Q. What is the total 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 total number of subsets of the set G = {1, 2, 3, 4, 5}?
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Solution
The total number of subsets is 2^5 = 32.
Correct Answer:
A
— 32
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Q. What is the union of the sets I = {1, 2} and J = {2, 3}?
A.
{1, 2, 3}
B.
{1, 2}
C.
{2, 3}
D.
{1, 2, 2, 3}
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Solution
The union of two sets includes all unique elements from both sets. Thus, I ∪ J = {1, 2, 3}.
Correct Answer:
A
— {1, 2, 3}
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Q. What is the union of the sets {1, 2} and {2, 3}?
A.
{1, 2}
B.
{1, 2, 3}
C.
{2, 3}
D.
{1, 2, 2, 3}
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Solution
The union of two sets includes all elements from both sets without duplicates. Thus, {1, 2} ∪ {2, 3} = {1, 2, 3}.
Correct Answer:
B
— {1, 2, 3}
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Q. What is the union of the sets {a, b} and {b, c}?
A.
{a, b}
B.
{a, b, c}
C.
{b, c}
D.
{a, c}
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Solution
The union of two sets is the set of elements that are in either set. Here, {a, b} ∪ {b, c} = {a, b, c}.
Correct Answer:
B
— {a, b, c}
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Q. What is the value of cos^(-1)(0)?
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Solution
cos^(-1)(0) corresponds to the angle where the cosine value is 0, which is π.
Correct Answer:
C
— π
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Q. What is the value of cot(cos^(-1)(1/2))?
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Solution
cot(cos^(-1)(1/2)) = √3
Correct Answer:
A
— √3
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Q. What is the value of f(2) if f(x) = x^2 - 2x + 1?
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Solution
f(2) = 2^2 - 2(2) + 1 = 4 - 4 + 1 = 1.
Correct Answer:
A
— 1
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Q. What is the value of k if f(x) = kx^2 + 2x + 1 has a minimum value of -3?
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Solution
The minimum value occurs at x = -b/(2a) = -2/(2k). Setting f(-1) = -3 gives k = -2.
Correct Answer:
B
— -2
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Q. What is the value of sec^(-1)(2)?
A.
π/3
B.
π/4
C.
π/6
D.
0
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Solution
sec^(-1)(2) = π/3, since sec(π/3) = 2.
Correct Answer:
A
— π/3
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Q. What is the value of sin(tan^(-1)(1))?
A.
1/√2
B.
1/2
C.
1
D.
√2/2
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Solution
sin(tan^(-1)(1)) = √2/2
Correct Answer:
D
— √2/2
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Q. What is the value of sin(tan^(-1)(√3))?
A.
√3/2
B.
1/2
C.
1
D.
√2/2
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Solution
sin(tan^(-1)(√3)) = √3/2
Correct Answer:
A
— √3/2
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Q. What is the value of sin^(-1)(sin(π/4))?
A.
π/4
B.
3π/4
C.
π/2
D.
0
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Solution
sin^(-1)(sin(π/4)) = π/4, as π/4 is in the range of sin^(-1).
Correct Answer:
A
— π/4
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Q. What is the value of tan^(-1)(√3)?
A.
π/3
B.
π/4
C.
π/6
D.
π/2
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Solution
tan^(-1)(√3) corresponds to the angle π/3.
Correct Answer:
A
— π/3
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Q. What is the value of \( \tan(\tan^{-1}(3)) \)?
A.
0
B.
1
C.
3
D.
undefined
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Solution
By definition, \( \tan(\tan^{-1}(x)) = x \). Therefore, \( \tan(\tan^{-1}(3)) = 3 \).
Correct Answer:
C
— 3
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Q. Which of the following functions is an even function?
A.
f(x) = x^3
B.
f(x) = x^2
C.
f(x) = x + 1
D.
f(x) = sin(x)
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Solution
An even function satisfies f(-x) = f(x). Here, f(x) = x^2 is even.
Correct Answer:
B
— f(x) = x^2
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Q. Which of the following functions is even?
A.
f(x) = x^3
B.
f(x) = x^2
C.
f(x) = x + 1
D.
f(x) = sin(x)
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Solution
A function is even if f(-x) = f(x). Here, f(x) = x^2 is even.
Correct Answer:
B
— f(x) = x^2
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Q. Which of the following functions is not a polynomial function?
A.
f(x) = x^2 + 3x + 1
B.
g(x) = 2x^3 - 4
C.
h(x) = sqrt(x)
D.
k(x) = 5
Show solution
Solution
h(x) = sqrt(x) is not a polynomial function.
Correct Answer:
C
— h(x) = sqrt(x)
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Q. Which of the following functions is not a polynomial?
A.
f(x) = x^3 + 2x^2 - 5
B.
g(x) = 1/x
C.
h(x) = 4x - 7
D.
k(x) = 2
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Solution
g(x) = 1/x is not a polynomial because it has a negative exponent.
Correct Answer:
B
— g(x) = 1/x
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Q. Which of the following functions is not continuous?
A.
f(x) = x^2
B.
f(x) = 1/x
C.
f(x) = sin(x)
D.
f(x) = e^x
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Solution
f(x) = 1/x is not continuous at x = 0.
Correct Answer:
B
— f(x) = 1/x
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Q. Which of the following functions is periodic?
A.
f(x) = x
B.
f(x) = cos(x)
C.
f(x) = e^x
D.
f(x) = ln(x)
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Solution
The cosine function is periodic with a period of 2π.
Correct Answer:
B
— f(x) = cos(x)
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Q. Which of the following is a one-to-one function?
A.
f(x) = x^2
B.
f(x) = 2x + 3
C.
f(x) = sin(x)
D.
f(x) = e^x
Show solution
Solution
A function is one-to-one if it never takes the same value twice. f(x) = 2x + 3 and f(x) = e^x are one-to-one.
Correct Answer:
B
— f(x) = 2x + 3
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Q. Which of the following is a subset of {1, 2, 3}?
A.
{1, 2}
B.
{4}
C.
{1, 2, 3, 4}
D.
{2, 3, 4}
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Solution
A subset is a set that contains elements from another set. {1, 2} is a subset of {1, 2, 3}.
Correct Answer:
A
— {1, 2}
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Q. Which of the following is a subset of {x, y, z}?
A.
{x, y}
B.
{x, y, z, w}
C.
{y, z, w}
D.
{x, y, z, x}
Show solution
Solution
A subset can only contain elements that are in the original set. {x, y} is a valid subset of {x, y, z}.
Correct Answer:
A
— {x, y}
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Q. Which of the following is an even function?
A.
f(x) = x^3
B.
f(x) = x^2
C.
f(x) = sin(x)
D.
f(x) = tan(x)
Show solution
Solution
An even function satisfies f(x) = f(-x). f(x) = x^2 is even.
Correct Answer:
B
— f(x) = x^2
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Q. Which of the following is not a subset of {a, b, c}?
A.
{a}
B.
{b, c}
C.
{a, b, c}
D.
{d}
Show solution
Solution
The set {d} does not contain any elements from {a, b, c}, so it is not a subset.
Correct Answer:
D
— {d}
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Q. Which of the following is the range of the function f(x) = x^2 - 4?
A.
(-∞, -4]
B.
[-4, ∞)
C.
(-4, ∞)
D.
[0, ∞)
Show solution
Solution
The minimum value of f(x) is -4 when x=0, so the range is [-4, ∞).
Correct Answer:
B
— [-4, ∞)
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Q. Which of the following is the range of the function f(x) = x^2?
A.
All real numbers
B.
All positive real numbers
C.
All non-negative real numbers
D.
All integers
Show solution
Solution
The function f(x) = x^2 is non-negative for all real x, hence the range is all non-negative real numbers.
Correct Answer:
C
— All non-negative real numbers
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Q. Which of the following is the range of the function f(x) = |x - 1|?
A.
(-∞, 1)
B.
[0, ∞)
C.
(-1, 1)
D.
[1, ∞)
Show solution
Solution
The minimum value of |x - 1| is 0 when x = 1, so the range is [0, ∞).
Correct Answer:
B
— [0, ∞)
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Q. Which of the following relations is an equivalence relation on the set of integers?
A.
x ~ y if x + y is even
B.
x ~ y if x - y is prime
C.
x ~ y if x > y
D.
x ~ y if x = y
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Solution
The relation x ~ y if x + y is even is reflexive, symmetric, and transitive, thus it is an equivalence relation.
Correct Answer:
A
— x ~ y if x + y is even
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Showing 181 to 210 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!
