Q. The function f(x) = x^2 - 2x + 1 is differentiable at x = 2?
A.
Yes
B.
No
C.
Only left
D.
Only right
Show solution
Solution
f(x) is a polynomial function, hence it is differentiable everywhere including at x = 2.
Correct Answer:
A
— Yes
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Q. The function f(x) = x^2 - 4 is:
A.
Always increasing
B.
Always decreasing
C.
Neither increasing nor decreasing
D.
Both increasing and decreasing
Show solution
Solution
The function has a minimum at x = 0, hence it is neither always increasing nor decreasing.
Correct Answer:
C
— Neither increasing nor decreasing
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Q. The function f(x) = x^2 - 4x + 4 can be expressed in which form?
A.
(x - 2)^2
B.
(x + 2)^2
C.
(x - 4)^2
D.
(x + 4)^2
Show solution
Solution
f(x) = (x - 2)^2 is the completed square form.
Correct Answer:
A
— (x - 2)^2
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Q. The function f(x) = x^2 - 4x + 4 is differentiable at x = 2?
A.
Yes
B.
No
C.
Only left
D.
Only right
Show solution
Solution
f(x) is a polynomial function, hence differentiable everywhere including at x = 2.
Correct Answer:
A
— Yes
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Q. The function f(x) = x^2 - 4x + 4 is differentiable everywhere?
A.
True
B.
False
C.
Only at x = 0
D.
Only at x = 2
Show solution
Solution
f(x) is a polynomial function, hence it is differentiable everywhere.
Correct Answer:
A
— True
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Q. The function f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x ≥ 1 is differentiable at x = 1?
A.
Yes
B.
No
C.
Only continuous
D.
Only from the left
Show solution
Solution
f'(1) from left = 2 and from right = 2; hence, f is continuous but not differentiable at x = 1.
Correct Answer:
B
— No
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Q. The function f(x) = x^2 is continuous at which of the following points? (2023)
A.
x = -1
B.
x = 0
C.
x = 1
D.
All of the above
Show solution
Solution
The function f(x) = x^2 is a polynomial function and is continuous at all points, including -1, 0, and 1.
Correct Answer:
D
— All of the above
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Q. The function f(x) = x^2 sin(1/x) for x ≠ 0 and f(0) = 0 is differentiable at x = 0. True or False?
A.
True
B.
False
C.
Depends on x
D.
Not enough information
Show solution
Solution
True, as the limit of f'(x) as x approaches 0 exists and equals 0.
Correct Answer:
A
— True
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Q. The function f(x) = x^3 - 3x + 2 is differentiable at x = 1?
A.
Yes
B.
No
C.
Only left
D.
Only right
Show solution
Solution
f(x) is a polynomial function, hence it is differentiable everywhere including at x = 1.
Correct Answer:
A
— Yes
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Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. Find its critical points.
Show solution
Solution
f'(x) = 3x^2 - 3 = 0 gives x = ±1, thus critical points are x = -1 and x = 1.
Correct Answer:
B
— 0
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Q. The function f(x) = x^3 - 3x + 2 is differentiable everywhere. What is f'(1)?
Show solution
Solution
f'(x) = 3x^2 - 3, thus f'(1) = 0.
Correct Answer:
A
— 0
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Q. The function f(x) = x^3 - 3x is continuous at which of the following points? (2023)
A.
x = -2
B.
x = 0
C.
x = 2
D.
All of the above
Show solution
Solution
The function f(x) = x^3 - 3x is a polynomial function and is continuous at all points, including -2, 0, and 2.
Correct Answer:
D
— All of the above
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Q. The function f(x) = x^3 - 6x^2 + 9x has how many local extrema?
Show solution
Solution
Finding f'(x) = 3x^2 - 12x + 9. Setting f'(x) = 0 gives x = 1 and x = 3. Checking the second derivative shows one local maximum and one local minimum.
Correct Answer:
B
— 1
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Q. The function f(x) = { 1/x, x != 0; 0, x = 0 } is continuous at x = 0?
A.
Yes
B.
No
C.
Only from the right
D.
Only from the left
Show solution
Solution
The limit as x approaches 0 does not equal f(0) = 0, hence it is not continuous at x = 0.
Correct Answer:
B
— No
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Q. The function f(x) = { 1/x, x ≠ 0; 0, x = 0 } is:
A.
Continuous at x = 0
B.
Not continuous at x = 0
C.
Continuous everywhere
D.
None of the above
Show solution
Solution
The function is not continuous at x = 0 since the limit does not equal f(0).
Correct Answer:
B
— Not continuous at x = 0
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Q. The function f(x) = { 2x + 3, x < 1; x^2 + 1, x >= 1 } is continuous at x = ?
Show solution
Solution
To check continuity at x = 1, we find the left limit (5) and the right limit (2). They are not equal, hence f(x) is not continuous at x = 1.
Correct Answer:
B
— 1
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Q. The function f(x) = { 3x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1 if which condition holds?
A.
3 = 2
B.
1 = 2
C.
2 = 1
D.
2 = 4
Show solution
Solution
For continuity at x = 1, the left limit (3) must equal f(1) (2), which is not true.
Correct Answer:
A
— 3 = 2
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Q. The function f(x) = { 3x + 1, x < 1; 2x + 3, x >= 1 } is continuous at x = 1 if:
Show solution
Solution
For continuity at x = 1, both pieces must equal 4, hence the function is continuous.
Correct Answer:
A
— 3
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Q. The function f(x) = { x + 1, x < 1; 2, x = 1; x^2, x > 1 } is continuous at x = 1?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
Left limit as x approaches 1 is 2, right limit is 1, but f(1) = 2. Hence, it is discontinuous at x = 1.
Correct Answer:
B
— No
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Q. The function f(x) = { x + 2, x < 1; 3, x = 1; x^2, x > 1 } is continuous at x = ?
Show solution
Solution
To check continuity at x = 1, we find the left limit (3) and the right limit (3). Both equal 3, hence f(x) is continuous at x = 1.
Correct Answer:
B
— 1
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Q. The function f(x) = { x^2, x < 0; 0, x = 0; x + 1, x > 0 } is:
A.
Continuous
B.
Not continuous
C.
Continuous from the left
D.
Continuous from the right
Show solution
Solution
The left limit as x approaches 0 is 0, but the right limit is 1. Hence, it is not continuous at x = 0.
Correct Answer:
B
— Not continuous
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Q. The function f(x) = { x^2, x < 0; 1, x = 0; x + 1, x > 0 } is continuous at x = 0?
A.
Yes
B.
No
C.
Only from the right
D.
Only from the left
Show solution
Solution
Limit as x approaches 0 from left is 0, and f(0) = 1, hence it is not continuous at x = 0.
Correct Answer:
A
— Yes
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Q. The function f(x) = { x^2, x < 0; 2, x = 0; x + 1, x > 0 } is continuous at x = 0?
A.
Yes
B.
No
C.
Only left continuous
D.
Only right continuous
Show solution
Solution
At x = 0, lim x→0- f(x) = 0 and lim x→0+ f(x) = 1, hence it is discontinuous at x = 0.
Correct Answer:
B
— No
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Q. The function f(x) = { x^2, x < 0; 2x + 1, x >= 0 } is continuous at which point?
A.
x = -1
B.
x = 0
C.
x = 1
D.
x = 2
Show solution
Solution
To check continuity at x = 0, we find f(0) = 1 and limit as x approaches 0 is also 1.
Correct Answer:
B
— x = 0
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Q. The function f(x) = { x^2, x < 1; 2, x = 1; x + 1, x > 1 } is:
A.
Continuous everywhere
B.
Continuous at x = 1
C.
Not continuous at x = 1
D.
Continuous for x < 1
Show solution
Solution
The function is not continuous at x = 1 because the left-hand limit does not equal the function value.
Correct Answer:
C
— Not continuous at x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at which point?
A.
x = 0
B.
x = 1
C.
x = 2
D.
x = -1
Show solution
Solution
To check continuity at x = 1, we find f(1) = 1, limit as x approaches 1 from left is 1, and from right is also 1.
Correct Answer:
B
— x = 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x >= 1 } is continuous at x = ?
Show solution
Solution
To check continuity at x = 1, we find the limit from both sides. Both limits equal 1, hence f(x) is continuous at x = 1.
Correct Answer:
B
— 1
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Q. The function f(x) = { x^2, x < 1; 2x - 1, x ≥ 1 } is differentiable at x = 1 if which condition holds?
A.
f(1) = 1
B.
f'(1) = 1
C.
f'(1) = 2
D.
f(1) = 2
Show solution
Solution
For differentiability, the left and right derivatives must equal at x = 1, hence f'(1) = 1.
Correct Answer:
B
— f'(1) = 1
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Q. The function f(x) = { x^2, x < 2; 4, x = 2; 2x, x > 2 } is continuous at x = 2 if:
A.
f(2) = 4
B.
lim x->2 f(x) = 4
C.
Both a and b
D.
None of the above
Show solution
Solution
Both conditions must hold true for continuity at x = 2.
Correct Answer:
C
— Both a and b
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Q. The function f(x) = { x^2, x < 2; k, x = 2; 3x - 4, x > 2 } is continuous at x = 2 for which value of k?
Show solution
Solution
To be continuous at x = 2, k must equal f(2) = 2^2 = 4.
Correct Answer:
C
— 4
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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!
