Question 1 :
If
{tex} f ( x ) = \left\{ \begin{array} { l } { x , \text { when } x \text { is rational } } \\ { 0 , \text { when } x \text { is irrational } } \end{array} ;
g ( x ) = \left\{ \begin{array} { l } { 0 , \text { when } x \text { is rational } } \\ { x , \text { when } x \text { is irrational } } \end{array} \right. \right. {/tex}<br>then {tex} ( f - g ) {/tex} is<br>
Question 2 :
Let {tex} f . R \rightarrow R {/tex} and {tex} g : R \rightarrow R {/tex} be two one-to-one and onto func- tions such that they are the mirror images of each other about<br>the line {tex} y = a . {/tex} If {tex} h ( x ) = f ( x ) + g ( x ) , {/tex} then {tex} h ( x ) {/tex} is<br>
Question 3 :
The domain of {tex} f ( x ) = \left( x ^ { 2 } - 1 \right) ^ { - 1 / 2 } {/tex} is
Question 5 :
If {tex} f : R \rightarrow R , {/tex} where {tex} f ( x ) = a x + \cos x . {/tex} If {tex} f ( x ) {/tex} is bijective, then
Question 6 :
Let {tex} f ( x ) = \frac { 3 } { x - 2 } + \frac { 4 } { x - 3 } + \frac { 5 } { x - 4 } . {/tex} Then {tex} f ( x ) = 0 {/tex} has
Question 7 :
If {tex} f ( x ) = x / \sqrt { 1 + x ^ { 2 } } , {/tex} then {tex} f {/tex}{tex} ( x ) {/tex} equals to
Question 8 :
If A, B, C be three sets such that A ∪ B = A ∪ C and A ∩ B = A ∩ C, then
Question 9 :
Let R be an equivalence relation on a finite set A having n elements. Then the number of ordered pairs in R is
Question 10 :
A relation on the set {tex} A = \{ x : | x | < 3 , \in x Z \} , {/tex} where {tex} Z {/tex} is the set of integers is defined by {tex} R = \{ ( x , y ) : y = | x | , x \neq - 1 \} . {/tex} Then the number of elements in the power set of {tex} R {/tex} is
Question 11 :
If $U = \left \{x|x\epsilon N, x < 5\right \}, A = \left \{x|x\epsilon N, x\leq 2\right \}$ then $A' =$ __________.
Question 12 :
The range of the function {tex} f ( x ) = ^{7 - x} P _ { x - 3 } {/tex} is
Question 15 :
{tex} f ( x , y ) = 1 / ( x + y ) {/tex} is a homogeneous function of degree
Question 16 : Let<img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e71c2e982d3c8134c8680fd' height='18' width='285' > be a relation on the set <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e71c2ea083ed11396865e5b' height='18' width='83' >. The relation is
Question 18 :
The number of proper subsets of the set {1, 2, 3} is
Question 19 :
The period of the function {tex} f ( x ) = [ 5 x + 7 ] + \cos \pi x - 5 x , {/tex} where {tex} [ \cdot ] {/tex} denotes the greatest integer function, is
Question 20 :
Let $A$ and $B$ are two finite sets such that $n(A)=3$ and $n(B)=4$ then the number of elements in $A\Delta B$.
Question 21 :
If {tex} f ( x ) = \cos | x | + \left[ \left| \frac { \sin x } { 2 } \right| \right] {/tex} , (where [.] denotes the greatest integer function), then<br>
Question 23 :
If {tex} a , b \in R , {/tex} then the period of {tex} f ( x ) = x - [ x + a ] - b , {/tex} where {tex} [ \cdot ] {/tex} denotes the greatest integer function, is
Question 24 :
The smallest set A such that A ∪ {1, 2} = {1, 2, 3, 5, 9} is
Question 26 :
If {tex} f ( x ) = \ln \left( \frac { x ^ { 2 } + e } { x ^ { 2 } + 1 } \right) , {/tex} then the range of {tex} f ( x ) {/tex} is
Question 27 :
The number of non-empty subsets of the set {1, 2, 3, 4} is
Question 28 :
$AB=A$ and $BA=B$, then which of the following is not true?
Question 29 :
If {tex} 5 ^ { x } + ( 2 \sqrt { 3 } ) ^ { 2 x } \geq 13 ^ { x } , {/tex} then the solution set for {tex} x {/tex} is
Question 30 :
The values of b and c for which the identity {tex} f ( x + 1 ) - f ( x ) = 8 x + 3 {/tex} is satisfied, where {tex} f ( x ) = b x ^ { 2 } + c x + d {/tex} are<br>
Question 31 :
If {tex} f ( x ) = 3 x - 5 , {/tex} then {tex} f ^ { - 1 } ( x ) {/tex}
Question 32 : <img style='object-fit:contain' src='https://storage.googleapis.com/teachmint/question_assets/JEE%20Main/5e71c35482d3c8134c868160' height='89' width='132' ><br>The shaded region in the given figure is
Question 33 :
The domain of the function {tex} f ( x ) = \log _ { 3 + x } \left( x ^ { 2 } - 1 \right) {/tex} is
Question 34 :
The interval for which {tex} \sin ^ { - 1 } \sqrt { x } + \cos ^ { - 1 } \sqrt { x } = \frac { \pi } { 2 } {/tex} holds
Question 35 :
Let {tex} A = \{ a , b , c \} {/tex} and {tex} B = \{ 1,2 \} . {/tex} Consider a relation {tex} R {/tex} defined from set {tex} A {/tex} to set {tex} B {/tex} . Then {tex} R {/tex} is equal to set
Question 36 :
The period of the function {tex} f ( x ) = [ x ] + [ 2 x ] + [ 3 x ] + \cdots + [ n x ] - {/tex} {tex} \frac { n ( n + 1 ) } { 2 } x , {/tex} when {tex} x \in N {/tex} is
Question 37 :
A relation {tex} R {/tex} is defined from {tex} \{ 2,3,4,5 \} {/tex} to {tex} \{ 3,6,7,10 \} {/tex} by {tex} x R y \Leftrightarrow x {/tex} is relatively prime to {tex} y . {/tex} Then domain of {tex} R {/tex} is<br>
Question 38 :
The domain of the function {tex} f ( x ) = \frac { 1 } { \sqrt { | x | - x } } {/tex} is
Question 39 :
If A, B and C are any three sets, then A × (B∩C) is equal to
Question 40 :
The range of {tex} f ( x ) = \cos 2 x - \sin 2 x {/tex} contains the set
Question 41 :
If A and B are sets, then A ∩ (B - A) is
Question 42 :
The domain of the function {tex} f ( x ) = \sqrt { x ^ { 12 } - x ^ { 3 } + x ^ { 4 } - x + 1 } {/tex} is
Question 43 :
Let {tex} f: N \rightarrow Y {/tex} be a function defined as <br> {tex} f ( x ) = 4 x + 3 {/tex} where {tex} Y = \{ y \in N: y = 4 x + 3 \text { for some } x \in N \} {/tex}<br> Show that {tex} f {/tex} is invertible and its inverse is
Question 44 :
The domain of the function {tex} f ( x ) = \sqrt { 2 - 2 x - x ^ { 2 } } {/tex} is
Question 45 :
If A has 5 elements and B has 8 elements such that $\displaystyle A\subset B,$ then the number of elements in $\displaystyle A\cap B,$ and $\displaystyle A\cup B,$ are respectively :
Question 46 :
If {tex} f ( x ) = \left\{ \begin{array} { l l } { x , } & { \text { when } x \text { is rational } } \\ { 1 - x , } & { \text { when } x \text { is irrational } } \end{array}, \right. {/tex} then {tex} f\circ f{/tex} (x) is given as
Question 47 :
If A = {0, 1), and B = {1, 0}, then A × B is equal to
Question 48 :
If {tex} ( x , y ) \in R {/tex} and {tex} x , y \neq 0 ; f ( x , y ) \rightarrow ( x / y ) , {/tex} then this function is a/an
Question 49 :
The period of {tex} \sin ^ { 2 } \theta {/tex} is
Question 50 :
If {tex} f {/tex} is a function such that {tex} f ( 0 ) = 2 , f ( 1 ) = 3 {/tex} and {tex} f ( x + 2 ) = 2 f ( x ) -{/tex} {tex} f ( x + 1 ) \forall x \in R , {/tex} then {tex} f ( 5 ) {/tex} is
Question 52 :
If $X$ and $Y$ are two sets, $X\cap { \left( Y\cup X \right) }^{ C }$ is equal to
Question 53 :
The domain of {tex} f ( x ) = \frac { \log _ { 2 } ( x + 3 ) } { x ^ { 2 } + 3 x + 2 } {/tex} is
Question 54 :
The domain of {tex} f ( x ) = \sqrt { ( x - 1 ) / ( x - 2 \{ x \} ) } {/tex} , where {tex} \{ x \} {/tex} denotes the fractional part of {tex} x , {/tex} is
Question 55 :
If {tex} X {/tex} and {tex} Y {/tex} are two non-empty sets, where {tex} f : X \rightarrow Y {/tex} is the function, is defined such that {tex} f ( c ) = \{ f ( x ) : x \in C \} {/tex} for {tex} C \subseteq X {/tex} and {tex} f ^ { - 1 } ( D ) = \{ x : f ( x ) \in D \} {/tex} for {tex} D \subseteq Y {/tex} for any {tex} A \subseteq X {/tex} and {tex} B \subseteq Y {/tex} then<br>
Question 56 :
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
Question 57 :
If {tex} f : [ 0 , \infty ) \rightarrow [ 0 , \infty ) {/tex} and {tex} f ( x ) = \frac { x } { 1 + x } , {/tex} then {tex} f {/tex} is
Question 58 :
The number of {tex} x \in [ 0,2 \pi ] {/tex} for which {tex} | \sqrt { 2 \sin ^ { 4 } + 18 \cos ^ { 2 } x } - {/tex} {tex} \sqrt { 2 \cos ^ { 4 } + 18 \sin ^ { 2 } x } | = 1 {/tex} is
Question 59 :
The domain of the function {tex} f ( x ) = \frac { \sin ^ { - 1 } ( x - 3 ) } { \sqrt { 9 - x ^ { 2 } } } {/tex} is
Question 60 :
Let {tex} f ( x ) = \int _ { 0 } ^ { x } \log \frac { 1 - \tan t } { 1 + \tan t } d t . {/tex} Then {tex} f ( x ) {/tex} is
Question 61 :
Let {tex} S {/tex} be a non-empty subset of {tex} R {/tex} . Consider the following statement: {tex} P : {/tex} There is a rational number {tex} x \in S {/tex} such that {tex} x > 0 {/tex} . Which of the following statements is the negation of the statement {tex} P ? {/tex}
Question 62 :
If {tex} f ( x + y , x - y ) = x y , {/tex} then the arithmetic mean of {tex} f ( x , y ) {/tex} and {tex} f ( y , x ) {/tex} is
Question 63 :
If {tex} f ( x ) = \sqrt { 4 - x ^ { 2 } } + ( 1 / \sqrt { | \sin x | - \sin x } ) , {/tex} then the domain of {tex} f ( x ) {/tex}
Question 64 :
The range of the function {tex} f ( x ) = \sqrt { x ^ { 2 } + 4 x } \mathrm { C } _ { 2 x ^ { 2 } + 3 } {/tex} is
Question 65 :
The number of real roots of {tex} 3 ^ { x } + 4 ^ { x } + 5 ^ { x } - 6 ^ { x } = 0 {/tex} is/are
Question 66 :
Let {tex} A = \left\{ x _ { 1 } , x _ { 2 } , \ldots , x _ { 7 } \right\} {/tex} and {tex} B = \left\{ y _ { 1 } , y _ { 2 } , y _ { 3 } \right\} {/tex} be two sets containing seven and three distinct elements respectively. Then the total number of functions {tex} f : A \rightarrow B {/tex} that are onto, if there exists exactly three elements {tex} x {/tex} in {tex} A {/tex} such that {tex} f ( x ) = y _ { 2 } , {/tex} is equal to<br>
Question 67 :
Let {tex} f ( x ) = ( x + 1 ) ^ { 2 } - 1 , x \geq - 1 {/tex}<br>{tex} {tex}\mathrm{Statement:1}{/tex} The set{tex}{\ x: } f ( x ) = f ^ { - 1 } ( x )\} = \{ 0 , - 1 \} {/tex}<br> {tex}\mathrm{Statement:2}{/tex}: {tex} f {/tex} is a bijection.<br>
Question 68 :
If {tex} f ( x ) = 1 / [ | \sin x | + | \cos x | ] {/tex} (where [.] denotes the greatest integer function), then
Question 69 :
The largest value of {tex} r {/tex} for which the region represented by the set {tex} \{ \omega \in C : | \omega - 4 - i | \leq r \} {/tex} is contained in the region represented by the set {tex} \{ z \in C / | z - 1 | z + i | \} , {/tex} is equal to
Question 70 :
Let {tex} k {/tex} be a non-zero real number. If {tex} f ( x ) = \left\{ \begin{array} { l l } { {\frac{(e^x-1)^2}{(sin(\frac{x}{k})log(1+\frac{x}{4})}} ,} & { \text { x≠0 } } \\ { 12} ,& { \text {x=0} } \end{array} \right. {/tex} is a continuous function, then the value of k is
Question 71 :
Let {tex} f : R \rightarrow R {/tex} be defined by {tex} f ( x ) = \frac { | x | - 1 } { | x | + 1 } {/tex} then {tex} f {/tex} is
Question 72 :
Let {tex} P = \left\{ ( x , y ) | x ^ { 2 } + y ^ { 2 } = 1 , x , y \in R \right\} . {/tex} Then {tex} P {/tex} is
Question 73 :
If the function {tex} f : [ 1 , \infty ) \rightarrow [ 1 , \infty ) {/tex} is defined by {tex} f ( x ) = 2 ^ { x ( x - 1 ) } {/tex} then {tex} f ^ { - 1 } ( x ) {/tex} is
Question 74 :
Let {tex} f ( \theta ) = \sin \theta ( \sin \theta + \sin 3 \theta ) , {/tex} then
Question 75 :
The function {tex} f : R \rightarrow R {/tex} defined by {tex} f ( x ) = ( x - 1 ) ( x - 2 ) ( x - 3 ) {/tex} is
Question 76 :
Let {tex} P = \{ \theta : \sin \theta - \cos \theta = \sqrt { 2 } \cos \theta \} {/tex} and {tex} Q = \{ \theta : \sin \theta + \cos \theta = {/tex} {tex} \sqrt { 2 } \sin \theta \} {/tex} be two sets. Then
Question 77 :
Let {tex} g ( x ) = 1 + x - [ x ] {/tex} and {tex} f ( x ) = \left\{ \begin{array} { l } { - 1 , x < 0 } \\ { 0 , x = 0 } \\\end{array} \right. {/tex}. Then {tex} \forall x , f [ g ( x )]{/tex} is equal to
Question 78 :
The function {tex} f ( x ) = | \sin 4 x | + | \cos 2 x | , {/tex} is a periodic function with period
Question 79 :
Let {tex} R {/tex} be the real line. Consider the following subsets of the plane <br> {tex} R \times R: {/tex} {tex} S = \{ ( x , y ): y = x + 1 \text { and } 0 < x < 2 \} {/tex} <br> {tex} T = \{ ( x , y ): x - y \text { is an integer } \} {/tex}<br> Which one of the following is true?
Question 80 :
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?
Question 81 :
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
Question 82 :
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
Question 83 :
The domain of definition of the function
{tex} f ( x ) = \sqrt { 4 ^ { x } + ( 64 ) ^ { ( x - 2 ) / 3 } - \left[ ( 1 / 2 ) \left( 72 + 2 ^ { 2 x } \right) \right] } {/tex} is<br>
Question 84 :
If {tex} f ( x ) = \cos \left[ \pi ^ { 2 } \right] x + \cos \left[ - \pi ^ { 2 } \right] x , {/tex} then
Question 85 :
Let {tex} f : ( - 1,1 ) \rightarrow R {/tex} be a continuous function. If {tex} \int _ { 0 } ^ { \sin x } f ( t ) d t = \frac { \sqrt { 3 } } { 2 } x {/tex} then {tex} f \left( \frac { \sqrt { 3 } } { 2 } \right) {/tex} is equal to<br>
Question 86 :
If {tex} f ( x ) {/tex} is defined on domain {tex} [ 0,1 ] , {/tex} then {tex} f ( 2 \sin x ) {/tex} is defined on
Question 87 :
Letf {tex} ( x ) = \sin x + \cos x , g ( x ) = x ^ { 2 } - 1 . {/tex} Thus, {tex} g [ f ( x ) ] {/tex} is invertible for {tex} x \in R {/tex}
Question 88 :
If {tex} f ( x ) {/tex} and {tex} g ( x ) {/tex} be two given functions with all real numbers as their domain, then {tex} h ( x ) = [ f ( x ) + f ( - x ) ] [ g ( x ) - g ( - x ) ] {/tex} is
Question 89 :
In a town of 10,000 families it was found that 40% family buy newspaper A, 20% buy newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers, then number of families which buy A only is
Question 90 :
Let R = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set A = {1, 2, 3, 4}. The relation R is
Question 91 :
The domain of the derivative of the function<br>{tex} f ( x ) = \left\{ \begin{array} { l l } { \tan ^ { - 1 } x , } & { | x | \leq 1 } \\ { \frac { 1 } { 2 } ( | x | - 1 ) , } & { | x | > 1 } \end{array} \right. {/tex}<br>
Question 92 :
In a class of 55 students, the number of students studying different subjects are 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. The number of students who have taken exactly one subject is
Question 93 :
The number of solution(s) of the equation {tex} x ^ { 2 } - 2 - 2 [ x ] = 0 {/tex} ([.] denotes the greatest integer function) is (are)
Question 94 :
If the function {tex} g ( x ) = \left\{ \begin{array} { l l } { k \sqrt { x + 1 } , } & { 0 \leq x \leq 3 } \\ { m x + 2 , } & { 3 < x \leq 5 } \end{array} \text { } \right. {/tex} is differentiable, then the value of {tex} k + m {/tex} is<br>
Question 95 :
Let {tex} A \equiv \{ 1,2,3,4 \} , B = \{ a , b , c \} , {/tex} then the number of function from {tex} A \rightarrow B , {/tex} which are not onto is
Question 96 :
A function {tex} f ( x ) {/tex} is defined for all real {tex} x {/tex} and satisfied {tex} f ( x + y ) = {/tex} {tex} f ( x y ) \forall x , y . {/tex} If {tex} f ( 1 ) = - 1 , {/tex} then {tex} f ( 2006 ) {/tex} equals
Question 97 :
The function {tex} f ( x ) = \left( x ^ { 2 } + 2 x + c \right) / \left( x ^ { 2 } + 4 x + 3 c \right) {/tex} has the range {tex} ( - \infty , \infty ) {/tex} for the allowed values of {tex} x \in R {/tex} if<br>
Question 98 :
If A, B and C are non-empty sets, then (A - B) ∪ (B - A) equals
Question 99 :
For {tex} x \in R , x \neq 0 , x \neq 1 , {/tex} let {tex} f _ { 0 } ( x ) = \frac { 1 } { 1 - x } {/tex} and {tex} f _ { n + 1 } ( x ) = f _ { 0 } \left( f _ { n } ( x ) \right) {/tex} {tex} n = 0,1,2 , \ldots . {/tex} Then, the value of {tex} f _ { 100 } ( 3 ) + f _ { 1 } \left( \frac { 2 } { 3 } \right) + f _ { 2 } \left( \frac { 3 } { 2 } \right) {/tex} is equal to
Question 100 :
Let {tex} f : R \rightarrow R {/tex} be a function such that {tex} f ( 2 - x ) = f ( 2 + x ) {/tex} and {tex} f ( 4 - x ) = f ( 4 + x ) , {/tex} for all {tex} x \in R {/tex} and {tex} \int \limits_ { 0 } ^ { 2 } f ( x ) d x = 5 . {/tex} Then the value of {tex} \int \limits_ { 10 } ^ { 50 } f ( x ) d x {/tex} is<br>
