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 angle between the vectors A = (3, -2, 1) and B = (1, 1, 1).
A.
60°
B.
45°
C.
90°
D.
30°
Show solution
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 3*1 + (-2)*1 + 1*1 = 2. |A| = √(3^2 + (-2)^2 + 1^2) = √14, |B| = √3. θ = cos^(-1)(2/(√14 * √3)).
Correct Answer:
A
— 60°
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Q. Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
180 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|). A · B = 0, hence θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. Find the angle between the vectors A = i + j and B = 2i + 2j.
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
60 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 1*2 = 4; |A| = √2, |B| = 2√2. Thus, cos(θ) = 4 / (√2 * 2√2) = 1, θ = 0 degrees.
Correct Answer:
A
— 0 degrees
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Q. Find the angle between the vectors A = i + j and B = i - j.
A.
0 degrees
B.
45 degrees
C.
90 degrees
D.
135 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A||B|) = (1 - 1) / (√2 * √2) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. Find the angle between the vectors A = i + j and B = j - i. (2022)
A.
90 degrees
B.
45 degrees
C.
60 degrees
D.
30 degrees
Show solution
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 0, hence θ = 90 degrees.
Correct Answer:
A
— 90 degrees
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Q. Find the angle θ between the vectors A = i + 2j and B = 2i + 3j if A · B = |A||B|cos(θ).
A.
60°
B.
45°
C.
30°
D.
90°
Show solution
Solution
A · B = 1*2 + 2*3 = 8. |A| = √(1^2 + 2^2) = √5, |B| = √(2^2 + 3^2) = √13. cos(θ) = 8/(√5*√13).
Correct Answer:
B
— 45°
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Q. Find the angle θ between vectors A = 4i + 3j and B = 1i + 2j if A · B = |A||B|cos(θ).
A.
60°
B.
45°
C.
30°
D.
90°
Show solution
Solution
A · B = 4*1 + 3*2 = 10; |A| = √(4^2 + 3^2) = 5; |B| = √(1^2 + 2^2) = √5; cos(θ) = 10/(5√5) = 2/√5; θ = 45°.
Correct Answer:
B
— 45°
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Q. Find the area between the curves y = x and y = x^2 from x = 0 to x = 1.
A.
0.5
B.
1
C.
0.25
D.
0.75
Show solution
Solution
The area between the curves is given by ∫(from 0 to 1) (x - x^2) dx = [x^2/2 - x^3/3] from 0 to 1 = (1/2 - 1/3) = 1/6 = 0.5.
Correct Answer:
A
— 0.5
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Q. Find the area between the curves y = x^2 and y = 4 from x = -2 to x = 2.
Show solution
Solution
The area between the curves is given by ∫(from -2 to 2) (4 - x^2) dx = [4x - x^3/3] from -2 to 2 = 16/3.
Correct Answer:
B
— 16/3
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Q. Find the area between the curves y = x^2 and y = 4 from x = 0 to x = 2.
Show solution
Solution
The area between the curves y = x^2 and y = 4 is given by ∫(from 0 to 2) (4 - x^2) dx = [4x - x^3/3] from 0 to 2 = (8 - 8/3) = 4/3.
Correct Answer:
A
— 4
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Q. Find the area between the curves y = x^3 and y = x from x = 0 to x = 1.
A.
1/4
B.
1/3
C.
1/2
D.
1/6
Show solution
Solution
The area between the curves is given by ∫(from 0 to 1) (x - x^3) dx = [x^2/2 - x^4/4] from 0 to 1 = (1/2 - 1/4) = 1/4.
Correct Answer:
B
— 1/3
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Q. Find the area of a triangle with vertices at A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0). (2023)
Show solution
Solution
Area = 0.5 * base * height = 0.5 * 1 * 1 = 0.5 square units.
Correct Answer:
A
— 0.5
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Q. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3).
Show solution
Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer:
A
— 6
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Q. Find the area of the triangle formed by the points (0, 0), (4, 0), and (0, 3). (2022) 2022
Show solution
Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer:
A
— 6
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Q. Find the area of the triangle formed by the points A(0, 0, 0), B(1, 0, 0), and C(0, 1, 0). (2023)
Show solution
Solution
Area = 0.5 * base * height = 0.5 * 1 * 1 = 0.5 square units.
Correct Answer:
A
— 0.5
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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9) using the vector product.
Show solution
Solution
Area = 0.5 * |AB × AC| = 0, as points are collinear.
Correct Answer:
A
— 0
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Q. Find the area of the triangle formed by the points A(1, 2, 3), B(4, 5, 6), and C(7, 8, 9). (2022)
Show solution
Solution
The points are collinear, hence the area = 0.
Correct Answer:
A
— 0
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Q. Find the area of the triangle with vertices (0,0), (4,0), (0,3).
Show solution
Solution
Area = 0.5 * base * height = 0.5 * 4 * 3 = 6.
Correct Answer:
A
— 6
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Q. Find the area of the triangle with vertices (0,0), (4,0), and (4,3).
Show solution
Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer:
B
— 12
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Q. Find the area of the triangle with vertices at (0,0), (4,0), and (0,3).
Show solution
Solution
Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.
Correct Answer:
A
— 6
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Q. Find the area under the curve y = 3x^2 from x = 1 to x = 2.
Show solution
Solution
The area under the curve is given by ∫(from 1 to 2) 3x^2 dx = [x^3] from 1 to 2 = (8 - 1) = 7.
Correct Answer:
B
— 6
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Q. Find the area under the curve y = e^x from x = 0 to x = 1.
Show solution
Solution
The area is given by the integral from 0 to 1 of e^x dx. This evaluates to [e^x] from 0 to 1 = e - 1.
Correct Answer:
A
— e - 1
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Q. Find the area under the curve y = x^2 + 2x from x = 0 to x = 3.
Show solution
Solution
The area under the curve is given by ∫(from 0 to 3) (x^2 + 2x) dx = [x^3/3 + x^2] from 0 to 3 = (27/3 + 9) = 18.
Correct Answer:
C
— 15
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Q. Find the area under the curve y = x^2 from x = 0 to x = 2.
Show solution
Solution
The area under the curve y = x^2 from 0 to 2 is given by the integral ∫(from 0 to 2) x^2 dx = [x^3/3] from 0 to 2 = (2^3/3) - (0^3/3) = 8/3.
Correct Answer:
C
— 8/3
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Q. Find the area under the curve y = x^2 from x = 0 to x = 3.
Show solution
Solution
Area = ∫ from 0 to 3 of x^2 dx = [1/3 * x^3] from 0 to 3 = 9.
Correct Answer:
A
— 9
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Q. Find the area under the curve y = x^2 from x = 1 to x = 3.
A.
8/3
B.
10/3
C.
9/3
D.
7/3
Show solution
Solution
The area is given by the integral ∫ (x^2) dx from 1 to 3. This evaluates to [x^3/3] from 1 to 3 = (27/3 - 1/3) = 26/3.
Correct Answer:
B
— 10/3
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Q. Find the area under the curve y = x^4 from x = 0 to x = 1.
A.
1/5
B.
1/4
C.
1/3
D.
1/2
Show solution
Solution
The area under the curve y = x^4 from 0 to 1 is given by ∫(from 0 to 1) x^4 dx = [x^5/5] from 0 to 1 = 1/5.
Correct Answer:
A
— 1/5
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Q. Find the area under the curve y = x^4 from x = 0 to x = 2.
Show solution
Solution
The area is given by the integral from 0 to 2 of x^4 dx. This evaluates to [x^5/5] from 0 to 2 = (32/5) = 16.
Correct Answer:
C
— 16
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Q. Find the argument of the complex number z = -1 - i.
A.
-3π/4
B.
3π/4
C.
π/4
D.
-π/4
Show solution
Solution
The argument of z = -1 - i is θ = tan^(-1)(-1/-1) = 3π/4.
Correct Answer:
A
— -3π/4
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Q. Find the arithmetic mean of the first five prime numbers.
Show solution
Solution
First five primes: 2, 3, 5, 7, 11. Mean = (2 + 3 + 5 + 7 + 11) / 5 = 28 / 5 = 5.6.
Correct Answer:
C
— 7
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