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. Evaluate the integral ∫_0^π/2 cos^2(x) dx.
Show solution
Solution
∫_0^π/2 cos^2(x) dx = π/4.
Correct Answer:
A
— π/4
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Q. Evaluate the integral ∫_1^2 (3x^2 - 2) dx.
Show solution
Solution
∫_1^2 (3x^2 - 2) dx = [x^3 - 2x] from 1 to 2 = (8 - 4) - (1 - 2) = 3.
Correct Answer:
A
— 1
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Q. Evaluate the integral: ∫ (1/(x^2 + 1)) dx
A.
tan^(-1)(x) + C
B.
sin^(-1)(x) + C
C.
ln
D.
x
.
+ C
.
cos^(-1)(x) + C
Show solution
Solution
The integral of 1/(x^2 + 1) is tan^(-1)(x) + C.
Correct Answer:
A
— tan^(-1)(x) + C
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Q. Evaluate the integral: ∫ (2x^3 - 3x^2 + 4) dx
A.
(1/2)x^4 - x^3 + 4x + C
B.
(1/4)x^4 - (1/3)x^3 + 4x + C
C.
(1/2)x^4 - (1/3)x^3 + 4x + C
D.
(1/4)x^4 - x^3 + 4x + C
Show solution
Solution
Integrating term by term gives (1/4)x^4 - (1/3)x^3 + 4x + C.
Correct Answer:
A
— (1/2)x^4 - x^3 + 4x + C
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine continuity. (2021)
A.
5, Continuous
B.
0, Not continuous
C.
5, Not continuous
D.
0, Continuous
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(kx)/x) = k. Here, k = 5, so the limit is 5, and the function is continuous at x = 0.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x) and determine its continuity.
A.
5, Continuous
B.
0, Continuous
C.
5, Not Continuous
D.
0, Not Continuous
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(5x)/x) = 5. The function is continuous at x = 0.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(5x)/x). Is the function continuous at x = 0?
A.
5, Continuous
B.
5, Discontinuous
C.
0, Continuous
D.
0, Discontinuous
Show solution
Solution
Using the limit property, lim (x -> 0) (sin(5x)/x) = 5. The function is continuous at x = 0 if defined as f(0) = 5.
Correct Answer:
A
— 5, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(x)/x) and determine its continuity.
A.
1, Continuous
B.
0, Continuous
C.
1, Discontinuous
D.
0, Discontinuous
Show solution
Solution
The limit lim (x -> 0) (sin(x)/x) = 1. Since the limit exists and equals the function value at x = 0, it is continuous.
Correct Answer:
A
— 1, Continuous
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Q. Evaluate the limit lim (x -> 0) (sin(x)/x). Is the function continuous at x = 0?
A.
1, Continuous
B.
0, Continuous
C.
1, Discontinuous
D.
0, Discontinuous
Show solution
Solution
The limit is 1, and if we define f(0) = 1, then f(x) is continuous at x = 0.
Correct Answer:
A
— 1, Continuous
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Q. Evaluate the limit lim (x -> 3) (x^2 - 9)/(x - 3). Is the function continuous at x = 3? (2021)
A.
0, Yes
B.
0, No
C.
6, Yes
D.
6, No
Show solution
Solution
lim (x -> 3) (x^2 - 9)/(x - 3) = lim (x -> 3) (x + 3) = 6. The function is not defined at x = 3, hence not continuous.
Correct Answer:
C
— 6, Yes
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Q. Evaluate the limit lim x->1 (x^3 - 1)/(x - 1).
Show solution
Solution
Factoring gives (x-1)(x^2 + x + 1)/(x - 1). Canceling (x - 1) gives lim x->1 (x^2 + x + 1) = 3.
Correct Answer:
C
— 2
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Q. Evaluate the limit lim x->1 of (x^3 - 1)/(x - 1).
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Solution
Factoring gives (x-1)(x^2 + x + 1)/(x-1) = x^2 + x + 1, thus limit is 3.
Correct Answer:
C
— 3
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Q. Evaluate the limit lim x->2 (x^2 - 4)/(x - 2).
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Solution
Factoring gives (x-2)(x+2)/(x-2). Canceling gives lim x->2 (x + 2) = 4.
Correct Answer:
C
— 2
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Q. Evaluate the limit lim x->2 of (x^2 - 4)/(x - 2).
A.
0
B.
2
C.
4
D.
undefined
Show solution
Solution
Factoring gives (x-2)(x+2)/(x-2) = x + 2, thus limit is 4.
Correct Answer:
C
— 4
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Q. Evaluate the limit lim x→2 (x^2 - 4)/(x - 2).
A.
0
B.
2
C.
4
D.
Undefined
Show solution
Solution
Using L'Hôpital's Rule, lim x→2 (x^2 - 4)/(x - 2) = lim x→2 (2x)/(1) = 4.
Correct Answer:
C
— 4
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Q. Evaluate the limit lim(x→∞) (3x^2 + 2)/(5x^2 - 4).
Show solution
Solution
Dividing by x^2, lim(x→∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer:
A
— 3/5
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Q. Evaluate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
A.
0
B.
1/2
C.
1
D.
Undefined
Show solution
Solution
Using the identity 1 - cos(x) = 2sin^2(x/2), we have lim (x -> 0) (2sin^2(x/2))/(x^2) = 1/2.
Correct Answer:
B
— 1/2
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Q. Evaluate the limit: lim (x -> 0) (e^x - 1)/x
A.
0
B.
1
C.
e
D.
Infinity
Show solution
Solution
Using L'Hôpital's Rule, we differentiate the numerator and denominator: lim (x -> 0) (e^x)/(1) = e^0 = 1.
Correct Answer:
B
— 1
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Q. Evaluate the limit: lim (x -> 0) (ln(1 + x)/x)
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
Using L'Hôpital's Rule, differentiate the numerator and denominator: lim (x -> 0) (1/(1 + x))/(1) = 1.
Correct Answer:
B
— 1
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Q. Evaluate the limit: lim (x -> 0) (sin(5x)/x)
A.
0
B.
5
C.
1
D.
Infinity
Show solution
Solution
Using the standard limit lim (x -> 0) (sin(x)/x) = 1, we have lim (x -> 0) (sin(5x)/x) = 5 * lim (x -> 0) (sin(5x)/(5x)) = 5 * 1 = 5.
Correct Answer:
B
— 5
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Q. Evaluate the limit: lim (x -> 0) (tan(3x)/x)
A.
0
B.
3
C.
1
D.
Infinity
Show solution
Solution
Using the standard limit lim (x -> 0) (tan(x)/x) = 1, we have lim (x -> 0) (tan(3x)/x) = 3 * lim (x -> 0) (tan(3x)/(3x)) = 3 * 1 = 3.
Correct Answer:
B
— 3
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Q. Evaluate the limit: lim (x -> 0) (tan(x)/x) (2023)
A.
0
B.
1
C.
∞
D.
Undefined
Show solution
Solution
Using the limit property lim (x -> 0) (tan(x)/x) = 1, we find that the limit is 1.
Correct Answer:
B
— 1
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Q. Evaluate the limit: lim (x -> 0) (x - sin(x))/x^3 (2022)
A.
0
B.
1/6
C.
1/3
D.
1/2
Show solution
Solution
Using the Taylor series expansion for sin(x), we find that lim (x -> 0) (x - sin(x))/x^3 = 1/6.
Correct Answer:
B
— 1/6
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Q. Evaluate the limit: lim (x -> 0) (x^2 * sin(1/x))
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
Since |sin(1/x)| <= 1, we have |x^2 * sin(1/x)| <= x^2, and thus lim (x -> 0) x^2 * sin(1/x) = 0.
Correct Answer:
A
— 0
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Q. Evaluate the limit: lim (x -> 0) (x^2)/(sin(x))
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
As x approaches 0, sin(x) approaches x, thus lim (x -> 0) (x^2)/(sin(x)) = 0.
Correct Answer:
A
— 0
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Q. Evaluate the limit: lim (x -> 0) (x^3)/(sin(x)) (2022)
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
As x approaches 0, x^3 approaches 0 and sin(x) approaches 0, thus the limit is 0.
Correct Answer:
A
— 0
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Q. Evaluate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)
A.
2
B.
0
C.
1
D.
Infinity
Show solution
Solution
Using L'Hôpital's Rule, the limit evaluates to 2.
Correct Answer:
A
— 2
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Q. Evaluate the limit: lim (x -> 3) (x^2 - 9)/(x - 3) (2020)
A.
3
B.
6
C.
9
D.
Undefined
Show solution
Solution
Factoring gives (x - 3)(x + 3)/(x - 3). Canceling (x - 3) gives lim (x -> 3) (x + 3) = 6.
Correct Answer:
B
— 6
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Q. Evaluate the limit: lim (x -> ∞) (2x^2 + 3)/(5x^2 - 4x + 1)
A.
2/5
B.
3/5
C.
1/2
D.
Infinity
Show solution
Solution
Divide numerator and denominator by x^2. The limit becomes lim (x -> ∞) (2 + 3/x^2)/(5 - 4/x + 1/x^2) = 2/5.
Correct Answer:
A
— 2/5
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Q. Evaluate the limit: lim (x -> ∞) (2x^3 - 3x)/(4x^3 + 5)
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Solution
Dividing numerator and denominator by x^3 gives lim (x -> ∞) (2 - 3/x^2)/(4 + 5/x^3) = 2/4 = 1/2.
Correct Answer:
B
— 1/2
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