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. Calculate the determinant of the matrix F = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]. (2022)

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

Solution

The determinant of F is 0 because the rows are linearly dependent.

Correct Answer: A — 0

Q. Calculate the determinant of the matrix J = [[1, 2, 1], [0, 1, 0], [2, 3, 1]]. (2023)

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

Solution

The determinant of J is calculated as 1*(1*1 - 0*3) - 2*(0*1 - 0*2) + 1*(0*3 - 1*2) = 1 - 0 - 2 = -1.

Correct Answer: B — 1

Q. Calculate the determinant of the matrix [[1, 2], [3, 4]].

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

Solution

Determinant = (1*4) - (2*3) = 4 - 6 = -2.

Correct Answer: A — -2

Q. Calculate the determinant of the matrix \( B = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).

A. -1
B. 1
C. 7
D. 10

Solution

The determinant is calculated as \( 2*7 - 3*5 = 14 - 15 = -1 \).

Correct Answer: D — 10

Q. Calculate the determinant of the matrix \( C = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \). (2020)

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

Solution

The determinant is \( 5*8 - 6*7 = 40 - 42 = -2 \).

Correct Answer: A — -2

Q. Calculate the determinant of the matrix \( G = \begin{pmatrix} 2 & 1 & 3 \\ 1 & 0 & 1 \\ 3 & 1 & 2 \end{pmatrix} \).

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

Solution

The determinant is calculated as \( 2(0*2 - 1*1) - 1(1*2 - 3*1) + 3(1*1 - 3*0) = 0 \).

Correct Answer: A — -1

Q. Calculate the determinant of the matrix \( G = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \). (2020)

A. 5
B. 10
C. 1
D. 8

Solution

The determinant is \( 2*4 - 1*3 = 8 - 3 = 5 \).

Correct Answer: A — 5

Q. Calculate the determinant of the matrix \( H = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 3 \\ 0 & 0 & 1 \end{pmatrix} \). (2020)

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

Solution

The determinant of an upper triangular matrix is the product of its diagonal elements: \( 1*1*1 = 1 \).

Correct Answer: A — 1

Q. Calculate the determinant of the matrix \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2023)

A. 10
B. 11
C. 12
D. 13

Solution

Det(I) = (3*4) - (2*1) = 12 - 2 = 10.

Correct Answer: A — 10

Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).

A. 5
B. 10
C. 7
D. 8

Solution

The determinant is calculated as \( 2*4 - 3*1 = 8 - 3 = 5 \).

Correct Answer: A — 5

Q. Calculate the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \).

A. 1
B. 2
C. 3
D. 4

Solution

The determinant is calculated as (2*7) - (3*5) = 14 - 15 = -1.

Correct Answer: A — 1

Q. Calculate the determinant of the matrix: | 1 1 1 | | 2 2 2 | | 3 3 3 |

A. 0
B. 3
C. 6
D. 9

Solution

The rows are linearly dependent, hence the determinant is 0.

Correct Answer: A — 0

Q. Calculate the determinant of \( D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \). (2021)

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

Solution

The determinant is \( 0*0 - 1*1 = 0 - 1 = -1 \).

Correct Answer: A — 1

Q. Calculate the determinant of \( D = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 2 & 3 \\ 1 & 3 & 6 \end{pmatrix} \). (2021)

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

Solution

The determinant of this matrix is 0 because the rows are linearly dependent.

Correct Answer: A — 0

Q. Calculate the determinant of \( I = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix} \). (2022)

A. 10
B. 8
C. 6
D. 12

Solution

The determinant is calculated as \( 3*4 - 2*1 = 12 - 2 = 10 \).

Correct Answer: B — 8

Q. Calculate the determinant \( \begin{vmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 3 & 4 & 1 \end{vmatrix} \).

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

Solution

The determinant is 0 because the rows are linearly dependent.

Correct Answer: A — 0

Q. Calculate the determinant \( \begin{vmatrix} 2 & 3 \\ 5 & 7 \end{vmatrix} \)

A. 1
B. 2
C. 3
D. 4

Solution

The determinant is \( 2*7 - 3*5 = 14 - 15 = -1 \).

Correct Answer: A — 1

Q. Calculate the determinant \( \begin{vmatrix} a & b \\ c & d \end{vmatrix} \)

A. ad - bc
B. ab + cd
C. ac - bd
D. bc - ad

Solution

The determinant is calculated as \( ad - bc \).

Correct Answer: A — ad - bc

Q. Calculate the determinant | 1 0 0 | | 0 1 0 | | 0 0 1 |.

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

Solution

The determinant of the identity matrix is 1.

Correct Answer: B — 1

Q. Calculate the determinant | 2 3 | | 4 5 | + | 1 1 | | 1 1 |.

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

Solution

The first determinant is -2 and the second is 0, so the total is -2 + 0 = -2.

Correct Answer: B — 1

Q. Calculate the determinant: | 2 3 1 | | 1 0 2 | | 0 1 3 |.

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

Solution

The determinant evaluates to 0 as the rows are linearly dependent.

Correct Answer: A — -1

Q. Calculate the determinant: | 2 3 1 | | 1 0 4 | | 0 5 2 |.

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

Solution

Using the determinant formula, we find that the determinant evaluates to 0.

Correct Answer: A — -1

Q. Calculate the distance between the points (1, 1) and (4, 5).

A. 4
B. 5
C. 6
D. 7

Solution

Using the distance formula: d = √[(4 - 1)² + (5 - 1)²] = √[9 + 16] = √25 = 5.

Correct Answer: B — 5

Q. Calculate the distance between the points (1, 2) and (1, 5).

A. 3
B. 4
C. 2
D. 5

Solution

Using the distance formula: d = √[(1 - 1)² + (5 - 2)²] = √[0 + 9] = √9 = 3.

Correct Answer: A — 3

Q. Calculate the distance between the points (6, 8) and (2, 3).

A. 5
B. 6
C. 7
D. 8

Solution

Using the distance formula: d = √[(2 - 6)² + (3 - 8)²] = √[16 + 25] = √41.

Correct Answer: B — 6

Q. Calculate the distance between the points (6, 8) and (6, 2).

A. 6
B. 5
C. 4
D. 3

Solution

Using the distance formula: d = √((6 - 6)² + (2 - 8)²) = √(0 + 36) = √36 = 6.

Correct Answer: A — 6

Q. Calculate the distance from the point (1, 2, 3) to the origin (0, 0, 0). (2021)

A. √14
B. √6
C. √9
D. √12

Solution

Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.

Correct Answer: A — √14

Q. Calculate the distance from the point P(1, 2, 3) to the origin O(0, 0, 0). (2023)

A. 3
B. √14
C. √6
D. √9

Solution

Distance = √[(1-0)² + (2-0)² + (3-0)²] = √[1 + 4 + 9] = √14.

Correct Answer: B — √14

Q. Calculate the gravitational potential energy of a 2 kg mass at a height of 5 m. (g = 9.8 m/s²)

A. 98 J
B. 19.6 J
C. 39.2 J
D. 49 J

Solution

Potential Energy (PE) = m * g * h = 2 kg * 9.8 m/s² * 5 m = 98 J

Correct Answer: C — 39.2 J

Q. Calculate the integral ∫ (x^2 + 2x + 1) dx.

A. (1/3)x^3 + x^2 + x + C
B. (1/3)x^3 + x^2 + C
C. (1/3)x^3 + 2x^2 + C
D. (1/3)x^3 + x^2 + x

Solution

The integral of x^2 is (1/3)x^3, the integral of 2x is x^2, and the integral of 1 is x. Thus, ∫ (x^2 + 2x + 1) dx = (1/3)x^3 + x^2 + x + C.

Correct Answer: A — (1/3)x^3 + x^2 + x + C

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