Algebra Study Resources for Competitive Exam Success

Algebra

Binomial theorem Complex numbers Determinants Inverse Trigonometric Functions Linear Inequalities Logarithms Matrices Permutations & combinations Quadratic equations Sequence & series Sets & Relations

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 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}, 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 = {3, 4, 5}, what is A - B?

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

Solution

The difference of sets A and B, A - B, contains elements that are in A but not in B. Here, A - B = {1, 2}.

Correct Answer: A — {1, 2}

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Q. If A = {1, 2, 3} and B = {3, 4, 5}, what is A Δ B (symmetric difference)?

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

Solution

The symmetric difference A Δ B includes elements in either A or B but not in both, which are {1, 2, 4, 5}.

Correct Answer: C — {1, 2, 4, 5}

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Q. If A = {1, 2, 3} and B = {3, 4, 5}, what is A ∪ (A ∩ B)?

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

Solution

First, A ∩ B = {3}. Then, A ∪ {3} = {1, 2, 3}.

Correct Answer: A — {1, 2, 3}

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Q. If A = {x | x is a letter in the word 'MATH'} and B = {x | x is a letter in the word 'SCIENCE'}, what is A ∩ B?

A. {A}
B. {M, A, T}
C. {A, C, E}
D. {A, T}

Solution

The intersection A ∩ B includes letters that are common in both words. The only common letter is 'A', so A ∩ B = {A}.

Correct Answer: A — {A}

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Q. If A = {x | x is a letter in the word 'MATH'} and B = {x | x is a letter in the word 'SET'}, what is A ∩ B?

A. {M, A, T}
B. {A, T}
C. {T}
D. {}

Solution

The intersection A ∩ B consists of common letters, which is {T}.

Correct Answer: C — {T}

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Q. If A = {x | x is a multiple of 3} and B = {x | x is a multiple of 5}, what is A ∩ B?

A. {15}
B. {3, 5}
C. {0}
D. {}

Solution

The intersection A ∩ B consists of common multiples, which is {0}.

Correct Answer: C — {0}

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Q. If A = {x | x is a natural number and x < 5} and B = {x | x is a natural number and x > 2}, what is A ∩ B?

A. {1, 2}
B. {3, 4}
C. {2, 3, 4}
D. {1, 2, 3, 4}

Solution

Set A = {1, 2, 3, 4} and set B = {3, 4, 5, ...}. The intersection A ∩ B = {3, 4}.

Correct Answer: B — {3, 4}

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Q. If A = {x | x is a natural number less than 5} and B = {x | x is a natural number less than 3}, what is A - B?

A. {1, 2}
B. {3, 4}
C. {1, 2, 3, 4}
D. {2, 3, 4}

Solution

Set A = {1, 2, 3, 4} and set B = {1, 2}. Thus, A - B = {3, 4}.

Correct Answer: B — {3, 4}

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Q. If A = {x | x is a natural number less than 5} and B = {x | x is an odd natural number}, what is A ∩ B?

A. {1, 2, 3, 4}
B. {1, 3}
C. {2, 4}
D. {1, 2, 3}

Solution

The intersection A ∩ B includes elements that are in both A and B. Here, A = {1, 2, 3, 4} and B = {1, 3}, so A ∩ B = {1, 3}.

Correct Answer: B — {1, 3}

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Q. If A = {x | x is a prime number less than 10} and B = {2, 3, 5, 7}, what is A = B?

A. True
B. False
C. Cannot be determined
D. None of the above

Solution

Both sets A and B contain the same elements: {2, 3, 5, 7}. Therefore, A = B is True.

Correct Answer: A — True

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Q. If A = {x | x is a prime number less than 10} and B = {2, 3, 5, 7}, what is A?

A. {2, 3, 5, 7}
B. {1, 2, 3, 4, 5, 6, 7, 8, 9}
C. {2, 3, 5, 7, 11}
D. {2, 3, 5, 7, 9}

Solution

The set A consists of all prime numbers less than 10, which are {2, 3, 5, 7}.

Correct Answer: A — {2, 3, 5, 7}

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Q. If A = {x | x is a vowel} and B = {x | x is a consonant}, what is A ∩ B?

A. {a, e, i, o, u}
B. {}
C. {a, b, c}
D. {a, e, i}

Solution

The intersection A ∩ B is empty because no letter can be both a vowel and a consonant.

Correct Answer: B — {}

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Q. If A = {x | x is an even integer} and B = {x | x is a multiple of 3}, what is A ∪ B?

A. {0, 2, 4, 6, ...}
B. {0, 3, 6, 9, ...}
C. {0, 2, 3, 4, 6, 9, ...}
D. {0, 2, 3, 4, 6, 8, 9, ...}

Solution

The union of sets A and B, A ∪ B, includes all even integers and all multiples of 3. Thus, A ∪ B = {0, 2, 3, 4, 6, 9, ...}.

Correct Answer: C — {0, 2, 3, 4, 6, 9, ...}

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Q. If A = {x | x is an even integer} and B = {x | x is a prime number}, what is A ∩ B?

A. {2}
B. {2, 3}
C. {2, 4}
D. {}

Solution

The only even prime number is 2, so A ∩ B = {2}.

Correct Answer: A — {2}

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Algebra

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