Question 1 :
For real number $x$ and $y$, define $xRy$ iff $x-y+\sqrt{2}$ is an irrational number. Then the relation $R$ is
Question 2 :
$N$ is the set of positive integers. The relation $R$ is defined on N x N as follows: $(a,b) R (c,d)\Longleftrightarrow ad=bc$ Prove that
Question 3 :
Assertion: Domain of $f(x)$ is singleton.
Reason: Range of $f(x)$ is singleton.
Question 4 :
Let $A=\left\{ 2,3,4,5,....,17,18 \right\} $. Let $\simeq $ be the equivalence relation on $A\times A$, cartesian product of $A$ with itself, defined by $(a,b)\simeq (c,d)$, iff $ad=bc$. The the number of ordered pairs of the equivalence class of $(3,2)$ is
Question 5 :
Let $A=\left\{ 1,2,3 \right\} $ and $R=\left\{ \left( 1,2 \right) ,\left( 2,3 \right) ,\left( 1,3 \right) \right\} $ be a relation on $A$. Then $R$ is
Question 6 :
Let $f:R\rightarrow R$ be defined as $f(x)=x^{3}+2x^{2}+4x+\sin \left(\dfrac{\pi}{2}\right)$ and $g(x)$ be the inverse function of $f(x)$, then $g'(8)$ is equal to
Question 7 :
The number of reflexive relation in set A = {a, b, c} is equal to
Question 9 :
If $A=\left\{ 1,2,3 \right\} $, then a relation $R=\left\{ \left( 2,3 \right) \right\} $ on $A$ is
Question 10 :
Let $A=\left\{ 1,2,3 \right\} $. Then, the number of equivalence relations containing $(1,2)$ over set A is
Question 11 :
If the relation is defined on $R-\left\{ 0 \right\} $ by $\left( x,y \right) \in S\Leftrightarrow xy>0$, then $S$ is ________
Question 12 :
If $A=\left\{ a,b,c \right\} $, then the relation $R=\left\{ \left( b,c \right) \right\} $ on $A$ is
Question 13 :
If $f: A \rightarrow B$is a bijective function and if n(A) = 5, then n(B) is equal to
Question 14 :
The relation $R$ in $N\times N$ such that $(a,b)R(c,d)\Leftrightarrow a+d=b+c$ is
Question 15 :
The relation $R=\left\{ \left( 1,1 \right) ,\left( 2,2 \right) \left( 3,3 \right)  \right\} $ on the set $A=\left\{ 1,2,3 \right\} $ is
Question 16 :
If $R$ is the largest equivalence relation on a set $A$ and $S$ is any relation on $A$, then
Question 17 :
Let A={ 1, 2, 3, 4} and R= {( 2, 2), (3, 3), (4, 4), (1, 2)} be a relation on A. Then R is
Question 18 :
$\displaystyle x^{2} = xy$ is a relation (defined on set R) which is<br/> <br/>
Question 19 :
The true set of real value of $x$ for which the function, $f(x)=x\ \mathrm{ln}\ x-x+1$ is positive is
Question 20 :
Find number of all such functions $y = f(x)$ which are one-one?
Question 22 :
The relation $R$ on the set $Z$ of all integer numbers defined by $(x,y)\ \epsilon \ R\\Leftrightarrow x-y$ is divisible by $n$ is
Question 23 :
Let $f: N\rightarrow R$ such that $f(x)=\dfrac{2x-1}{2}$ and $g: Q\rightarrow R$such that $g(x)=x+2$ be two function. Then $(gof)\left(\dfrac{3}{2}\right)$ is equal to
Question 24 :
Given the relation R= {(1,2), (2,3) } on the set {1, 2, 3}, the minimum number of ordered pairs which when added to R make it an equivalence relation is
Question 25 :
Which of the following is not an equivalence relation on $Z$?
Question 26 :
If $A=\left\{ a,b,c,d \right\} $, then a relation $R=\left\{ \left( a,b \right) ,\left( b,a \right) ,\left( a,a \right) \right\} $ on $A$ is
Question 28 :
Let $f(x,y)=xy^{2}$ if $x$ and $y$ satisfy $x^{2}+y^{2}=9$ then the minimum value of $f(x,y)$ is
Question 29 :
Let $L$ denote the set of all straight lines in a plane, Let a relation $R$ be defined by $lRm$, iff $l$ is perpendicular to $m$ for all $l \in L$. Then, $R$ is
Question 30 :
Which one of the following relations on R (set of real numbers) is an equivalence relation
Question 31 :
Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $aRb$, if $a$ is congruent to $b$ for all $a,b\in T$. Then, $R$ is
Question 32 :
Let E = {1, 2, 3, 4} and F {1, 2}. Then the number of onto functions from E to F is
Question 33 :
Let $R$ be the relation over the set of all straight lines in a plane such that ${l}_{1}$ $R$ ${l}_{2}\Leftrightarrow {l}_{1}\bot {l}_{2}$. Then, $R$ is
Question 34 :
If $f:\mathbb{N} \rightarrow \mathbb{N}$ and $f(x) = x^{2}$ then the function is<br/>
Question 36 :
Consider the following two binary relations on the set $A = \left \{a, b, c\right \} : R_{1} = \left \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\right \}$<br>and $R_{2} = \left \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\right \}$ Then<br>
Question 37 :
The number of reflexive relations of a set with four elements is equal to
Question 38 :
Let N denote the set of all natural numbers and R a relation on $N\times N$. Which of the following is an equivalence relation?
Question 39 :
On the set $N$ of all natural numbers define the relation $R$ by $a R b$ if and only if the G.C.D. of $a$ and $b$ is $2$. Then $R$ is:
Question 40 :
Let $A = \left\{ {1,2,3} \right\}$ and $R = \left\{ {\left( {1,1} \right),\left( {1,3} \right),\left( {3,1} \right),\left( {2,2} \right),\left( {2,1} \right),\left( {3,3} \right)} \right\}$, then the relation $R$ and $A$ is
Question 42 :
In the set $Z$ of all integers, which of the following relation $R$ is an equivalence relation?
Question 43 :
Let $A = [-1,1]= B $then which of the following functions from $A$ to $B$ is bijective function?
Question 44 :
Let $f:R\rightarrow R$ be defined by $f(x)=2x+6$ which is a bijective mapping then ${ f }^{ -1 }(x)\quad $ is given by
Question 45 :
If $f:[2, \infty)\rightarrow B$ defined by $f(x)=x^2-4x+5$ is a bijection, then$B=$
Question 46 :
A relation $\rho$ on the set of real number $R$ is defined as follows:<br>$x\rho y$ if any only if $xy > 0$. Then which of the following is/are true?
Question 47 :
Let X be the set of all citizens of India. Elements x, y in X are said to be related if the difference of their age is 5 years. Which one of the following is correct ?
Question 48 :
The minimum number of elements that must be added to the relation $R=\{(1,2,),(2,3)\} $ on the set of natural numbers so that it is an equivalence is
Question 49 :
Let f :$R \to R$ and g :$R \to R$ be two one-one and onto functions such that they are the mirror images of each other about the line y = 2. If h(x) = f(x) + g(x), then h(0) equal to
Question 51 :
Assertion: Let $f\, :\, R\, \rightarrow\, R$, $f(x)\, =\, x^{3}\, +\, x^{2}\, +\, 100x\, +\, 5\sin x$, then $f(x)$ is bijective.
Reason: $3x^{2}\, +\, 2x\, +\, 95\, >\, 0\, \, x\, \in\, R$.
Question 52 :
Write the properties that the relation "is greter that" satisfies in the set of all positive integers
Question 54 :
The minimum number of elements that must be added to the relation $R=\left \{ (1, 2),(2, 3) \right \}$ on the set of natural numbers, so that it is an equivalence is:
Question 55 :
 $f : R \rightarrow R , f ( x ) = e ^ { | x | } - e ^ { - x }$  is many-one into function.
Question 56 :
If $A =\{1, 2, 3\}$ and $ B = \{4, 5\}$ then the number of function $f : A \rightarrow B$ which is not onto is ______
Question 57 :
Let $A=\left\{ x\in R:-1\le x\le 1 \right\} =B$, then the mapping $f:A\rightarrow B$ given by $f(x)=x\left| x \right| $ is
Question 58 :
If $f:R\rightarrow R$ defined by $f(x)=x, $ if $ x>2; f(x)=5x-2 $ if $ x\leq 2$ then $f$ is
Question 59 :
Let $R$ be a relation defined as $aRb$ if $1 + ab > 0$, then the relation $R$ is:
Question 60 :
The function $f:A\rightarrow B$ given by $f(x),x\in A$, is one to one but not onto. Then;
Question 61 :
If the real-valued function $f(x)=px+\sin x$ is a bijective function, then the set of all possible values of $p\epsilon R$ is?
Question 62 :
If the function $f:R\rightarrow A$ given by $f(x)=\cfrac { { x }^{ 2 } }{ { x }^{ 2 }+1 } $ is surjection, then $A=$
Question 63 :
Let $\displaystyle f:R\rightarrow A=\left \{ y: 0\leq y< \dfrac{\pi}{2} \right \}$ be a function such that $\displaystyle f(x)=\tan^{-1}(x^{2}+x+k),$ where $k$ is a constant. The value of $k$ for which $f$ is an onto function is
Question 66 :
The Set $A$ has $4$ elements and the Set $B$ has $5$ elements then the number of injective mappings that can be defined from $A$ to $B$ is
Question 67 :
Let S, T, U be three non-void sets and $f:S\rightarrow T, g:T\rightarrow U$ be so that g o f$:S\rightarrow$ U is surjective. Then?
Question 68 :
The function $f$ defined by $f\left( x \right) = {x^3} - 3{x^2} + 5x + 7,$ is:
Question 69 :
If $f(x)=\left | x-1 \right |+\left | x-2 \right |+\left | x-3 \right |$ when $2<x<3$ is
Question 70 :
If $f\left( x+y+1 \right) =\left( \sqrt { f\left( x \right)  }  \right)^2 +\left( \sqrt { f\left( y \right)  }  \right) ^{ 2 }\forall x,y\epsilon N$ and $f\left( 0 \right) =1,$ then $f\left( +3 \right) =$ 
Question 71 :
Let $E=\{1, 2, 3, 4\}$ and $F=\{1, 2\}$ then the number of onto functions from E to F is
Question 72 :
If A ={1, 3, 5, 7} and B = {1, 2, 3, 4, 5, 6, 7, 8}, then the number of one-to-one functions from A into B is
Question 74 :
Let $L$ be the set of all straight lines in the Euclidean plane. Two lines ${l}_{1}$ and ${l}_{2}$ are said to be related by the relation $R$ if ${l}_{1}$ is parallel to ${l}_{2}$. Then the relation $R$ is-
Question 75 :
$f:\left ( -\dfrac{\pi }{2},\dfrac{\pi }{2} \right )\rightarrow \left ( -\infty ,\infty \right )$defined by $f(x)=1+3x$ is
Question 76 :
The distinct linear functions which maps from $[-1,1]$ onto $[0,2]$ are
Question 77 :
The function $f: R\rightarrow R$ given by $f(x) = x^{3} - 1$ is
Question 79 :
Let $R$ be a reflexive on a finite set $A$ having $n$ elements, and let there be $m$ ordered pairs in $R$. Then
Question 80 :
If $f:R\rightarrow C$ is defined by $f(x)=e^{2ix}$ for $x\in R$ then, $f$ is (where $C$ denotes the set of all Complex numbers)
Question 81 :
The relation 'has the same father as' over the set of children is:
Question 82 :
Assertion: Let $f\left ( x \right )=x^{3}+ax^{2}+bx+5\sin ^{2}x$, then the condition that $f(x)$ is always one-one function is $a^{2}-3b+15< 0$
Reason: $f(x)$ to be one one either $f$ is entirely increasing or entirely decreasing
Question 84 :
The function $f:(0,\infty )\rightarrow R$ given by $ f(x)=\cfrac { x }{ x+1 } $ is<br>
Question 85 :
Let $f : R \rightarrow R$ be defined by $f(x) = x^4$ then
Question 86 :
The set onto which the derivative of the function $f(x)=x(\log x-1)$ maps the range $[1,\infty )$ is
Question 87 :
The relation $"$iscongruent to$"$ on the set of all triangles in a plane is
Question 88 :
Let $f: N \rightarrow N$ be defined by $f(x) = x^2 + x + 1, x \in N$. Then $f$ is
Question 89 :
Let the function $f:R-\left\{ -b \right\} \rightarrow R-\left\{ 1 \right\} $ be defined by $f(x)=\cfrac { x+a }{ x+b } ,a\neq b$, then
Question 90 :
Let $A = \left\{p,q,r\right\}$. Which of the following is an equivalence relation in $A$?
Question 92 :
Let$\displaystyle f:R \rightarrow R, g(x) = f(x) + 3x - 1$, then the least value of function$\displaystyle y = g(|x|)$ is
Question 93 :
Let $f:N\rightarrow N$ ($N$ being the set of positive integers) be a function defined by $f(x)=$ the biggest positive integer obtained by reshuffling the digits of $x$. For example, $f(296)=962$<br>$f$ is
Question 94 :
Let $f:\{x, y , z\} \rightarrow \{1, 2, 3\}$ be a one-one mapping such that only one of the following three statements and remaining two are false : $f(x) \neq 2, f(y) =2, f(z) \neq 1$, then
Question 95 :
Consider the functions<br>$\displaystyle f: X\rightarrow Y$ and$\displaystyle g: Y\rightarrow Z$<br>then which of the following is/are incorrect?
Question 96 :
Which of the following functions from $Z$ to itself are bijections?
Question 97 :
If $f:R\rightarrow S$ defined by<br/>$f(x)=4\sin { x } -3\cos { x } +1$ is onto, then $S$ is equal to
Question 98 :
The function $f:\left[ -\dfrac {1}{2},\dfrac {1}{2} \right] \rightarrow \left[ -\dfrac {\pi }{2},\dfrac {\pi }{2} \right] $ defined by $f(x)=\sin ^{ -1 }{ \left( 3x-4{ x }^{ 3 } \right)  } $ is
Question 99 :
Let $\displaystyle f\left ( x \right )=\frac{ax^{2}+2x+1}{2x^{2}-2x+1}$, the value of $a$ for which $\displaystyle f:R\rightarrow \left [ -1,2 \right ]$ is onto , is<br>
Question 100 :
$f\left( x \right) =\begin{cases} x\left( \dfrac { { ae }^{ \dfrac { 1 }{ \left| x \right| } }+{ 3.e }^{ \dfrac { -1 }{ x } } }{ \left( a+2 \right) { e }^{ \dfrac { 1 }{ \left| x \right| } }-{ e }^{ \dfrac { -1 }{ x } } } \right) \\ 0 \end{cases},\begin{matrix} x\neq 0 \\ x=0 \end{matrix}$ is differentiable at $x=0$ then $[a]=$__ ([] denotes greatest integers function )
Question 101 :
Let $R$ a relation on the set $N$ be defined by $\left\{ \left( x,y \right) |x,y\in N,2x+y=41 \right\}$. Then $R$ is
Question 103 :
Let $f:{x, y, z}\rightarrow (a, b, c)$ be a one-one function. It is known that only one of the following statements is true:(i) $f(x)\neq b$<br/>(ii)$f(y)=b$<br/>(iii)$f(z)\neq  a$
Question 106 :
Let $f : R\rightarrow R$ is defined by $f(x)=\dfrac {|x|-1}{|x|+1}$ then $f$ is :
Question 107 :
If $f:R\rightarrow \left [\dfrac {\pi}{6}, \dfrac {\pi}{2}\right ), f(x)=\sin^{-1}\left (\dfrac {x^2-a}{x^2+1}\right )$ is a onto function, then set of values of $a$ is
Question 108 :
Let f: $X\rightarrow Y$ be a function defined by $f(x)=a \sin \left (x+\dfrac {\pi}{4}\right )+b \cos x+c$. If f is both one-one and onto, then find the sets $X$ and $Y$
Question 109 :
In the following functions defined from $[-1, 1]$ to $[-1, 1]$, then functions which are not bijective are
Question 111 :
The function $f: [0, 3]$ $\rightarrow$ $[1, 29]$, defined by $f(x) = 2x^3-15x^2 + 36x+ 1$, is<br>
Question 112 :
Let $f:R\rightarrow R$ be a function defined by $f(x)=\cfrac { { e }^{ \left| x \right|  }-{ e }^{ -x } }{ { e }^{ x }+{ e }^{ -x } } $, then
Question 113 :
If $n \geq 2$ then the number of surjections that can be defined from $\{1, 2, 3, .......  n\}$ onto $\{1, 2\}$ is<br/>
Question 114 :
If $f:A\rightarrow B$ given by ${ 3 }^{ f(x) }+{ 2 }^{ -x }=4$ is a bijection, then
