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JEE Advanced > Sets, Relations and Functions

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

Q 1.

Correct4

Incorrect-1

If {tex} \tan \theta = - \frac { 4 } { 3 } , {/tex} then {tex} \sin \theta {/tex} is

A

{tex} - \frac { 4 } { 5 } {/tex} but not {tex} \frac { 4 } { 5 } {/tex}

{tex} - \frac { 4 } { 5 } {/tex} or {tex} \frac { 4 } { 5 } {/tex}

C

{tex} \frac { 4 } { 5 } {/tex} but not {tex} - \frac { 4 } { 5 } {/tex}

D

None of these

Explanation

Q 2.

Correct4

Incorrect-1

If {tex} \alpha + \beta + \gamma = 2 \pi , {/tex} then

{tex} \tan \frac { \alpha } { 2 } + \tan \frac { \beta } { 2 } + \tan \frac { \gamma } { 2 } = \tan \frac { \alpha } { 2 } \tan \frac { \beta } { 2 } \tan \frac { \gamma } { 2 } {/tex}

B

{tex} \tan \frac { \alpha } { 2 } \tan \frac { \beta } { 2 } + \tan \frac { \beta } { 2 } \tan \frac { \gamma } { 2 } + \tan \frac { \gamma } { 2 } \tan \frac { \alpha } { 2 } = 1 {/tex}

C

{tex} \tan \frac { \alpha } { 2 } + \tan \frac { \beta } { 2 } + \tan \frac { \gamma } { 2 } = - \tan \frac { \alpha } { 2 } \tan \frac { \beta } { 2 } \tan \frac { \gamma } { 2 } {/tex}

D

None of these

Explanation

Q 3.

Correct4

Incorrect-1

Given {tex} A = \sin ^ { 2 } \theta + \cos ^ { 4 } \theta {/tex} then for all real values of {tex} \theta {/tex}

A

{tex} 1 \leq A \leq 2 {/tex}

{tex} \frac { 3 } { 4 } \leq A \leq 1 {/tex}

C

{tex} \frac { 13 } { 16 } \leq A \leq 1 {/tex}

D

{tex} \frac { 3 } { 4 } \leq A \leq \frac { 13 } { 16 } {/tex}

Explanation

Q 4.

Correct4

Incorrect-1

The equation {tex} 2 \cos ^ { 2 } \frac { x } { 2 } \sin ^ { 2 } x = x ^ { 2 } + x ^ { - 2 } ; 0 < x \leq \frac { \pi } { 2 } {/tex} has

no real solution

B

one real solution

C

more than one solution

D

none of these

Explanation

Q 5.

Correct4

Incorrect-1

Let {tex} 0 < x < \frac { \pi } { 4 } {/tex} then {tex} ( \sec 2 x - \tan 2 x ) {/tex} equals

A

{tex} \tan \left( x - \frac { \pi } { 4 } \right) {/tex}

{tex} \tan \left( \frac { \pi } { 4 } - x \right) {/tex}

C

{tex} \tan \left( x + \frac { \pi } { 4 } \right) {/tex}

D

{tex} \tan ^ { 2 } \left( x + \frac { \pi } { 4 } \right) {/tex}

Explanation

Q 6.

Correct4

Incorrect-1

Let {tex} n {/tex} be a positive integer such that
{tex} \sin \frac { \pi } { 2 n } + \cos \frac { \pi } { 2 n } = \frac { \sqrt { n } } { 2 } . {/tex} Then

A

{tex} 6 \leq n \leq 8 {/tex}

B

{tex} 4 < n \leq 8 {/tex}

C

{tex} 4 \leq n \leq 8 {/tex}

{tex} 4 < n < 8 {/tex}

Explanation

Q 7.

Correct4

Incorrect-1

If {tex} \omega {/tex} is an imaginary cube root of unity then the value of
{tex} \sin \left\{ \left( \omega ^ { 10 } + \omega ^ { 23 } \right) \pi - \frac { \pi } { 4 } \right\} {/tex} is

A

{tex} - \frac { \sqrt { 3 } } { 2 } {/tex}

B

{tex} - \frac { 1 } { \sqrt { 2 } } {/tex}

{tex} \frac { 1 } { \sqrt { 2 } } {/tex}

D

{tex} \frac { \sqrt { 3 } } { 2 } {/tex}

Explanation

Q 8.

Correct4

Incorrect-1

{tex} 3 ( \sin x - \cos x ) ^ { 4 } + 6 ( \sin x + \cos x ) ^ { 2 } + 4 \left( \sin ^ { 6 } x + \cos ^ { 6 } x \right) = {/tex}

A

11

B

12

13

D

14

Explanation

Q 9.

Correct4

Incorrect-1

The general values of {tex} \theta {/tex} satisfying the equation {tex} 2 \sin ^ { 2 } \theta - 3 \sin \theta - 2 = 0 {/tex} is

A

{tex} n \pi + ( - 1 ) ^ { n } \pi / 6 {/tex}

B

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

C

{tex} n \pi + ( - 1 ) ^ { n } 5 \pi / 6 {/tex}

{tex} n \pi + ( - 1 ) ^ { n } 7 \pi / 6 {/tex}

Explanation

Q 10.

Correct4

Incorrect-1

{tex} \sec ^ { 2 } \theta = \frac { 4 x y } { ( x + y ) ^ { 2 } } {/tex} is true if and only if

A

{tex} x + y \neq 0 {/tex}

{tex} x = y , x \neq 0 {/tex}

C

{tex} x = y {/tex}

D

{tex} x \neq 0 , y \neq 0 {/tex}

Explanation



Q 11.

Correct4

Incorrect-1

If {tex} \alpha + \beta = \pi / 2 {/tex} and {tex} \beta + \gamma = \alpha , {/tex} then tan {tex} \alpha {/tex} equals

A

{tex} 2 ( \tan \beta + \tan \gamma ) {/tex}

B

{tex} \tan \beta + \tan \gamma {/tex}

{tex} \tan \beta + 2 \tan \gamma {/tex}

D

{tex} 2 \tan \beta + \tan \gamma {/tex}

Explanation

Q 12.

Correct4

Incorrect-1

The number of integral values of {tex} k {/tex} for which the equation {tex} 7 \cos x + 5 \sin x = 2 k + 1 {/tex} has a solution is

A

4

8

C

10

D

12

Explanation

Q 13.

Correct4

Incorrect-1

Given both {tex} \theta {/tex} and {tex} \phi {/tex} are acute angles and {tex} \sin \theta = \frac { 1 } { 2 } {/tex}, {tex} \cos \phi = \frac { 1 } { 3 } , {/tex} then the value of {tex} \theta + \phi {/tex} belongs to

A

{tex} \left( \frac { \pi } { 3 } , \frac { \pi } { 2 } \right] {/tex}

{tex} \left( \frac { \pi } { 2 } , \frac { 2 \pi } { 3 } \right) {/tex}

C

{tex} \left( \frac { 2 \pi } { 3 } , \frac { 5 \pi } { 6 } \right] {/tex}

D

{tex} \left( \frac { 5 \pi } { 6 } , \pi \right] {/tex}

Explanation

Q 14.

Correct4

Incorrect-1

{tex} \cos ( \alpha - \beta ) = 1 {/tex} and {tex} \cos ( \alpha + \beta ) = 1 / e {/tex} where {tex} \alpha , \beta \in [ - \pi , \pi ] {/tex} Pairs of {tex} \alpha , \beta {/tex} which satisfy both the equations is/are

A

0

B

1

C

2

4

Explanation

Q 15.

Correct4

Incorrect-1

The values of {tex} \theta \in ( 0,2 \pi ) {/tex} for which {tex} 2 \sin ^ { 2 } \theta - 5 \sin \theta + 2 > 0 {/tex} , are

{tex} \left( 0 , \frac { \pi } { 6 } \right) \cup \left( \frac { 5 \pi } { 6 } , 2 \pi \right) {/tex}

B

{tex} \left( \frac { \pi } { 8 } , \frac { 5 \pi } { 6 } \right) {/tex}

C

{tex} \left( 0 , \frac { \pi } { 8 } \right) \cup \left( \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } \right) {/tex}

D

{tex} \left( \frac { 41 \pi } { 48 } , \pi \right) {/tex}

Explanation





Q 16.

Correct4

Incorrect-1

The number of solutions of the pair of equations
{tex} 2 \sin ^ { 2 } \theta - \cos 2 \theta = 0 {/tex}
{tex} 2 \cos ^ { 2 } \theta - 3 \sin \theta = 0 {/tex}
in the interval {tex} [ 0,2 \pi ] {/tex} is

A

zero

B

one

two

D

four

Explanation





Q 17.

Correct4

Incorrect-1

For {tex} x \in ( 0 , \pi ) , {/tex} the equation {tex} \sin x + 2 \sin 2 x - \sin 3 x = 3 {/tex} has

A

infinitely many solutions

B

three solutions

C

one solution

no solution

Explanation



Q 18.

Correct4

Incorrect-1

Let {tex} \mathrm { S } = \left\{ \mathrm { x } \in ( - \pi , \pi ): \mathrm { x } \neq 0 , \pm \frac { \pi } { 2 } \right\} . {/tex} The sum of all distinct
solutions of the equation {tex} \sqrt { 3 } \sec x + \mathrm {cosec} x + 2 ( \tan x -\cot x ) {/tex} {tex} = 0 {/tex} in the set {tex} S {/tex} is equal to

A

{tex} - \frac { 7 \pi } { 9 } {/tex}

B

{tex} - \frac { 2 \pi } { 9 } {/tex}

0

D

{tex} \frac { 5 \pi } { 9 } {/tex}

Explanation



Q 19.

Correct4

Incorrect-1

The value of {tex} \displaystyle \sum _ { k = 1 } ^ { 13 } \frac { 1 } { \sin \left( \frac { \pi } { 4 } + \frac { ( k - 1 ) \pi } { 6 } \right) \sin \left( \frac { \pi } { 4 } + \frac { k \pi } { 6 } \right) } {/tex} is equal to

A

{tex} 3 - \sqrt { 3 } {/tex}

B

{tex} 2 ( 3 - \sqrt { 3 } ) {/tex}

{tex} 2 ( \sqrt { 3 } - 1 ) {/tex}

D

{tex} 2 ( 2 - \sqrt { 3 } ) {/tex}

Explanation



Q 20.

Correct4

Incorrect-1

The value of {tex} \tan \left[ \cos ^ { - 1 } \left( \frac { 4 } { 5 } \right) + \tan ^ { - 1 } \left( \frac { 2 } { 3 } \right) \right] {/tex} is

A

{tex} \frac { 6 } { 17 } {/tex}

B

{tex} \frac { 7 } { 16 } {/tex}

C

{tex} \frac { 16 } { 7 } {/tex}

none

Explanation

Q 21.

Correct4

Incorrect-1

If we consider only the principle values of the inverse trigonometric functions then the value of
{tex} \tan \left( \cos ^ { - 1 } \frac { 1 } { 5 \sqrt { 2 } } - \sin ^ { - 1 } \frac { 4 } { \sqrt { 17 } } \right) {/tex} is

A

{tex} \frac { \sqrt { 29 } } { 3 } {/tex}

B

{tex} \frac { 29 } { 3 } {/tex}

C

{tex} \frac { \sqrt { 3 } } { 29 } {/tex}

{tex} \frac { 3 } { 29 } {/tex}

Explanation

Q 22.

Correct4

Incorrect-1

The number of real solutions of

{tex} \tan ^ { - 1 } \sqrt { x ( x + 1 ) } + \sin ^ { - 1 } \sqrt { x ^ { 2 } + x + 1 } = \pi / 2 {/tex} is

A

zero

B

one

two

D

infinite

Explanation

Q 23.

Correct4

Incorrect-1


for {tex} 0 < | x | < \sqrt { 2 } , {/tex} then {tex} x {/tex} equals

A

{tex} 1 / 2 {/tex}

1

C

{tex} - 1 / 2 {/tex}

D

{tex} -1 {/tex}

Explanation

Q 24.

Correct4

Incorrect-1

The value of {tex} x {/tex} for which {tex} \sin \left( \cot ^ { - 1 } ( 1 + x ) \right) = \cos \left( \tan ^ { - 1 } x \right) {/tex} is

A

{tex} 1 / 2 {/tex}

B

{tex}1{/tex}

C

{tex}0{/tex}

{tex} - 1 / 2 {/tex}

Explanation

Q 25.

Correct4

Incorrect-1

If {tex} 0 < x < 1 , {/tex} then
{tex} \sqrt { 1 + x ^ { 2 } } \left[ \left\{ x \cos \left( \cot ^ { - 1 } x \right) + \sin \left( \cot ^ { - 1 } x \right) \right\} ^ { 2 } - 1 \right] ^ { 1 / 2 } = {/tex}

A

{tex} \frac { x } { \sqrt { 1 + x ^ { 2 } } } {/tex}

B

{tex} x {/tex}

{tex} x \sqrt { 1 + x ^ { 2 } } {/tex}

D

{tex} \sqrt { 1 + x ^ { 2 } } {/tex}

Explanation