Sets, Relations & Functions for Competitive Exam Prep

Sets, Relations & Functions

Functions & types Inverse trigonometric functions Sets, subsets, power set Types of relations

Q. If E = {a, b, c}, how many subsets of E contain the element 'a'?

A. 2
B. 3
C. 4
D. 5

Solution

If 'a' is included, we can choose from the remaining elements {b, c}. The subsets containing 'a' are {a}, {a, b}, {a, c}, and {a, b, c}, totaling 4 subsets.

Correct Answer: C — 4

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Q. If E = {a, b, c}, what is the union of E and F = {b, c, d}?

A. {a, b, c}
B. {a, b, c, d}
C. {b, c}
D. {a, d}

Solution

The union of two sets includes all unique elements from both sets. Here, the union is {a, b, c, d}.

Correct Answer: B — {a, b, c, d}

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Q. If E = {a, b}, what is the size of the power set of E?

A. 1
B. 2
C. 3
D. 4

Solution

The power set of a set with n elements has 2^n subsets. For E, n = 2, so the size of the power set is 2^2 = 4.

Correct Answer: D — 4

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Q. If E = {x | x is a prime number less than 10}, what is E?

A. {2, 3, 5, 7}
B. {1, 2, 3, 4}
C. {2, 4, 6, 8}
D. {3, 5, 7}

Solution

The prime numbers less than 10 are 2, 3, 5, and 7. Therefore, E = {2, 3, 5, 7}.

Correct Answer: A — {2, 3, 5, 7}

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Q. If E = {x | x is a prime number less than 10}, what is the power set of E?

A. {∅, {2}, {3}, {5}, {7}, {2, 3}, {2, 5}, {2, 7}, {3, 5}, {3, 7}, {5, 7}, {2, 3, 5}, {2, 3, 7}, {2, 5, 7}, {3, 5, 7}, {2, 3, 5, 7}}
B. {∅, {2, 3, 5, 7}}
C. {∅, {2}, {3}, {5}, {7}}
D. {∅, {2, 3}, {5, 7}}

Solution

The prime numbers less than 10 are {2, 3, 5, 7}. The power set has 2^4 = 16 elements.

Correct Answer: A — {∅, {2}, {3}, {5}, {7}, {2, 3}, {2, 5}, {2, 7}, {3, 5}, {3, 7}, {5, 7}, {2, 3, 5}, {2, 3, 7}, {2, 5, 7}, {3, 5, 7}, {2, 3, 5, 7}}

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Q. If F = {1, 2, 3, 4}, how many subsets of F contain the element 1?

A. 4
B. 8
C. 6
D. 2

Solution

If 1 is included, we can choose from the remaining elements {2, 3, 4}, which has 2^3 = 8 subsets.

Correct Answer: C — 6

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Q. If F = {1, 2, 3, 4}, what is the number of proper subsets of F?

A. 4
B. 8
C. 15
D. 16

Solution

The total number of subsets of a set with n elements is 2^n. For F, n = 4, so there are 2^4 = 16 subsets. Proper subsets are total subsets minus the set itself, so 16 - 1 = 15.

Correct Answer: B — 8

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Q. If F = {1, 2, 3}, what is the number of elements in the power set of F?

A. 3
B. 4
C. 6
D. 8

Solution

The power set of a set with n elements has 2^n elements. For F, n = 3, so the power set has 2^3 = 8 elements.

Correct Answer: D — 8

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Q. If F = {a, b, c}, how many subsets contain exactly 2 elements?

A. 3
B. 4
C. 2
D. 1

Solution

The subsets containing exactly 2 elements are {a, b}, {a, c}, and {b, c}. So, there are 3 such subsets.

Correct Answer: A — 3

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Q. If F = {x | x is an even number between 1 and 10}, what is F?

A. {2, 4, 6, 8}
B. {1, 3, 5, 7, 9}
C. {2, 4, 6, 8, 10}
D. {0, 2, 4, 6, 8}

Solution

The even numbers between 1 and 10 are 2, 4, 6, and 8. Therefore, F = {2, 4, 6, 8}.

Correct Answer: A — {2, 4, 6, 8}

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Q. If F = {x, y, z}, what is the size of the power set of F?

A. 3
B. 6
C. 8
D. 12

Solution

The size of the power set is 2^n where n is the number of elements in the set. Here, n = 3, so the size is 2^3 = 8.

Correct Answer: C — 8

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Q. If F = {x, y}, how many subsets of F are there that do not contain 'y'?

A. 1
B. 2
C. 3
D. 4

Solution

The subsets of F that do not contain 'y' are {∅, {x}}, totaling 2 subsets.

Correct Answer: A — 1

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Q. If f(x) = 2x + 3, what is f(4)?

A. 8
B. 11
C. 12
D. 10

Solution

Substituting x = 4 into the function gives f(4) = 2(4) + 3 = 8 + 3 = 11.

Correct Answer: B — 11

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Q. If f(x) = 2x + 3, what is f(5)?

A. 10
B. 13
C. 15
D. 8

Solution

f(5) = 2(5) + 3 = 10 + 3 = 13.

Correct Answer: B — 13

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Q. If f(x) = 2x^2 - 3x + 1, what is f(2)?

A. 1
B. 3
C. 5
D. 7

Solution

f(2) = 2(2^2) - 3(2) + 1 = 8 - 6 + 1 = 3.

Correct Answer: C — 5

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Q. If f(x) = 2^x, what is f(3)?

A. 6
B. 8
C. 9
D. 10

Solution

f(3) = 2^3 = 8.

Correct Answer: B — 8

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Q. If f(x) = 3x - 4, find f(-2).

A. -10
B. -8
C. -6
D. -4

Solution

f(-2) = 3(-2) - 4 = -6 - 4 = -10.

Correct Answer: A — -10

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Q. If f(x) = 3x - 4, what is f(-2)?

A. -10
B. -8
C. -6
D. -4

Solution

f(-2) = 3(-2) - 4 = -6 - 4 = -10.

Correct Answer: A — -10

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Q. If f(x) = 3x - 4, what is the inverse function f^(-1)(x)?

A. (x + 4)/3
B. 3x + 4
C. 3(x - 4)
D. x/3 + 4

Solution

To find the inverse, set y = 3x - 4, solve for x: x = (y + 4)/3, thus f^(-1)(x) = (x + 4)/3.

Correct Answer: A — (x + 4)/3

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Q. If f(x) = 3x - 5, what is f(f(1))?

A. -2
B. 1
C. 4
D. 7

Solution

First, find f(1) = 3(1) - 5 = -2. Then, f(-2) = 3(-2) - 5 = -6 - 5 = -11.

Correct Answer: D — 7

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Q. If f(x) = sin(x), what is f(π/2)?

A. 0
B. 1
C. -1
D. undefined

Solution

f(π/2) = sin(π/2) = 1.

Correct Answer: B — 1

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Q. If f(x) = x^2 + 2x + 1, what is the vertex of the parabola?

A. (-1, 0)
B. (0, 1)
C. (-1, 1)
D. (1, 0)

Solution

The vertex form is f(x) = (x + 1)^2, so the vertex is (-1, 0).

Correct Answer: C — (-1, 1)

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Q. If f(x) = x^2 - 4, what are the x-intercepts?

A. -2, 2
B. 0, 4
C. 2, 4
D. None

Solution

To find x-intercepts, set f(x) = 0: x^2 - 4 = 0, which gives x = ±2.

Correct Answer: A — -2, 2

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Q. If f(x) = x^2 - 4x + 3, what is the value of f(2)?

A. 0
B. 1
C. 2
D. 3

Solution

f(2) = 2^2 - 4*2 + 3 = 4 - 8 + 3 = -1.

Correct Answer: A — 0

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Q. If f(x) = x^2 and g(x) = x + 1, what is (f ∘ g)(2)?

A. 4
B. 9
C. 16
D. 25

Solution

(f ∘ g)(2) = f(g(2)) = f(2 + 1) = f(3) = 3^2 = 9.

Correct Answer: B — 9

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Q. If f(x) = x^2, what is f(-3)?

A. 3
B. 6
C. 9
D. 12

Solution

f(-3) = (-3)^2 = 9.

Correct Answer: C — 9

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Q. If f(x) = x^3 - 3x + 2, what is f(1)?

A. 0
B. 1
C. 2
D. 3

Solution

f(1) = 1^3 - 3(1) + 2 = 1 - 3 + 2 = 0.

Correct Answer: A — 0

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Q. If f(x) = x^3 - 3x + 2, what is the value of f(1)?

A. 0
B. 1
C. 2
D. 3

Solution

f(1) = 1^3 - 3(1) + 2 = 1 - 3 + 2 = 0.

Correct Answer: A — 0

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Q. If f(x) = |x - 2|, what is f(2)?

A. 0
B. 1
C. 2
D. 3

Solution

f(2) = |2 - 2| = |0| = 0.

Correct Answer: A — 0

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Q. If f(x) = |x|, what is f(-3)?

A. -3
B. 3
C. 0
D. undefined

Solution

f(-3) = |-3| = 3.

Correct Answer: B — 3

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Showing 61 to 90 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!

Sets, Relations & Functions

Sets, Relations & Functions MCQ & Objective Questions

What You Will Practise Here

Exam Relevance

Common Mistakes Students Make

FAQs

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