empty if there are no common elements.
D.
Both B and C are correct.
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Solution
The intersection of two sets contains only the elements that are common to both sets, and it can be empty if there are no common elements.
Correct Answer:
D
— Both B and C are correct.
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Q. If a hybrid set is defined as the union of sets A = {a, b} and B = {b, c}, what is the resulting hybrid set? (2023)
A.
{a, b, c}
B.
{b}
C.
{a, b, b, c}
D.
{a, c}
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Solution
The union of sets A and B combines all unique elements, resulting in the hybrid set {a, b, c}.
Correct Answer:
A
— {a, b, c}
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Q. If A invests $50,000 and B invests $70,000 in a partnership, and they agree to share profits in the ratio of their investments, what will be A's share if the total profit is $30,000?
A.
$12,000
B.
$15,000
C.
$18,000
D.
$20,000
Show solution
Solution
Total investment = 50000 + 70000 = 120000. A's share = (50000/120000) * 30000 = $12,500.
Correct Answer:
B
— $15,000
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Q. If A is a 2x2 matrix and B is a 2x2 matrix, what is the order of the product AB? (2019)
A.
2x2
B.
2x3
C.
3x2
D.
3x3
Show solution
Solution
The order of the product of two matrices is determined by the outer dimensions. Since both A and B are 2x2 matrices, their product AB will also be a 2x2 matrix.
Correct Answer:
A
— 2x2
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Q. If A is a 2x2 matrix and B is a 2x3 matrix, what is the order of the product AB? (2019)
A.
2x2
B.
2x3
C.
3x2
D.
2x5
Show solution
Solution
The order of the product of two matrices is determined by the outer dimensions. Here, A (2x2) and B (2x3) can be multiplied, resulting in a matrix of order 2x3.
Correct Answer:
B
— 2x3
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Q. If A is a 2x2 matrix such that A^2 = I, where I is the identity matrix, then which of the following is true?
A.
A is invertible
B.
A is singular
C.
A is a zero matrix
D.
A is a diagonal matrix
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Solution
Since A^2 = I, A is invertible because the inverse of A is A itself.
Correct Answer:
A
— A is invertible
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Q. If A is a 2x2 matrix with eigenvalues 1 and -1, what is the determinant of A?
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Solution
The determinant of a matrix is the product of its eigenvalues. Thus, det(A) = 1 * (-1) = -1.
Correct Answer:
C
— -1
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Q. If A is a 2x2 matrix with eigenvalues 3 and 5, what is the trace of A?
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Solution
The trace of a matrix is the sum of its eigenvalues. Therefore, trace(A) = 3 + 5 = 8.
Correct Answer:
A
— 8
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Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the maximum number of non-zero elements in A + B? (2021)
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Solution
The maximum number of non-zero elements in the sum of two matrices occurs when all elements of both matrices are non-zero. Therefore, A + B can have a maximum of 9 non-zero elements.
Correct Answer:
A
— 9
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Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the maximum order of the resultant matrix when A is added to B? (2021)
A.
3x3
B.
3x2
C.
2x3
D.
3x1
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Solution
The sum of two matrices of the same order results in a matrix of the same order. Therefore, A + B will be a 3x3 matrix.
Correct Answer:
A
— 3x3
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Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the maximum order of the resultant matrix AB? (2023)
A.
3x3
B.
2x2
C.
3x2
D.
2x3
Show solution
Solution
The product of two matrices A and B, both of order 3x3, will also be a matrix of order 3x3.
Correct Answer:
A
— 3x3
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Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the maximum order of the resultant matrix when A is multiplied by B? (2022)
A.
3x3
B.
6x6
C.
9x9
D.
3x6
Show solution
Solution
The order of the resultant matrix when two matrices are multiplied is determined by the outer dimensions. Here, both A and B are 3x3, so the product AB is also 3x3.
Correct Answer:
A
— 3x3
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Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the order of A + B? (2023)
A.
3x3
B.
3x2
C.
2x3
D.
3x1
Show solution
Solution
The order of the sum of two matrices is the same as the order of the individual matrices. Therefore, A + B is a 3x3 matrix.
Correct Answer:
A
— 3x3
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Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the order of the matrix A + B? (2019)
A.
3x3
B.
3x2
C.
2x3
D.
3x1
Show solution
Solution
The sum of two matrices is defined only when they have the same order. Since both A and B are 3x3 matrices, A + B will also be a 3x3 matrix.
Correct Answer:
A
— 3x3
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Q. If A is a 3x3 matrix and B is a 3x3 matrix, what is the order of the matrix product AB? (2021)
A.
3x3
B.
2x2
C.
3x2
D.
2x3
Show solution
Solution
The product of two matrices A and B, both of order 3x3, will also be a matrix of order 3x3.
Correct Answer:
A
— 3x3
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Q. If A is a 3x3 matrix with eigenvalues 2, 3, and 4, what is the trace of A?
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Solution
The trace of a matrix is the sum of its eigenvalues. Thus, trace(A) = 2 + 3 + 4 = 9.
Correct Answer:
A
— 9
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