JEE Main > Sets, Relations and Functions

Explore popular questions from Sets, Relations and Functions for JEE Main. This collection covers Sets, Relations and Functions previous year JEE Main questions hand picked by popular teachers.


Q 1.    

Correct4

Incorrect-1

If {tex} \tan x = n \cdot \tan y , n \in R ^ { + } , {/tex} then maximum value of {tex} \sec ^ { 2 } ( x - y ) {/tex} is equal to

A

{tex} \frac { ( n + 1 ) ^ { 2 } } { 2 n } {/tex}

B

{tex} \frac { ( n + 1 ) ^ { 2 } } { n } {/tex}

C

{tex} \frac { ( n + 1 ) ^ { 2 } } { 2 } {/tex}

{tex} \frac { ( n + 1 ) ^ { 2 } } { 4 n } {/tex}

Explanation


Q 2.    

Correct4

Incorrect-1

Which one is not periodic?

A

{tex} | \sin 3 x | + \sin ^ { 2 } x {/tex}

{tex} \cos \sqrt { x } + \cos ^ { 2 } x {/tex}

C

{tex} \cos 4 x + \tan ^ { 2 } x {/tex}

D

{tex} \cos 2 x + \sin x {/tex}

Explanation





Q 3.    

Correct4

Incorrect-1

The period of {tex} \sin ^ { 2 } \theta {/tex} is

A

{tex} \pi ^ { 2 } {/tex}

{tex} \pi {/tex}

C

{tex} \pi ^ { 3 } {/tex}

D

{tex} \pi / 2 {/tex}

Explanation


Q 4.    

Correct4

Incorrect-1

The domain of {tex} \sin ^ { - 1 } \left[ \log _ { 3 } \left( \frac { x } { 3 } \right) \right] {/tex} is

{tex} [ 1,9 ] {/tex}

B

{tex} [ - 1,9 ] {/tex}

C

{tex} [ - 9,1 ] {/tex}

D

{tex} [ - 9 , - 1 ] {/tex}

Explanation



Q 5.    

Correct4

Incorrect-1

The function {tex} f ( x ) = \log ( x + \sqrt { x ^ { 2 } + 1 } ) {/tex} is

an odd function

B

a periodic function

C

neither an even nor an odd function

D

an even function.

Explanation




Q 6.    

Correct4

Incorrect-1

A function {tex} f {/tex} from the set of natural numbers to integers defined by \[ f ( n ) = \left\{ \begin{array} { l } { \frac { n - 1 } { 2 } , \text { when } n \text { is odd } } \\ { - \frac { n } { 2 } , \text { when } n \text { is even } } \end{array} \right. \] is an

A

onto but not one-one

one-one and onto both

C

neither one-one nor onto

D

one-one but not onto.

Explanation





Q 7.    

Correct4

Incorrect-1

Domain of definition of the function \[ f ( x ) = \frac { 3 } { 4 - x ^ { 2 } } + \log _ { 10 } \left( x ^ { 3 } - x \right) , \text { is } \]

A

{tex} ( - 1,0 ) \cup ( 1,2 ) {/tex}

B

{tex} ( 1,2 ) \cup ( 2 , \infty ) {/tex}

{tex} ( - 1,0 ) \cup ( 1,2 ) \cup ( 2 , \infty ) {/tex}

D

{tex} ( 1,2 ) {/tex}

Explanation



Q 8.    

Correct4

Incorrect-1

If {tex} f: R \rightarrow R {/tex} satisfies {tex} f ( x + y ) = f ( x ) + f ( y ) , {/tex} for all {tex} x , y \in R {/tex} and {tex} f ( 1 ) = 7 , {/tex} then {tex} \sum _ { r = 1 } ^ { n } f ( r ) {/tex} is

A

{tex} \frac { 7 ( n + 1 ) } { 2 } {/tex}

B

{tex} 7 n ( n + 1 ) {/tex}

{tex} \frac { 7 n ( n + 1 ) } { 2 } {/tex}

D

{tex} \frac { 7 n } { 2 } {/tex}

Explanation





Q 9.    

Correct4

Incorrect-1

Let {tex} R = \{ ( 1,3 ) , ( 4,2 ) , ( 2,4 ) , ( 2,3 ) , ( 3,1 ) \} {/tex} be a relation on the set {tex} A = \{ 1,2,3,4 \} . {/tex} The relation {tex} R {/tex} is

not symmetric

B

transitive

C

a function

D

reflexive.

Explanation



Q 10.    

Correct4

Incorrect-1

The range of the function {tex} f ( x ) = ^{7 - x} P _ { x - 3 } {/tex} is

A

{tex} \{ 1,2,3,4 \} {/tex}

B

{tex} \{ 1,2,3,4,5,6 \} {/tex}

{tex} \{ 1,2,3 \} {/tex}

D

{tex} \{ 1,2,3,4,5 \} {/tex}

Explanation



Q 11.    

Correct4

Incorrect-1

If {tex} f: R \rightarrow S , {/tex} defined by {tex} f ( x ) = \sin x - \sqrt { 3 } \cos x + 1 {/tex} is onto, then the interval of {tex} S {/tex} is

A

{tex} [ 0,1 ] {/tex}

B

{tex} [ - 1,1 ] {/tex}

C

{tex} [ 0,3 ] {/tex}

{tex} [ - 1,3 ] {/tex}

Explanation



Q 12.    

Correct4

Incorrect-1

The graph of the function {tex} y = f ( x ) {/tex} is symmetrical about the line {tex} x = 2 , {/tex} then

A

{tex} f ( x ) = f ( - x ) {/tex}

B

{tex} f ( 2 + x ) = f ( 2 - x ) {/tex}

{tex} f ( x + 2 ) = f ( x - 2 ) {/tex}

D

{tex} f ( x ) = - f ( - x ) {/tex}

Explanation



Q 13.    

Correct4

Incorrect-1

The domain of the function {tex} f ( x ) = \frac { \sin ^ { - 1 } ( x - 3 ) } { \sqrt { 9 - x ^ { 2 } } } {/tex} is

A

{tex} [ 1,2 ] {/tex}

{tex} [ 2,3 ) {/tex}

C

{tex} [ 2,3 ] {/tex}

D

{tex} [ 1,2 ) {/tex}

Explanation



Q 14.    

Correct4

Incorrect-1

Let {tex} R = \{ ( 3,3 ) ( 6,6 ) ( 9,9 ) , ( 12,12 ) , ( 6,12 ) , {/tex} {tex} ( 3,9 ) , ( 3,12 ) , ( 3,6 ) \} {/tex} be a relation on the set {tex} A = \{ 3,6,9,12 \} . {/tex} The relation is

A

reflexive and symmetric only

B

an equivalence relation

C

reflexive only

reflexive and transitive only.

Explanation



Q 15.    

Correct4

Incorrect-1

Let {tex} f: ( - 1,1 ) \rightarrow B , {/tex} be a function defined by {tex} f ( x ) = \tan ^ { - 1 } \left( \frac { 2 x } { 1 - x ^ { 2 } } \right) , {/tex} then {tex} f {/tex} is both one-one and onto when {tex} B {/tex} is the interval

A

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

B

{tex} \left( 0 , \frac { \pi } { 2 } \right) {/tex}

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

D

{tex} \left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right] {/tex}

Explanation



Q 16.    

Correct4

Incorrect-1

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched in the pair Interval : Function?

A

{tex} [ 2 , \infty ) : 2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + 6 {/tex}

B

{tex} ( - \infty , \infty ) : x ^ { 3 } - 3 x ^ { 2 } + 3 x + 3 {/tex}

C

{tex} ( - \infty , - 4 ] : x ^ { 3 } + 6 x ^ { 2 } + 6 {/tex}

{tex} \left( - \infty , \frac { 1 } { 3 } \right] : 3 x ^ { 2 } - 2 x + 1 {/tex}

Explanation


Q 17.    

Correct4

Incorrect-1

A real valued function {tex} f ( x ) {/tex} satisfies the functional equation
{tex} f ( x - y ) = f ( x ) f ( y ) - f ( a - x ) f ( a + y ) {/tex}
where {tex} a {/tex} is a given constant and {tex} f ( 0 ) = 1 {/tex} {tex} f ( 2 a - x ) {/tex} is equal to

A

{tex} f ( x ) {/tex}

{tex} - f ( x ) {/tex}

C

{tex} f ( - x ) {/tex}

D

{tex} f ( a ) + f ( a - x ) {/tex}

Explanation



Q 18.    

Correct4

Incorrect-1

Let {tex} W {/tex} denote the words in the English dictionary. Define the relation {tex} R {/tex} by: {tex} R = \{ ( x , y ) \in W \times W | \text { the words } x \text { and } y {/tex} have at least one letter in common}.
Then {tex} R {/tex} is

A

not reflexive, symmetric and transitive

reflexive, symmetric and not transitive

C

reflexive, symmetric and transitive

D

reflexive, not symmetric and transitive.

Explanation


Q 19.    

Correct4

Incorrect-1

The set {tex} S = \{ 1,2,3 , \ldots \ldots , 12 \} {/tex} is to be partitioned into three sets {tex} A , B , C {/tex} of equal size. Thus {tex} A \cup B \cup C = S , A \cap B = B \cap C = A \cap C = \phi . {/tex} The number of ways to partition {tex} S {/tex} is

{tex} \frac { 12 ! } { ( 4 ! ) ^ { 3 } } {/tex}

B

{tex} \frac { 12 ! } { ( 4 ! ) ^ { 4 } } {/tex}

C

{tex} \frac { 12 ! } { 3 ! ( 4 ! ) ^ { 3 } } {/tex}

D

{tex} \frac { 12 ! } { 3 ! ( 4 ! ) ^ { 4 } } {/tex}

Explanation

Q 20.    

Correct4

Incorrect-1

Let {tex} R {/tex} be the real line. Consider the following subsets of the plane
{tex} R \times R: {/tex} {tex} S = \{ ( x , y ): y = x + 1 \text { and } 0 < x < 2 \} {/tex}
{tex} T = \{ ( x , y ): x - y \text { is an integer } \} {/tex}
Which one of the following is true?

{tex} T {/tex} is an equivalence relation on {tex} R {/tex} but {tex} S {/tex} is not

B

Neither {tex} S {/tex} nor {tex} T {/tex} is an equivalence relation on {tex} R {/tex}

C

Both {tex} S {/tex} and {tex} T {/tex} are equivalence relations on {tex} R {/tex}

D

{tex} S {/tex} is an equivalence relation on {tex} R {/tex} but {tex} T {/tex} is not

Explanation



Q 21.    

Correct4

Incorrect-1

Let {tex} f: N \rightarrow Y {/tex} be a function defined as
{tex} f ( x ) = 4 x + 3 {/tex} where {tex} Y = \{ y \in N: y = 4 x + 3 \text { for some } x \in N \} {/tex}
Show that {tex} f {/tex} is invertible and its inverse is

{tex} g ( y ) = \frac { y - 3 } { 4 } {/tex}

B

{tex} g ( y ) = \frac { 3 y + 4 } { 3 } {/tex}

C

{tex} g ( y ) = 4 + \frac { y + 3 } { 4 } {/tex}

D

{tex} g ( y ) = \frac { y + 3 } { 4 } {/tex}

Explanation


Q 22.    

Correct4

Incorrect-1

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

A

{tex} [ 0 , \pi ] {/tex}

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

C

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

D

{tex} \left[ 0 , \frac { \pi } { 4} \right) {/tex}

Explanation









Q 23.    

Correct4

Incorrect-1

Let {tex} f : R \rightarrow R {/tex} be a function defined by {tex} f ( x ) = \operatorname { Min } \{ x + 1 , | x | + 1 \} {/tex} . Then which of the following is true?

A

{tex} f ( x ) \geq 1 {/tex} for all {tex} x \in R {/tex}

B

{tex} f ( x ) {/tex} is not differentiable at {tex} x = 1 {/tex}

{tex} f ( x ) {/tex} is differentiable everywhere

D

{tex} f ( x ) {/tex} is not differentiable at {tex} x = 0 {/tex}

Explanation







Q 24.    

Correct4

Incorrect-1

Let {tex} f : N \rightarrow Y {/tex} be a function defined as {tex} f ( x ) = 4 x + 3 , {/tex} where {tex} Y = \{ y \in N : y = 4 x + 3 \text { for some } x \in N \} . {/tex} Show that {tex} f {/tex} is invertible and its inverse is
and its inverse is

A

{tex} g ( y ) = \frac { 3 y + 4 } { 3 } {/tex}

B

{tex} g ( y ) = 4 + \frac { y + 3 } { 4 } {/tex}

C

{tex} g ( y ) = \frac { y + 3 } { 4 } {/tex}

{tex} g ( y ) = \frac { y - 3 } { 4 } {/tex}

Explanation









Q 25.    

Correct4

Incorrect-1

Let {tex} R {/tex} be the real line. Consider the following subsets of the plane {tex} R \times R . {/tex}
{tex} S = \{ ( x , y ) : y = x + 1 \text { and } 0 < x < 2 \} , T = \{ ( x , y ) : x - y \text { is an integer } \} {/tex} Which one of the following is true?

A

neither {tex} S {/tex} nor {tex} T {/tex} is an equivalence relation on {tex} R {/tex}

B

both {tex} S {/tex} and {tex} T {/tex} are equivalence relations on {tex} R {/tex}

C

{tex} S {/tex} is an equivalence relation on {tex} R {/tex} but {tex} T {/tex} is not

{tex} T {/tex} is an equivalence relation on {tex} R {/tex} but {tex} S {/tex} is not

Explanation