JEE Sets, Relations & Functions PYQs With Video Solutions PDF

JEE Sets, Relations & Functions PYQ

Solving JEE Sets, Relations & Functions PYQ problems helps students build a strong foundation for algebra and calculus. Questions from this chapter test set operations, Cartesian products, different types of relations, domain and range, composition of functions and inverse functions.

The chapter is concept-driven. Many questions do not require lengthy calculations, but they demand careful interpretation of definitions and conditions. A minor mistake while identifying the domain, checking whether a relation is transitive or determining whether a function is one-one can lead to an incorrect answer.

Sets, Relations and Functions also supports several other chapters in JEE Mathematics, including inverse trigonometric functions, limits, continuity, differentiation and probability. Regular practice of JEE Sets, Relations & Functions Questions helps students understand how these basic concepts are applied in more advanced problems.

Questions from JEE Mains Previous Year Papers may combine two or more concepts. For example, students may be asked to determine the domain of a composite function or count the number of relations satisfying a particular condition. Therefore, definitions should be understood clearly instead of being memorised without application.

JEE Sets, Relations & Functions Important PYQ PDF

The JEE Sets, Relations & Functions Important PYQ PDF provided below contains selected previous-year questions for structured chapter-wise practice. The questions cover set operations, Venn diagrams, Cartesian products, types of relations, domain and range, composition of functions and inverse functions.

Attempt every question independently before checking the answer or explanation. While analyzing an incorrect response, identify whether the mistake occurred because of an incorrect set operation, a missed restriction in the domain, confusion between the properties of relations or an error while composing functions.

Important Topics Covered in Sets, Relations & Functions PYQs

Sets, Relations and Functions form an important part of the JEE Mathematics syllabus. Students should be comfortable with both direct definition-based questions and problems that combine these concepts with equations, graphs and inequalities.

Important topics include:

How to Solve Sets, Relations & Functions PYQs Effectively

Begin by identifying whether the problem is based on sets, relations or functions. Although these concepts are connected, each requires a different method of analysis.

Follow these steps while solving JEE Mains Questions from this chapter:

• Write the universal set and the given subsets clearly.
• Translate set notation into words before simplifying it.
• Use a Venn diagram when multiple sets overlap.
• List ordered pairs for small Cartesian products.
• Check reflexivity, symmetry and transitivity separately.
• Identify the domain, codomain and range of a function.
• Apply all restrictions before determining the domain.
• Use the horizontal-line test when checking whether a graph is one-to-one.
• Confirm bijectivity before finding an inverse.
• Check the order and domain while composing functions.
• Test endpoints and excluded values carefully.
• Substitute the final result into the original condition.

For domain-based questions, consider every restriction introduced by denominators, square roots, logarithms and inverse trigonometric functions. When more than one restriction is present, the final domain is the intersection of all the permitted sets.

In relation-based problems, do not rely only on visual patterns. Test every required condition systematically. For a relation to be an equivalence relation, it must satisfy all three properties: reflexivity, symmetry and transitivity.

In function problems, students often confuse codomain with range. The codomain is the target set stated in the definition, while the range contains the values actually produced by the function. This distinction is particularly important when determining whether a function is onto.

After completing chapter-wise practice, attempt a JEE Mains Mock Test to assess conceptual recall, speed and accuracy. During analysis, review questions in which you misunderstood notation, ignored a domain restriction or applied the composition in the wrong order.

Maintain an error log for common mistakes such as:

• Confusing an element with a subset
• Incorrectly applying De Morgan’s laws
• Double-counting elements in Venn diagrams
• Mixing up reflexive, symmetric and transitive relations
• Confusing range with codomain
• Ignoring restrictions while finding the domain
• Assuming every function has an inverse
• Reversing the order of composite functions
• Failing to check whether a function is onto

List of JEE Sets, Relations & Functions PYQs

The questions listed below can be attempted as a timed chapter-wise test. They cover set operations, Venn diagrams, Cartesian products, relations, functions, domain and range, composition and inverse functions.

Solve all the questions without checking the answers. After completing the test, review every incorrect, guessed and skipped question. Revise the corresponding definition or property and attempt the question again to ensure that the concept has been understood correctly.

Question 1

Which of the following statements is a tautology?

A

$$(\sim p \lor q) \Rightarrow p$$

B

$$p \Rightarrow (\sim p \lor q)$$

C

$$(\sim p \lor q) \Rightarrow q$$

D

$$q \Rightarrow (\sim p \lor q)$$

Show Answer Explanation

correct answer:- 4

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Question 2

For a suitably chosen real constant $$a$$, let a function, $$f : \mathbb{R} - \{-a\} \to \mathbb{R}$$ be defined by $$f(x) = \frac{a-x}{a+x}$$. Further suppose that for any real number $$x \neq -a$$, and $$f(x) \neq -a$$, $$(f \circ f)(x) = x$$. Then $$f\left(-\frac{1}{2}\right)$$ is equal to:

A

$$\frac{1}{3}$$

B

$$-\frac{1}{3}$$

C

$$-3$$

D

3

Show Answer Explanation

correct answer:- 4

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Question 3

The number of functions $$f$$, from the set $$A = \{x \in \mathbb{N}: x^2 - 10x + 9 \leq 0\}$$ to the set $$B = \{n^2 : n \in \mathbb{N}\}$$ such that $$f(x) \leq (x-3)^2 + 1$$, for every $$x \in A$$, is _______.

Show Answer Explanation

correct answer:- 1440

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Question 4

Negation of the Boolean expression $$p \leftrightarrow (q \rightarrow p)$$ is

A

$$\sim p \wedge q$$

B

$$p \wedge \sim q$$

C

$$\sim p \vee \sim q$$

D

$$\sim p \wedge \sim q$$

Show Answer Explanation

correct answer:- 4

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Question 5

The statement $$p \rightarrow (q \rightarrow p)$$ is equivalent to :

A

$$p \rightarrow q$$

B

$$p \rightarrow (p \vee q)$$

C

$$p \rightarrow (p \rightarrow q)$$

D

$$p \rightarrow (p \wedge q)$$

Show Answer Explanation

correct answer:- 2

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Question 6

Let $$f : N \rightarrow N$$ be a function such that $$f(m+n) = f(m) + f(n)$$ for every $$m, n \in N$$. If $$f(6) = 18$$ then $$f(2) \cdot f(3)$$ is equal to:

A

54

B

6

C

36

D

18

Show Answer Explanation Video Solution

correct answer:- 1

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Question 7

Negation of the statement $$(p \vee r) \Rightarrow (q \vee r)$$ is:

A

$$\sim p \wedge q \wedge \sim r$$

B

$$\sim p \wedge q \wedge r$$

C

$$p \wedge \sim q \wedge \sim r$$

D

$$p \wedge q \wedge r$$

Show Answer Explanation Video Solution

correct answer:- 3

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Question 8

The function $$f : R \to \left(-\frac{1}{2}, \frac{1}{2}\right)$$ defined as $$f(x) = \frac{x}{1+x^{2}}$$, is:

A

Invertible

B

Injective but not surjective

C

Surjective but not injective

D

Neither injective nor surjective

Show Answer Explanation

correct answer:- 3

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Question 9

Let $$f : (0, \infty) \to \mathbb{R}$$ be a function defined by $$f(x) = \frac{e^x}{e^x - 1}$$. Let $$g(x) = f(x) + f(-x)$$. Consider the following two statements:

• (I) $$g(x)$$ is a strictly decreasing function in $$(0, \infty)$$.
• (II) $$g(x)$$ is a one-one function in $$(0, \infty)$$.

A

Only (I) is true

B

Only (II) is true

C

Neither (I) nor (II) is true

D

Both (I) and (II) are true

Show Answer Explanation

Solution

Step 1: Simplify the function $$g(x)$$

Given:

$$f(x) = \frac{e^x}{e^x - 1}$$

We find $$f(-x)$$:

$$f(-x) = \frac{e^{-x}}{e^{-x} - 1} = \frac{\frac{1}{e^x}}{\frac{1}{e^x} - 1} = \frac{1}{1 - e^x}$$

Now, compute $$g(x) = f(x) + f(-x)$$:

$$g(x) = \frac{e^x}{e^x - 1} + \frac{1}{1 - e^x}$$ $$g(x) = \frac{e^x}{e^x - 1} - \frac{1}{e^x - 1}$$ $$g(x) = \frac{e^x - 1}{e^x - 1} = 1$$

Step 2: Evaluate Statement (I)

From the simplification above, we see that $$g(x) = 1$$ for all $$x$$ in the domain where $$e^x \neq 1$$ (which is $$x \neq 0$$).

• A constant function is not strictly decreasing. Its derivative $$g'(x) = 0$$.
• Statement (I) is False.

Step 3: Evaluate Statement (II)

A function is "one-one" (injective) if $$g(x_1) = g(x_2)$$ implies $$x_1 = x_2$$.

• Since $$g(x)$$ is a constant function ($$g(x) = 1$$), any two distinct values $$x_1, x_2$$ will result in $$g(x_1) = g(x_2) = 1$$.
• Therefore, the function is many-to-one , not one-one.
• Statement (II) is False.

Conclusion

Both statements (I) and (II) are incorrect because the function simplifies to a constant.

Correct Option: (3) Neither (I) nor (II) is true

correct answer:- 3

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Question 10

Let $$f : R \to R$$ be defined by $$f(x) = \frac{x}{1+x^2}$$, $$x \in R$$. Then the range of $$f$$ is

A

$$\left[-\frac{1}{2}, \frac{1}{2}\right]$$

B

$$R - [-1, 1]$$

C

$$R - \left[-\frac{1}{2}, \frac{1}{2}\right]$$

D

$$(-1, 1) - \{0\}$$

Show Answer Explanation

correct answer:- 1

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Question 11

If the domain of the function $$\log_{5}(18x - x^{2} - 77)$$ is $$(\alpha,\beta)$$ and the domain of the function $$\log_{(x-1)}\left(\frac{2x^{2}+3x-2}{x^{2}-3x-4}\right)$$ is $$(\gamma,\delta)$$, then $$\alpha^{2}+\beta^{2}+\gamma^{2}$$ is equal to :

A

195

B

176

C

186

D

174

Show Answer Explanation Video Solution

correct answer:- 3

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Question 12

Let R be a relation defined on the set {1 , 2, 3, 4} x { l, 2, 3, 4} by R = {((a, b), (c, d)): 2a + 3b = 3c + 4d}.
Then the number of elements in R is

A

15

B

6

C

12

D

18

Show Answer Explanation

correct answer:- 3

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Question 13

Let p, q, r denote arbitrary statements. Then the logically equivalent of the statement $$p \Rightarrow (q \vee r)$$ is:

A

$$(p \vee q) \Rightarrow r$$

B

$$(p \Rightarrow q) \vee (p \Rightarrow r)$$

C

$$(p \Rightarrow \sim q) \wedge (p \Rightarrow r)$$

D

$$(p \Rightarrow q) \wedge (p \Rightarrow \sim r)$$

Show Answer Explanation

correct answer:- 2

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Question 14

Let $$A = \{x \in R : x$$ is not a positive integer$$\}$$. Define a function $$f: A \to R$$ as $$f(x) = \frac{2x}{x-1}$$, then $$f$$ is:

A

Injective but not surjective

B

Not injective

C

Surjective but not injective

D

Neither injective nor surjective

Show Answer Explanation Video Solution

correct answer:- 1

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Question 15

Which of the following Boolean expression is a tautology?

A

$$(p \wedge q) \vee (p \vee q)$$

B

$$(p \wedge q) \vee (p \to q)$$

C

$$(p \wedge q) \wedge (p \to q)$$

D

$$(p \wedge q) \to (p \to q)$$

Show Answer Explanation Video Solution

correct answer:- 4

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Question 16

The largest interval lying in $$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$ for which the function $$\left[f(x) = 4^{-x^2} + \cos^{-1}\left(\frac{x}{2} - 1\right) + \log(\cos x)\right]$$ is defined, is

A

$$[0, \pi]$$

B

$$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$

C

$$\left[-\frac{\pi}{4}, \frac{\pi}{2}\right)$$

D

$$\left[0, \frac{\pi}{2}\right)$$

Show Answer Explanation Video Solution

correct answer:- 4

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Question 17

Let $$S=\left\{x^{3}+ax^{2}+bx+c:a,b,c, \in N \text{ and }a,b,c \leq 20\right\}$$ be a set of polynomials. Then the number of polynomials in S, which are divisible by $$x^{2}+2$$, is

A

10

B

6

C

20

D

120

Show Answer Explanation Video Solution

correct answer:- 1

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Question 18

Let $$f: \mathbb{R} \setminus \{0, 1\} \to \mathbb{R}$$ be a function satisfying the functional equation:$$f(x) + f\left(\frac{x-1}{x}\right) = 1 + x$$What is the explicit expression for $$f(2)$$?

A

$$\frac{1}{4}$$

B

$$\frac{3}{4}$$

C

$$\frac{5}{4}$$

D

$$\frac{7}{4}$$

Show Answer Explanation

Solution

Let

$$T(x)=\frac{x-1}{x}$$

Then the given functional equation is

$$f(x)+f(T(x))=1+x$$

Now,

$$T(T(x))=\frac{\frac{x-1}{x}-1}{\frac{x-1}{x}}=-\frac1{x-1}$$

and

$$T(T(T(x)))=x$$

Thus,

$$x,\ \frac{x-1}{x},\ -\frac1{x-1}$$

form a cycle of length $$3$$.

Applying the given equation successively to these three values:

$$f(x)+f\left(\frac{x-1}{x}\right)=1+x \qquad (1)$$

$$f\left(\frac{x-1}{x}\right)+f\left(-\frac1{x-1}\right)=1+\frac{x-1}{x} \qquad (2)$$

$$f\left(-\frac1{x-1}\right)+f(x)=1-\frac1{x-1} \qquad (3)$$

Adding (1), (2), and (3),

$$2\left[f(x)+f\left(\frac{x-1}{x}\right)+f\left(-\frac1{x-1}\right)\right]=3+x+\frac{x-1}{x}-\frac1{x-1}$$

For $$x=2$$,

$$\frac{x-1}{x}=\frac12$$

and

$$-\frac1{x-1}=-1$$

Hence,

$$f(2)+f\left(\frac12\right)=3 \qquad (4)$$

$$f\left(\frac12\right)+f(-1)=\frac32 \qquad (5)$$

$$f(-1)+f(2)=0 \qquad (6)$$

Adding (4), (5), and (6),

$$2\left[f(2)+f\left(\frac12\right)+f(-1)\right]=\frac92$$

$$f(2)+f\left(\frac12\right)+f(-1)=\frac94$$

Using (5) and (6),

$$f\left(\frac12\right)+f(-1)=\frac32$$

Therefore,

$$f(2)=\frac94-\frac32$$

$$=\frac94-\frac64$$

$$=\frac34$$

Hence,

$$\boxed{\frac34}$$

correct answer:- 2

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Question 19

If $$f(x) + 2f\left(\frac{1}{x}\right) = 3x$$, $$x \neq 0$$, and $$S = \{x \in R : f(x) = f(-x)\}$$, then $$S$$

A

Contains exactly two elements

B

Contains more than two elements

C

Is an empty set

D

Contains exactly one element

Show Answer Explanation

correct answer:- 1

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Question 20

The number of values of $$r \in \{p, q, \sim p, \sim q\}$$ for which $$((p \wedge q) \Rightarrow (r \vee q)) \wedge ((p \wedge r) \Rightarrow q)$$ is a tautology, is:

A

1

B

2

C

4

D

3

Show Answer Explanation

correct answer:- 2

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Question 21

A

B

C

D

Show Answer Explanation

correct answer:- 2

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Question 22

Consider :
Statement - I : $$(p \wedge \sim q) \wedge (\sim p \wedge q)$$ is a fallacy.
Statement - II : $$(p \rightarrow q) \leftrightarrow (\sim q \rightarrow \sim p)$$ is a tautology.

A

Statement - I is true; statement - II is false.

B

Statement - I is false; Statement - II is true.

C

Statement - I true; Statement - II is true; Statement - II is a correct explanation for Statement - I.

D

Statement - I is true; Statement - II is true; Statement - II is not a correct explanation for Statement - I.

Show Answer Explanation

correct answer:- 4

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Question 23

Let $$a, b, c \in R$$. If $$f(x) = ax^{2} + bx + c$$ is such that $$a + b + c = 3$$ and $$f(x + y) = f(x) + f(y) + xy$$, $$\forall$$ $$x, y \in R$$, then $$\sum_{n=1}^{10} f(n)$$ is equal to:

A

330

B

165

C

190

D

255

Show Answer Explanation

correct answer:- 1

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Question 24

The statement $$\sim(p \leftrightarrow \sim q)$$ is:

A

A tautology

B

A fallacy

C

Equivalent to $$p \leftrightarrow q$$

D

Equivalent to $$\sim p \leftrightarrow q$$

Show Answer Explanation

correct answer:- 3

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Question 25

The contrapositive of the statement "I go to school if it does not rain" is:

A

If it rains, I go to school.

B

If it rains, I do not go to school.

C

If I go to school, it rains.

D

If I do not go to school, it rains.

Show Answer Explanation

correct answer:- 2

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Question 26

Let $$f(x)=\log_{e}x$$ and $$g(x)=\frac{x^{4}-2x^{3}+3x^{2}-2x+2}{2x^{2}-2x+1}$$. Then the domain of $$f \circ g$$ is

A

$$\left(0,\infty\right)$$

B

$$\left(1,\infty\right)$$

C

$$\left(0,-\infty\right)$$

D

$$\mathbb{R}$$

Show Answer Explanation Video Solution

correct answer:- 4

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Question 27

Let $$f(x) = 2^{10}x + 1$$ and $$g(x) = 3^{10}x - 1$$. If $$(fog)(x) = x$$, then $$x$$ is equal to:

A

$$\frac{2^{10} - 1}{2^{10} - 3^{-10}}$$

B

$$\frac{1 - 2^{-10}}{3^{10} - 2^{-10}}$$

C

$$\frac{3^{10} - 1}{3^{10} - 2^{-10}}$$

D

$$\frac{1 - 3^{-10}}{2^{10} - 3^{-10}}$$

Show Answer Explanation

correct answer:- 2

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Question 28

The Boolean Expression $$(p \wedge \sim q) \vee q \vee (\sim p \wedge q)$$ is equivalent to

A

$$p \vee q$$

B

$$p \vee \sim q$$

C

$$\sim p \wedge q$$

D

$$p \wedge q$$

Show Answer Explanation

correct answer:- 1

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Question 29

The domain of the definition of the function $$f(x) = \frac{1}{4 - x^2} + \log_{10}(x^3 - x)$$ is:

A

$$(-1, 0) \cup (1, 2) \cup (2, \infty)$$

B

$$(1, 2) \cup (2, \infty)$$

C

$$(-2, -1) \cup (-1, 0) \cup (2, \infty)$$

D

$$(-1, 0) \cup (1, 2) \cup (3, \infty)$$

Show Answer Explanation Video Solution

correct answer:- 1

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Question 30

Let $$f : A \to B$$ be a function defined as $$f(x) = \frac{x-1}{x-2}$$, where $$A = R - \{2\}$$ and $$B = R - \{1\}$$. Then f is:

A

invertible and $$f^{-1}(y) = \frac{2y+1}{y-1}$$

B

invertible and $$f^{-1}(y) = \frac{3y-1}{y-1}$$

C

not invertible

D

invertible and $$f^{-1}(y) = \frac{2y-1}{y-1}$$

Show Answer Explanation

correct answer:- 4

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Question 31

Let N denote the set of all natural numbers. Define two binary relations on N as $$R_1 = \{(x, y) \in N \times N : 2x + y = 10\}$$ and $$R_2 = \{(x, y) \in N \times N : x + 2y = 10\}$$. Then:

A

both $$R_1$$ and $$R_2$$ are transitive relations

B

range of $$R_2$$ is $$\{1, 2, 3, 4\}$$

C

range of $$R_1$$ is $$\{2, 4, 8\}$$

D

both $$R_1$$ and $$R_2$$ are symmetric relations

Show Answer Explanation

correct answer:- 2