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!
Q. Find the determinant of the matrix \( J = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \). (2022)
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Solution
The determinant is \( 2*7 - 3*5 = 14 - 15 = -1 \).
Correct Answer:
A
— 1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).
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Solution
The determinant of the identity matrix is always 1.
Correct Answer:
B
— 1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix} \).
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Solution
The determinant is calculated as \( 2*4 - 1*3 = 8 - 3 = 5 \).
Correct Answer:
A
— 5
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 0 \end{pmatrix} \).
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Solution
Using the determinant formula, we find it equals 10.
Correct Answer:
A
— -10
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 & 1 \\ 1 & 0 & 4 \\ 5 & 2 & 1 \end{pmatrix} \).
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Solution
The determinant evaluates to 0.
Correct Answer:
A
— -1
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Q. Find the determinant of the matrix \( \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \).
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Solution
The determinant is \( 2*4 - 3*1 = 8 - 3 = 5 \).
Correct Answer:
A
— 5
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Q. Find the determinant of the matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \).
A.
ad - bc
B.
bc - ad
C.
a + b + c + d
D.
a^2 + b^2
Show solution
Solution
The determinant is given by the formula \( ad - bc \).
Correct Answer:
A
— ad - bc
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Q. Find the determinant of the matrix | 1 0 0 | | 0 1 0 | | 0 0 1 |.
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Solution
This is the identity matrix, and its determinant is 1.
Correct Answer:
B
— 1
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Q. Find the determinant of the matrix | 1 2 3 | | 0 1 4 | | 5 6 0 |.
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Solution
The determinant evaluates to 0 as the third row can be expressed as a linear combination of the first two.
Correct Answer:
A
— -12
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Q. Find the determinant of the matrix: | 1 2 | | 3 5 |.
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Solution
det = (1*5) - (2*3) = 5 - 6 = -1.
Correct Answer:
A
— -1
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Q. Find the determinant of \( G = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \). (2021)
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Solution
The determinant is calculated as \( 2*7 - 3*5 = 14 - 15 = -1 \).
Correct Answer:
A
— 1
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Q. Find the determinant of \( G = \begin{pmatrix} 4 & 2 \\ 3 & 1 \end{pmatrix} \). (2020)
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Solution
The determinant is \( 4*1 - 2*3 = 4 - 6 = -2 \).
Correct Answer:
A
— -2
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Q. Find the dimensions of a box with a square base that maximizes volume given a surface area of 600 sq. units. (2020)
A.
10, 10
B.
15, 15
C.
12, 12
D.
20, 20
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Solution
Let x be the side of the base and h the height. The surface area constraint gives 2x^2 + 4xh = 600. Max volume occurs at x = 12.
Correct Answer:
C
— 12, 12
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Q. Find the dimensions of a rectangle with a fixed area of 50 m^2 that minimizes the perimeter. (2021)
A.
5, 10
B.
7, 7.14
C.
8, 6.25
D.
10, 5
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Solution
For a fixed area, the minimum perimeter occurs when the rectangle is a square. Thus, dimensions are approximately 7 m by 7.14 m.
Correct Answer:
B
— 7, 7.14
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Q. Find the dimensions of a rectangle with a fixed area of 50 square units that minimizes the perimeter. (2022) 2022
A.
5, 10
B.
7, 7.14
C.
10, 5
D.
8, 6.25
Show solution
Solution
For minimum perimeter, the rectangle should be a square. Thus, side = sqrt(50) ≈ 7.07.
Correct Answer:
B
— 7, 7.14
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Q. Find the dimensions of a rectangle with a fixed area of 50 square units that minimizes the perimeter. (2020)
A.
5, 10
B.
7, 7
C.
10, 5
D.
8, 6.25
Show solution
Solution
For a fixed area, the perimeter is minimized when the rectangle is a square. Thus, side = √50.
Correct Answer:
B
— 7, 7
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Q. Find the directrix of the parabola y^2 = -8x.
A.
x = 2
B.
x = -2
C.
x = 4
D.
x = -4
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Solution
For the parabola y^2 = 4px, here 4p = -8, so p = -2. The directrix is given by x = -p, which is x = 2.
Correct Answer:
B
— x = -2
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Q. Find the distance between the parallel planes 2x + 3y + 4z = 5 and 2x + 3y + 4z = 10. (2023)
A.
5/√29
B.
10/√29
C.
15/√29
D.
20/√29
Show solution
Solution
Distance = |d1 - d2| / √(a² + b² + c²) = |5 - 10| / √(2² + 3² + 4²) = 5 / √29.
Correct Answer:
B
— 10/√29
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Q. Find the distance between the parallel planes 2x + 3y + z = 5 and 2x + 3y + z = 10. (2022)
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Solution
Distance = |d1 - d2| / √(A² + B² + C²) = |5 - 10| / √(2² + 3² + 1²) = 5 / √14.
Correct Answer:
A
— 5
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Q. Find the distance between the parallel planes x + 2y + 3z = 4 and x + 2y + 3z = 10. (2023)
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Solution
Distance = |d1 - d2| / √(a² + b² + c²) = |4 - 10| / √(1² + 2² + 3²) = 6 / √14.
Correct Answer:
A
— 2
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Q. Find the distance between the points (-1, -1) and (2, 2).
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Solution
Using the distance formula: d = √[(2 - (-1))² + (2 - (-1))²] = √[9 + 9] = √18 = 3√2.
Correct Answer:
C
— 5
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Q. Find the distance between the points (-2, -3) and (4, 5).
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Solution
Using the distance formula: d = √[(4 - (-2))² + (5 - (-3))²] = √[(4 + 2)² + (5 + 3)²] = √[36 + 64] = √100 = 10.
Correct Answer:
B
— 7
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Q. Find the distance between the points (0, 0) and (x, y) where x = 6 and y = 8.
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Solution
Using the distance formula: d = √[(6 - 0)² + (8 - 0)²] = √[36 + 64] = √100 = 10.
Correct Answer:
A
— 10
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Q. Find the distance between the points (1, 1) and (4, 5). (2023)
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Solution
Using the distance formula: d = √[(4 - 1)² + (5 - 1)²] = √[9 + 16] = √25 = 5.
Correct Answer:
A
— 5
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Q. Find the distance between the points (1, 2) and (4, 6).
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Solution
Distance = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5.
Correct Answer:
A
— 5
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Q. Find the distance between the points (3, 3) and (3, 7).
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Solution
Using the distance formula: d = √[(3 - 3)² + (7 - 3)²] = √[0 + 16] = √16 = 4.
Correct Answer:
A
— 4
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Q. Find the distance between the points (3, 4) and (7, 1).
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Solution
Distance = √[(7-3)² + (1-4)²] = √[4 + 9] = √13 ≈ 3.6, closest option is 4.
Correct Answer:
A
— 5
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Q. Find the distance between the points (3, 7) and (3, 1).
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Solution
Using the distance formula: d = √((3 - 3)² + (1 - 7)²) = √(0 + 36) = √36 = 6.
Correct Answer:
A
— 6
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Q. Find the distance between the points (5, 5) and (5, 1).
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Solution
Using the distance formula: d = √[(5 - 5)² + (1 - 5)²] = √[0 + 16] = √16 = 4.
Correct Answer:
A
— 4
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Q. Find the distance between the points A(2, 3) and B(5, 7).
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Solution
Distance = √[(5-2)² + (7-3)²] = √[3² + 4²] = √[9 + 16] = √25 = 5.
Correct Answer:
C
— 5
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