Question 1 :
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 2 :
Let the function {tex} f : R \rightarrow R {/tex} be defined by {tex} f ( x ) = 2 x + \sin x , x \in R {/tex} . Then {tex} f {/tex} is
Question 3 :
The domain of {tex} \sin ^ { - 1 } \left[ \log _ { 3 } \left( \frac { x } { 3 } \right) \right] {/tex} is
Question 4 :
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 5 :
{tex} f ( x ) = x + \sqrt { x ^ { 2 } } {/tex} is a function from {tex} R \rightarrow R , {/tex} then {tex} f ( x ) {/tex} is
Question 6 :
Let $n(u)=700,n(A)=200,n(B)=300$<br>$n\left( A\cap B \right) =100,n\left( A^{\prime} \cap B^{\prime} \right) =$
Question 7 :
Let {tex} x {/tex} be a non-zero rational number and {tex} y {/tex} be an irrational number. Then {tex} x y {/tex} is
Question 8 :
For {tex} \theta > \frac { \pi } { 3 } , {/tex} the value of {tex} f ( \theta ) = \sec ^ { 2 } \theta + \cos ^ { 2 } \theta {/tex} always lies in theninterval<br>
Question 9 :
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 10 :
The relation R= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on set A = {1, 2, 3} is
Question 11 :
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 12 :
Let {tex} X {/tex} be a family of sets and {tex} R {/tex} be a relation on {tex} X {/tex} defined by {tex} A^ { \prime } {/tex} is disjoint from {tex} B ^ { \prime } {/tex} . Then {tex} R {/tex} is
Question 15 :
If A, B and C are any three sets, then A × (B∪C) is equal to
Question 16 :
The number of proper subsets of the set {1, 2, 3} is
Question 17 :
The range of {tex} f ( x ) = \cos 2 x - \sin 2 x {/tex} contains the set
Question 18 :
If {tex} y = f ( x ) = ( x + 2 ) / ( x - 1 ) , {/tex} then {tex} x {/tex} is
Question 19 :
The interval for which {tex} \sin ^ { - 1 } \sqrt { x } + \cos ^ { - 1 } \sqrt { x } = \frac { \pi } { 2 } {/tex} holds
Question 20 :
If {tex} f ( x ) = \left( a - x ^ { n } \right) ^ { v n } {/tex} where {tex} a > 0 {/tex} and {tex} n {/tex} is a positive integer, then {tex} f [ f ( x ) ] {/tex}=
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 22 :
The domain of {tex} f ( x ) = \left( x ^ { 2 } - 1 \right) ^ { - 1 / 2 } {/tex} is
Question 24 :
The smallest set A such that A ∪ {1, 2} = {1, 2, 3, 5, 9} is
Question 26 :
A relation from {tex} P {/tex} to {tex} Q {/tex} is
Question 28 :
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 29 :
If A, B and C are any three sets, then A × (B∩C) is equal to
Question 31 :
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 32 :
The function {tex} f : R \rightarrow R {/tex} defined by {tex} f ( x ) = ( x - 1 ) ( x - 2 ) ( x - 3 ) {/tex} is
Question 33 :
If {tex} f ( x ) {/tex} is defined on domain {tex} [ 0,1 ] , {/tex} then {tex} f ( 2 \sin x ) {/tex} is defined on
Question 34 :
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 35 :
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 36 :
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 37 :
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 38 :
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 39 :
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 40 :
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 41 :
Let {tex} f ( \theta ) = \sin \theta ( \sin \theta + \sin 3 \theta ) , {/tex} then
Question 42 :
The function {tex} f ( x ) = | \sin 4 x | + | \cos 2 x | , {/tex} is a periodic function with period
Question 43 :
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 44 :
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 45 :
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 46 :
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 47 :
If A, B and C are non-empty sets, then (A - B) ∪ (B - A) equals
Question 48 :
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 49 :
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 50 :
If {tex} a \in R {/tex} and the equation, {tex} - 3 ( x - [ x ] ) ^ { 2 } + 2 ( x - [ x ] ) + a ^ { 2 } = 0 {/tex} (where {tex} [ x ] {/tex} denotes the greatest integer {tex} \leq x {/tex} ) has no integral solution, then all possible values of {tex} a {/tex} lie in the interval<br>
