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Q. If a = 4 and b = 5, what is the value of a² + b²?

A. 41
B. 25
C. 16
D. 9

Solution

a² + b² = 4² + 5² = 16 + 25 = 41.

Correct Answer: A — 41

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Q. If A = 4i + 2j and B = -i + 3j, what is the scalar triple product A · (B × A)? (2023)

A. 0
B. 8
C. 12
D. 16

Solution

B × A = |i  j  k| |-1  3  0| |4  2  0| = 0, hence A · (B × A) = 0.

Correct Answer: A — 0

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Q. If A = 4i + 3j and B = 0i + 5j, what is A · B? (2019)

A. 0
B. 15
C. 20
D. 12

Solution

A · B = (4)(0) + (3)(5) = 0 + 15 = 15

Correct Answer: B — 15

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Q. If A = 5i + 12j, what is the unit vector in the direction of A? (2021)

A. (5/13)i + (12/13)j
B. (12/13)i + (5/13)j
C. (5/12)i + (12/5)j
D. (12/5)i + (5/12)j

Solution

Unit vector = A / |A| = (5i + 12j) / √(5^2 + 12^2) = (5/13)i + (12/13)j.

Correct Answer: A — (5/13)i + (12/13)j

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Q. If A = 5i - 2j and B = -3i + 4j, calculate A · B. (2023)

A. -23
B. -7
C. 7
D. 23

Solution

A · B = (5)(-3) + (-2)(4) = -15 - 8 = -23

Correct Answer: A — -23

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Q. If A = 6i + 8j and B = 3i + 4j, what is the scalar product A · B?

A. 50
B. 54
C. 60
D. 66

Solution

A · B = (6)(3) + (8)(4) = 18 + 32 = 50.

Correct Answer: B — 54

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Q. If A = 7i + 1j and B = 2i + 3j, what is the scalar product A · B? (2022)

A. 23
B. 21
C. 19
D. 17

Solution

A · B = (7)(2) + (1)(3) = 14 + 3 = 17

Correct Answer: A — 23

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Q. If A = 7i + 24j and B = 24i - 7j, calculate A · B.

A. 0
B. 1
C. 2
D. 3

Solution

A · B = (7)(24) + (24)(-7) = 168 - 168 = 0.

Correct Answer: A — 0

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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, find A · B.

A. 32
B. 30
C. 28
D. 34

Solution

A · B = (1)(4) + (2)(5) + (3)(6) = 4 + 10 + 18 = 32.

Correct Answer: B — 30

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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, what is A - B? (2023)

A. -3i - 3j - 3k
B. -3i - 3j + 3k
C. 3i + 3j + 3k
D. 3i + 3j - 3k

Solution

A - B = (1 - 4)i + (2 - 5)j + (3 - 6)k = -3i - 3j - 3k.

Correct Answer: A — -3i - 3j - 3k

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Q. If A = [[1, 0], [0, 1]] and B = [[2, 3], [4, 5]], what is AB?

A. [2, 3], [4, 5]
B. [1, 0], [0, 1]
C. [0, 0], [0, 0]
D. [6, 8], [12, 15]

Solution

AB = [[1*2 + 0*4, 1*3 + 0*5], [0*2 + 1*4, 0*3 + 1*5]] = [[2, 3], [4, 5]].

Correct Answer: A — [2, 3], [4, 5]

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Q. If A = [[1, 0], [0, 1]] is the identity matrix, what is A^n for any integer n?

A. A
B. 0
C. I
D. None of the above

Solution

A^n = I for any integer n, where I is the identity matrix.

Correct Answer: C — I

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Q. If A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], what is A + B?

A. [6, 8], [10, 12]
B. [1, 2], [3, 4]
C. [5, 6], [7, 8]
D. [8, 10], [10, 12]

Solution

A + B = [[1+5, 2+6], [3+7, 4+8]] = [[6, 8], [10, 12]].

Correct Answer: A — [6, 8], [10, 12]

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Q. If A = [[1, 2], [3, 4]], find A^2.

A. [7, 10], [15, 22]
B. [1, 2], [3, 4]
C. [10, 13], [22, 29]
D. [-1, -2], [-3, -4]

Solution

A^2 = A * A = [[1*1 + 2*3, 1*2 + 2*4], [3*1 + 4*3, 3*2 + 4*4]] = [[7, 10], [15, 22]].

Correct Answer: A — [7, 10], [15, 22]

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Q. If A = [[1, 2], [3, 4]], find the determinant of A.

A. -2
B. 2
C. 0
D. 4

Solution

The determinant of A is (1*4) - (2*3) = 4 - 6 = -2.

Correct Answer: B — 2

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Q. If A = [[1, 2], [3, 4]], what is A^2?

A. [7, 10; 15, 22]
B. [1, 2; 3, 4]
C. [10, 14; 22, 30]
D. [-1, -2; -3, -4]

Solution

A^2 = A * A = [[1*1 + 2*3, 1*2 + 2*4], [3*1 + 4*3, 3*2 + 4*4]] = [[7, 10], [15, 22]].

Correct Answer: A — [7, 10; 15, 22]

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Q. If A = [[1, 2], [3, 4]], what is the adjoint of A?

A. [[4, -2], [-3, 1]]
B. [[1, 3], [2, 4]]
C. [[2, 1], [4, 3]]
D. [[0, 0], [0, 0]]

Solution

The adjoint of A is [[4, -2], [-3, 1]].

Correct Answer: A — [[4, -2], [-3, 1]]

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Q. If A = [[1, 2], [3, 4]], what is the determinant of A?

A. -2
B. 2
C. 0
D. 4

Solution

The determinant of A is (1*4) - (2*3) = 4 - 6 = -2.

Correct Answer: A — -2

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Q. If A = [[1, 2], [3, 4]], what is the eigenvalue of A?

A. 5
B. 2
C. 3
D. 1

Solution

The eigenvalues are found from the characteristic polynomial λ^2 - 5λ + 2 = 0, which gives λ = 5.

Correct Answer: A — 5

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Q. If A = [[1, 2], [3, 4]], what is the inverse of A?

A. [[4, -2], [-3, 1]]
B. [[-2, 1], [1.5, -0.5]]
C. [[-2, 1], [1.5, -0.5]]
D. [[4, -2], [-3, 1]]

Solution

The inverse of A is (1/det(A)) * adj(A) = (1/(-2)) * [[4, -2], [-3, 1]] = [[-2, 1], [1.5, -0.5]].

Correct Answer: A — [[4, -2], [-3, 1]]

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Q. If A = [[1, 2], [3, 4]], what is the trace of A?

A. 5
B. 7
C. 3
D. 1

Solution

The trace of A is the sum of the diagonal elements: 1 + 4 = 5.

Correct Answer: A — 5

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Q. If A = [[2, 0], [0, 3]], what is the eigenvalue of A?

A. 2
B. 3
C. 0
D. 5

Solution

The eigenvalues of A are the diagonal elements: 2 and 3.

Correct Answer: B — 3

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Q. If A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, what is A ∪ B?

A. {1, 2, 3, 4, 6, 8}
B. {2, 4}
C. {1, 3, 5, 7}
D. {1, 2, 3, 4, 5}

Solution

The union A ∪ B includes all elements from both sets, which are {1, 2, 3, 4, 6, 8}.

Correct Answer: A — {1, 2, 3, 4, 6, 8}

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Q. If A = {1, 2, 3} and B = {1, 2, 3, 4, 5}, what is A ⊆ B?

A. True
B. False
C. Depends on the context
D. None of the above

Solution

Set A is a subset of set B because all elements of A are contained in B. Therefore, A ⊆ B is True.

Correct Answer: A — True

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Q. If A = {1, 2, 3} and B = {1, 2, 3, 4}, what is A ⊆ B?

A. True
B. False
C. Depends on A
D. Not enough information

Solution

A is a subset of B if all elements of A are in B. Here, all elements of A are in B, so A ⊆ B is true.

Correct Answer: A — True

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Q. If A = {1, 2, 3} and B = {1, 2, 3, 4}, what is the cardinality of A × B?

A. 3
B. 6
C. 9
D. 12

Solution

The cardinality of the Cartesian product A × B is given by |A| * |B|. Here, |A| = 3 and |B| = 4, so |A × B| = 3 * 4 = 12.

Correct Answer: B — 6

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Q. If A = {1, 2, 3} and B = {1, 2, 3, 4}, what is the number of subsets of A ∪ B?

A. 8
B. 16
C. 4
D. 12

Solution

A ∪ B = {1, 2, 3, 4}, which has 4 elements. The number of subsets is 2^4 = 16.

Correct Answer: B — 16

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Q. If A = {1, 2, 3} and B = {1, 2, 3}, what is A × B?

A. {(1,1), (2,2), (3,3)}
B. {(1,2), (2,3), (3,1)}
C. {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}
D. {}

Solution

A × B = {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)} as it is the Cartesian product of A and B.

Correct Answer: C — {(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)}

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Q. If A = {1, 2, 3} and B = {1, 2}, what is the number of elements in A × B?

A. 2
B. 3
C. 6
D. 4

Solution

The number of elements in the Cartesian product A × B is the product of the number of elements in A and B. Here, |A| = 3 and |B| = 2, so |A × B| = 3 * 2 = 6.

Correct Answer: D — 4

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Q. If A = {1, 2, 3} and B = {2, 3, 4}, what is |A ∩ B|?

A. 1
B. 2
C. 3
D. 0

Solution

A ∩ B = {2, 3}, so |A ∩ B| = 2.

Correct Answer: B — 2

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Major Competitive Exams

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