Q. What is the derivative of \( y = \tan^{-1}(x) \)?
A.
\( \frac{1}{1+x^2} \)
B.
\( \frac{1}{x^2+1} \)
C.
\( \frac{1}{x} \)
D.
0
Show solution
Solution
The derivative of \( y = \tan^{-1}(x) \) is \( \frac{1}{1+x^2} \).
Correct Answer:
A
— \( \frac{1}{1+x^2} \)
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Q. What is the domain of the function f(x) = 1/(x - 2)?
A.
x ≠ 2
B.
x > 2
C.
x < 2
D.
All real numbers
Show solution
Solution
The function is undefined at x = 2, so the domain is all real numbers except 2.
Correct Answer:
A
— x ≠ 2
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Q. What is the domain of the function f(x) = 1/(x-3)?
A.
x ≠ 3
B.
x > 3
C.
x < 3
D.
All real numbers
Show solution
Solution
The function is undefined at x = 3, so the domain is x ≠ 3.
Correct Answer:
A
— x ≠ 3
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Q. What is the domain of the function f(x) = sqrt(x - 1)?
A.
x >= 1
B.
x > 1
C.
x <= 1
D.
x < 1
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Solution
The expression under the square root must be non-negative, so x - 1 >= 0, hence x >= 1.
Correct Answer:
A
— x >= 1
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Q. What is the intersection of the sets A = {1, 2, 3} and B = {2, 3, 4}?
A.
{1, 2, 3}
B.
{2, 3}
C.
{4}
D.
{1, 4}
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Solution
The intersection of two sets includes only the elements that are present in both sets. Here, the intersection is {2, 3}.
Correct Answer:
B
— {2, 3}
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Q. What is the inverse of the function f(x) = 2x + 3?
A.
(x - 3)/2
B.
(x + 3)/2
C.
2x - 3
D.
2(x - 3)
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Solution
To find the inverse, set y = 2x + 3, solve for x: x = (y - 3)/2.
Correct Answer:
A
— (x - 3)/2
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Q. What is the inverse of the function f(x) = 2x + 5?
A.
f^-1(x) = (x - 5)/2
B.
f^-1(x) = 2x - 5
C.
f^-1(x) = (x + 5)/2
D.
f^-1(x) = 5 - 2x
Show solution
Solution
To find the inverse, set y = 2x + 5, solve for x: x = (y - 5)/2, thus f^-1(x) = (x - 5)/2.
Correct Answer:
A
— f^-1(x) = (x - 5)/2
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Q. What is the inverse of the function f(x) = 3x + 1?
A.
(x-1)/3
B.
(x-1)/3
C.
(x-3)/1
D.
3(x-1)
Show solution
Solution
To find the inverse, set y = 3x + 1, solve for x: x = (y - 1)/3.
Correct Answer:
A
— (x-1)/3
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Q. What is the inverse of the function f(x) = 3x + 4?
A.
(x - 4)/3
B.
(x + 4)/3
C.
3/x - 4
D.
3/x + 4
Show solution
Solution
To find the inverse, set y = 3x + 4, solve for x: x = (y - 4)/3, hence the inverse is (x - 4)/3.
Correct Answer:
A
— (x - 4)/3
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Q. What is the inverse of the function f(x) = 3x - 5?
A.
(x + 5)/3
B.
(x - 5)/3
C.
3(x + 5)
D.
3(x - 5)
Show solution
Solution
To find the inverse, set y = 3x - 5, solve for x: x = (y + 5)/3, hence f^(-1)(x) = (x + 5)/3.
Correct Answer:
A
— (x + 5)/3
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Q. What is the number of proper subsets of the set E = {a, b}?
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Solution
The total number of subsets is 2^n = 2^2 = 4. Proper subsets exclude the set itself, so there are 4 - 1 = 3 proper subsets.
Correct Answer:
B
— 3
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Q. What is the number of subsets of the empty set?
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Solution
The empty set has exactly one subset, which is itself (∅).
Correct Answer:
B
— 1
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Q. What is the number of subsets of the set F = {x, y, z}?
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Solution
The number of subsets of a set with n elements is 2^n. Here, n = 3, so 2^3 = 8.
Correct Answer:
D
— 8
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Q. What is the number of subsets of the set H = {1, 2, 3, 4, 5}?
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Solution
The number of subsets of a set with n elements is 2^n. Here, n = 5, so 2^5 = 32.
Correct Answer:
A
— 32
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Q. What is the number of subsets of the set H = {x, y, z, w, v}?
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Solution
The number of subsets of a set with n elements is 2^n. Here, n = 5, so the number of subsets is 2^5 = 32.
Correct Answer:
A
— 32
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Q. What is the number of subsets of the set {1, 2, 3, 4}?
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Solution
The number of subsets of a set with n elements is 2^n. Here, n = 4, so the number of subsets is 2^4 = 16.
Correct Answer:
C
— 16
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Q. What is the number of subsets of the set {a, b, c, d}?
Show solution
Solution
The number of subsets of a set with n elements is 2^n. Here, n = 4, so the number of subsets is 2^4 = 16.
Correct Answer:
B
— 8
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Q. What is the power set of A = {a, b}?
A.
{∅, {a}, {b}, {a, b}}
B.
{a, b}
C.
{∅, {a, b}}
D.
{a, b, {a, b}}
Show solution
Solution
The power set of a set is the set of all its subsets. For A = {a, b}, the power set is {∅, {a}, {b}, {a, b}}.
Correct Answer:
A
— {∅, {a}, {b}, {a, b}}
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Q. What is the power set of the empty set ∅?
A.
{∅}
B.
{∅, {∅}}
C.
∅
D.
{∅, ∅}
Show solution
Solution
The power set of the empty set contains only the empty set itself: {∅}.
Correct Answer:
A
— {∅}
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Q. What is the power set of the set F = {a}?
A.
{∅, {a}}
B.
{∅, a}
C.
{a}
D.
{∅}
Show solution
Solution
The power set of F = {a} is {∅, {a}}.
Correct Answer:
A
— {∅, {a}}
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Q. What is the power set of the set G = {1}?
A.
{∅, {1}}
B.
{1}
C.
{∅}
D.
{1, ∅}
Show solution
Solution
The power set of G = {1} is {∅, {1}}.
Correct Answer:
A
— {∅, {1}}
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Q. What is the power set of the set {1, 2}?
A.
{∅, {1}, {2}, {1, 2}}
B.
{∅, {1, 2}}
C.
{1, 2}
D.
{1, 2, ∅}
Show solution
Solution
The power set of a set with n elements has 2^n elements. For {1, 2}, the power set is {∅, {1}, {2}, {1, 2}}.
Correct Answer:
A
— {∅, {1}, {2}, {1, 2}}
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Q. What is the principal value of cot^(-1)(0)?
A.
0
B.
π/2
C.
π
D.
undefined
Show solution
Solution
cot^(-1)(0) = π/2
Correct Answer:
B
— π/2
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Q. What is the principal value of sec^(-1)(2)?
A.
π/3
B.
π/4
C.
π/6
D.
0
Show solution
Solution
sec^(-1)(2) = π/3, since sec(π/3) = 2.
Correct Answer:
A
— π/3
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Q. What is the range of the function f(x) = -x^2 + 4?
A.
(-∞, 4]
B.
[0, 4]
C.
[4, ∞)
D.
(-∞, 0)
Show solution
Solution
The function is a downward-opening parabola with a maximum value of 4, so the range is (-∞, 4].
Correct Answer:
A
— (-∞, 4]
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Q. What is the range of the function f(x) = x^2?
A.
(-∞, ∞)
B.
[0, ∞)
C.
(-∞, 0)
D.
[0, 1]
Show solution
Solution
The function f(x) = x^2 has a minimum value of 0, so its range is [0, ∞).
Correct Answer:
B
— [0, ∞)
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Q. What is the range of the function sin^(-1)(x)?
A.
(-π/2, π/2)
B.
(-1, 1)
C.
(0, π)
D.
(0, 1)
Show solution
Solution
The range of sin^(-1)(x) is (-π/2, π/2).
Correct Answer:
A
— (-π/2, π/2)
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Q. What is the sum of the roots of the equation f(x) = x^2 - 5x + 6?
Show solution
Solution
The sum of the roots of ax^2 + bx + c = 0 is -b/a. Here, sum = 5.
Correct Answer:
A
— 5
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Q. What is the sum of the roots of the quadratic equation f(x) = x^2 - 5x + 6?
Show solution
Solution
The sum of the roots is given by -b/a = 5/1 = 5.
Correct Answer:
A
— 5
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Q. What is the sum of the roots of the quadratic equation x^2 - 5x + 6 = 0?
Show solution
Solution
The sum of the roots is given by -b/a = 5/1 = 5.
Correct Answer:
A
— 5
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Showing 151 to 180 of 219 (8 Pages)
Sets, Relations & Functions MCQ & Objective Questions
Understanding "Sets, Relations & Functions" is crucial for students aiming to excel in their exams. This topic forms the foundation of many mathematical concepts and is frequently tested in various assessments. Practicing MCQs and objective questions not only enhances your grasp of the subject but also boosts your confidence in tackling important questions during exams.
What You Will Practise Here
Basic definitions and properties of sets
Types of relations and their characteristics
Functions: definitions, types, and notations
Operations on sets: union, intersection, and difference
Venn diagrams and their applications
Domain, range, and co-domain of functions
Important theorems related to sets and functions
Exam Relevance
The topic of "Sets, Relations & Functions" is integral to the curriculum of CBSE, State Boards, and competitive exams like NEET and JEE. You can expect questions that require you to apply concepts in problem-solving scenarios. Common question patterns include identifying properties of sets, solving problems involving relations, and interpreting functions graphically. Mastery of this topic can significantly enhance your performance in both objective and subjective formats.
Common Mistakes Students Make
Confusing the definitions of sets and subsets
Misunderstanding the types of relations (reflexive, symmetric, transitive)
Overlooking the importance of domain and range in functions
Errors in Venn diagram representations
Neglecting to apply the correct operations on sets
FAQs
Question: What are the different types of sets?Answer: The different types of sets include finite sets, infinite sets, equal sets, null sets, and singleton sets.
Question: How do I determine the domain and range of a function?Answer: The domain is the set of all possible input values, while the range is the set of all possible output values based on the function's definition.
Start solving practice MCQs today to solidify your understanding of "Sets, Relations & Functions". Testing your knowledge with objective questions will prepare you for success in your exams!
