Major Competitive Exams

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Major Competitive Exams

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Major Competitive Exams MCQ & Objective Questions

Major Competitive Exams play a crucial role in shaping the academic and professional futures of students in India. These exams not only assess knowledge but also test problem-solving skills and time management. Practicing MCQs and objective questions is essential for scoring better, as they help in familiarizing students with the exam format and identifying important questions that frequently appear in tests.

What You Will Practise Here

Key concepts and theories related to major subjects
Important formulas and their applications
Definitions of critical terms and terminologies
Diagrams and illustrations to enhance understanding
Practice questions that mirror actual exam patterns
Strategies for solving objective questions efficiently
Time management techniques for competitive exams

Exam Relevance

The topics covered under Major Competitive Exams are integral to various examinations such as CBSE, State Boards, NEET, and JEE. Students can expect to encounter a mix of conceptual and application-based questions that require a solid understanding of the subjects. Common question patterns include multiple-choice questions that test both knowledge and analytical skills, making it essential to be well-prepared with practice MCQs.

Common Mistakes Students Make

Rushing through questions without reading them carefully
Overlooking the negative marking scheme in MCQs
Confusing similar concepts or terms
Neglecting to review previous years’ question papers
Failing to manage time effectively during the exam

FAQs

Question: How can I improve my performance in Major Competitive Exams?
Answer: Regular practice of MCQs and understanding key concepts will significantly enhance your performance.

Question: What types of questions should I focus on for these exams?
Answer: Concentrate on important Major Competitive Exams questions that frequently appear in past papers and mock tests.

Question: Are there specific strategies for tackling objective questions?
Answer: Yes, practicing under timed conditions and reviewing mistakes can help develop effective strategies.

Start your journey towards success by solving practice MCQs today! Test your understanding and build confidence for your upcoming exams. Remember, consistent practice is the key to mastering Major Competitive Exams!

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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

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]

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]

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

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]

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]]

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

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

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]]

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

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

Q. If A = [[2, 3], [1, 4]], what is A^2? (2020)

A. [[7, 18], [18, 7]]
B. [[12, 21], [21, 12]]
C. [[12, 21], [21, 16]]
D. [[10, 21], [21, 10]]

Solution

A^2 = A * A = [[2*2 + 3*1, 2*3 + 3*4], [1*2 + 4*1, 1*3 + 4*4]] = [[10, 21], [21, 16]].

Correct Answer: C — [[12, 21], [21, 16]]

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

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

Solution

The inverse of A is given by (1/det(A)) * adj(A). Det(A) = (2*4) - (3*1) = 5. The adjoint is [[4, -3], [-1, 2]]. Thus, A^(-1) = (1/5) * [[4, -3], [-1, 2]].

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

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}

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

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

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

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

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)}

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

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

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}

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}

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}

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

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

Solution

The Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B, which is {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}.

Correct Answer: B — {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}

Q. If A = {1, 2, 3}, how many subsets does A have?

A. 2
B. 3
C. 4
D. 8

Solution

The number of subsets of a set with n elements is 2^n. Here, n = 3, so 2^3 = 8.

Correct Answer: D — 8

Q. If A = {1, 2, 3}, what is the power set of A?

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

Solution

The power set of a set with n elements has 2^n subsets. Here, n=3, so the power set has 2^3 = 8 subsets.

Correct Answer: A — {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}

Q. If A = {1, 2} and B = {x | x is an odd integer}, what is A × B?

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

Solution

The Cartesian product A × B consists of all ordered pairs (a, b) where a ∈ A and b ∈ B. Thus, A × B = {(1, x), (2, x)} for all odd integers x.

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

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

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

Solution

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

Correct Answer: A — {2}

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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