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. Calculate the integral ∫ (x^2 + 2x + 1)/(x + 1) dx.
A.
(1/3)x^3 + x^2 + C
B.
x^2 + 2x + C
C.
x^2 + x + C
D.
(1/3)x^3 + (1/2)x^2 + C
Show solution
Solution
The integrand simplifies to x + 1. Therefore, ∫ (x + 1) dx = (1/2)x^2 + x + C.
Correct Answer:
A
— (1/3)x^3 + x^2 + C
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Q. Calculate the integral ∫ (x^3 - 4x) dx.
A.
(1/4)x^4 - 2x^2 + C
B.
(1/4)x^4 - 2x^2
C.
(1/4)x^4 - 4x^2 + C
D.
(1/4)x^4 - 2x^2 + 1
Show solution
Solution
The integral of x^3 is (1/4)x^4 and the integral of -4x is -2x^2. Therefore, ∫ (x^3 - 4x) dx = (1/4)x^4 - 2x^2 + C.
Correct Answer:
A
— (1/4)x^4 - 2x^2 + C
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Q. Calculate the integral ∫ cos^2(x) dx.
A.
(1/2)x + (1/4)sin(2x) + C
B.
(1/2)x + C
C.
(1/2)x - (1/4)sin(2x) + C
D.
(1/2)x + (1/2)sin(2x) + C
Show solution
Solution
Using the identity cos^2(x) = (1 + cos(2x))/2, we find that ∫ cos^2(x) dx = (1/2)x + (1/4)sin(2x) + C.
Correct Answer:
A
— (1/2)x + (1/4)sin(2x) + C
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Q. Calculate the integral ∫ from 0 to π of sin(x) dx.
Show solution
Solution
The integral evaluates to [-cos(x)] from 0 to π = [1 - (-1)] = 2.
Correct Answer:
C
— 2
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Q. Calculate the integral ∫(2 to 3) (x^3) dx. (2023)
Show solution
Solution
∫(2 to 3) (x^3) dx = [x^4/4] from 2 to 3 = (81/4 - 16/4) = 65/4 = 16.25.
Correct Answer:
C
— 8
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Q. Calculate the integral ∫(2 to 5) (4x - 1) dx. (2023)
Show solution
Solution
∫(2 to 5) (4x - 1) dx = [2x^2 - x] from 2 to 5 = (50 - 5) - (8 - 2) = 40.
Correct Answer:
A
— 20
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Q. Calculate the interquartile range (IQR) for the data set: 1, 3, 7, 8, 9, 10.
Show solution
Solution
Q1 = 3, Q3 = 9; IQR = Q3 - Q1 = 9 - 3 = 6.
Correct Answer:
A
— 4
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Q. Calculate the limit: lim (x -> 0) (1 - cos(x))/(x^2)
A.
0
B.
1/2
C.
1
D.
Infinity
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.
Correct Answer:
B
— 1/2
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Q. Calculate the limit: lim (x -> 0) (e^x - 1)/x
A.
0
B.
1
C.
Infinity
D.
Undefined
Show solution
Solution
Using the definition of the derivative of e^x at x = 0, we find that lim (x -> 0) (e^x - 1)/x = e^0 = 1.
Correct Answer:
B
— 1
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Q. Calculate the limit: lim (x -> 0) (ln(1 + x)/x) (2023)
A.
1
B.
0
C.
Undefined
D.
Infinity
Show solution
Solution
Using L'Hôpital's Rule, we differentiate the numerator and denominator to find lim (x -> 0) (1/(1 + x)) = 1.
Correct Answer:
A
— 1
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Q. Calculate the limit: lim (x -> 0) (tan(3x)/x)
A.
3
B.
1
C.
0
D.
Infinity
Show solution
Solution
Using the standard limit lim (x -> 0) (tan(kx)/x) = k, we have k = 3, so the limit is 3.
Correct Answer:
A
— 3
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Q. Calculate the limit: lim (x -> 0) (tan(5x)/x) (2022)
A.
0
B.
1
C.
5
D.
Undefined
Show solution
Solution
Using the limit property lim (x -> 0) (tan(kx)/x) = k, we have lim (x -> 0) (tan(5x)/x) = 5.
Correct Answer:
C
— 5
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Q. Calculate the limit: lim (x -> 0) (x^2 sin(1/x))
A.
0
B.
1
C.
∞
D.
Undefined
Show solution
Solution
Since |sin(1/x)| ≤ 1, we have |x^2 sin(1/x)| ≤ |x^2|. Thus, lim (x -> 0) x^2 sin(1/x) = 0.
Correct Answer:
A
— 0
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Q. Calculate the limit: lim (x -> 0) (x^3)/(sin(x)) (2023)
A.
0
B.
1
C.
∞
D.
Undefined
Show solution
Solution
Using the fact that sin(x) ~ x as x approaches 0, we find that lim (x -> 0) (x^3)/(sin(x)) = 0.
Correct Answer:
A
— 0
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)
A.
0
B.
1
C.
2
D.
Undefined
Show solution
Solution
This is an indeterminate form (0/0). Factor the numerator: (x-1)(x+1)/(x-1) = x + 1. Thus, lim (x -> 1) (x + 1) = 2.
Correct Answer:
C
— 2
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Q. Calculate the limit: lim (x -> 1) (x^2 - 1)/(x - 1)^2
A.
0
B.
1
C.
2
D.
Undefined
Show solution
Solution
Factoring gives (x - 1)(x + 1)/(x - 1)^2 = (x + 1)/(x - 1). Thus, lim (x -> 1) (x + 1)/(x - 1) = 2.
Correct Answer:
C
— 2
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Q. Calculate the limit: lim (x -> 1) (x^3 - 1)/(x - 1)
A.
0
B.
1
C.
3
D.
Undefined
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
— 3
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Q. Calculate the limit: lim (x -> 1) (x^4 - 1)/(x - 1) (2021)
Show solution
Solution
Factoring gives (x - 1)(x^3 + x^2 + x + 1)/(x - 1). Canceling gives lim (x -> 1) (x^3 + x^2 + x + 1) = 4.
Correct Answer:
D
— 4
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Q. Calculate the limit: lim (x -> 2) (x^2 - 2x)/(x - 2)
A.
0
B.
2
C.
4
D.
Undefined
Show solution
Solution
Factoring gives (x(x - 2))/(x - 2), canceling gives lim (x -> 2) x = 2.
Correct Answer:
D
— Undefined
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Q. Calculate the limit: lim (x -> 2) (x^3 - 8)/(x - 2)
Show solution
Solution
Factoring gives lim (x -> 2) ((x - 2)(x^2 + 2x + 4))/(x - 2) = lim (x -> 2) (x^2 + 2x + 4) = 12.
Correct Answer:
A
— 4
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Q. Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4) (2023)
Show solution
Solution
Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x^2) = 3/5.
Correct Answer:
A
— 3/5
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Q. Calculate the limit: lim (x -> ∞) (3x^2 + 2)/(5x^2 - 4x + 1) (2023)
Show solution
Solution
Dividing numerator and denominator by x^2 gives lim (x -> ∞) (3 + 2/x^2)/(5 - 4/x + 1/x^2) = 3/5.
Correct Answer:
A
— 3/5
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Q. Calculate the limit: lim (x -> ∞) (5x^2 + 3)/(2x^2 + 1) (2023)
Show solution
Solution
Dividing the numerator and denominator by x^2, we get lim (x -> ∞) (5 + 3/x^2)/(2 + 1/x^2) = 5/2.
Correct Answer:
B
— 5/2
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Q. Calculate the mean absolute deviation for the data set: 1, 2, 3, 4, 5.
Show solution
Solution
Mean = 3. Mean Absolute Deviation = (|1-3| + |2-3| + |3-3| + |4-3| + |5-3|)/5 = (2 + 1 + 0 + 1 + 2)/5 = 1.5.
Correct Answer:
B
— 1.5
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Q. Calculate the mean of the following data: 5, 10, 15, 20.
A.
10
B.
12.5
C.
15
D.
17.5
Show solution
Solution
Mean = (5 + 10 + 15 + 20) / 4 = 50 / 4 = 12.5.
Correct Answer:
B
— 12.5
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Q. Calculate the mean of the following numbers: 10, 20, 30, 40, 50.
Show solution
Solution
Mean = (10 + 20 + 30 + 40 + 50) / 5 = 150 / 5 = 30.
Correct Answer:
A
— 30
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Q. Calculate the mean of the following numbers: 4, 8, 12, 16, 20.
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Solution
Mean = (4 + 8 + 12 + 16 + 20) / 5 = 60 / 5 = 12.
Correct Answer:
C
— 14
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Q. Calculate the median of the following set: 1, 2, 3, 4, 5, 6, 7, 8. (2020)
Show solution
Solution
Arranging the numbers: 1, 2, 3, 4, 5, 6, 7, 8. The median is the average of the 4th and 5th numbers: (4 + 5) / 2 = 4.5.
Correct Answer:
B
— 4.5
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Q. Calculate the median of the following set: 22, 19, 25, 30, 28, 24.
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Solution
Arrange the numbers: 19, 22, 24, 25, 28, 30. The median is the average of the 3rd and 4th values: (24 + 25) / 2 = 24.5.
Correct Answer:
A
— 24
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Q. Calculate the molality of a solution if the boiling point elevation is 1.024 °C. (K_b for water = 0.512 °C kg/mol)
A.
1 mol/kg
B.
2 mol/kg
C.
0.5 mol/kg
D.
0.25 mol/kg
Show solution
Solution
Molality = ΔT_b / (i * K_b) = 1.024 / (2 * 0.512) = 1 mol/kg
Correct Answer:
B
— 2 mol/kg
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