Q. The function f(x) = x^2 - 2x + 1 is differentiable at all points?
A.
True
B.
False
C.
Only at x = 0
D.
Only for x > 0
Show solution
Solution
f(x) is a polynomial function, which is differentiable everywhere.
Correct Answer:
A
— True
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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 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 - 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 + 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; 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; 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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Q. The function f(x) = |x - 3| is continuous at which of the following points?
A.
x = 1
B.
x = 2
C.
x = 3
D.
x = 4
Show solution
Solution
The function |x - 3| is continuous everywhere, including at x = 3.
Correct Answer:
C
— x = 3
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Q. The function f(x) = |x| is differentiable at x = 0?
A.
Yes
B.
No
C.
Only from the right
D.
Only from the left
Show solution
Solution
f(x) = |x| is not differentiable at x = 0 because the left-hand and right-hand derivatives do not match.
Correct Answer:
B
— No
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Q. The general form of the family of curves for circles is given by:
A.
(x - h)^2 + (y - k)^2 = r^2
B.
x^2 + y^2 = r^2
C.
x^2 + y^2 + Dx + Ey + F = 0
D.
y = mx + b
Show solution
Solution
The equation x^2 + y^2 + Dx + Ey + F = 0 represents a family of circles.
Correct Answer:
C
— x^2 + y^2 + Dx + Ey + F = 0
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Q. The general form of the family of curves y^2 = 4ax is known as:
A.
Circle
B.
Ellipse
C.
Parabola
D.
Hyperbola
Show solution
Solution
The equation y^2 = 4ax represents a parabola.
Correct Answer:
C
— Parabola
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Mathematics Syllabus (JEE Main) MCQ & Objective Questions
The Mathematics Syllabus for JEE Main is crucial for students aiming to excel in competitive exams. Understanding this syllabus not only helps in grasping key concepts but also enhances your ability to tackle objective questions effectively. Practicing MCQs and important questions from this syllabus is essential for solid exam preparation, ensuring you are well-equipped to score better in your exams.
What You Will Practise Here
Sets, Relations, and Functions
Complex Numbers and Quadratic Equations
Permutations and Combinations
Binomial Theorem
Sequences and Series
Limits and Derivatives
Statistics and Probability
Exam Relevance
The Mathematics Syllabus (JEE Main) is not only relevant for JEE but also appears in CBSE and State Board examinations. Students can expect a variety of question patterns, including direct MCQs, numerical problems, and conceptual questions. Mastery of this syllabus will prepare you for similar topics in NEET and other competitive exams, making it vital for your overall academic success.
Common Mistakes Students Make
Misinterpreting the questions, especially in word problems.
Overlooking the importance of units and dimensions in problems.
Confusing formulas related to sequences and series.
Neglecting to practice derivations, leading to errors in calculus.
Failing to apply the correct methods for solving probability questions.
FAQs
Question: What are the key topics in the Mathematics Syllabus for JEE Main? Answer: Key topics include Sets, Complex Numbers, Permutations, Binomial Theorem, and Calculus.
Question: How can I improve my performance in Mathematics MCQs? Answer: Regular practice of MCQs and understanding the underlying concepts are essential for improvement.
Now is the time to take charge of your exam preparation! Dive into solving practice MCQs and test your understanding of the Mathematics Syllabus (JEE Main). Your success is just a question away!
