Question 1 :
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 3 :
The true set of real value of $$x$$ for which the function, $$f(x)=x\ \mathrm{ln}\ x-x+1$$ is positive is
Question 4 :
$$N$$ 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 5 :
The relation $$R$$ in $$N\times N$$ such that $$(a,b)R(c,d)\Leftrightarrow a+d=b+c$$ is
Question 6 :
The number of reflexive relations of a set with four elements is equal to
Question 7 :
If $$A=\left\{ 1,2,3 \right\} $$, then a relation $$R=\left\{ \left( 2,3 \right) \right\} $$ on $$A$$ is
Question 8 :
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 9 :
Which one of the following relations on R (set of real numbers) is an equivalence relation
Question 10 :
If $$A=\left\{ a,b,c \right\} $$, then the relation $$R=\left\{ \left( b,c \right) \right\} $$ on $$A$$ is
Question 11 :
A function $$f$$ from the set of natural numbers to the set of integers defined by<br/>$$f(n)=\begin{cases} \cfrac { n-1 }{ 2 } ,\quad \text{when n is odd} \\ -\cfrac { n }{ 2 } ,\quad \text{when n is even} \end{cases}$$
Question 12 :
Which one of the following relations on $$R$$ is equivalence redlation-
Question 13 :
Let $$A=\{1,2,3\}, B =\{a, b, c\}$$ and If $$f=\{(1,a),(2,b),(3,c)\}, g=\{(1,b),(2,a),(3,b)\}, h=\{(1,b)(2,c),(3,a)\}$$ then
Question 14 :
If the function $$f : R \rightarrow R$$ is defined by $$f(x) = (x^2+1)^{35} \forall \in R$$, then $$f$$ is
Question 15 :
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 16 :
Let $$R$$ be a relation defined on the set $$Z$$ of all integers and $$xRy$$ when $$x + 2y$$ is divisible by $$3$$. Then
Question 17 :
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 18 :
Let $$S$$ be a relation on $$\mathbb{R}^{+}$$ defined by $$xSy\Leftrightarrow { x }^{ 2 }-{ y }^{ 2 }=2\left( y-x \right)$$, then $$S$$ is
Question 19 :
Let $$f ( x ) = \left\{ \begin{array} { c l } { ( x - 1 ) \sin \dfrac { 1 } { x - 1 } } & { \text { if } x \neq 1 } \\ { 0 } & { \text { if } x = 1 } \end{array} \right.$$Then which one of the following is true?
Question 21 :
Let$$\displaystyle f:R \rightarrow R, g(x) = f(x) + 3x - 1$$, then the least value of function$$\displaystyle y = g(|x|)$$ is
Question 22 :
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 23 :
If $$n \geq 2$$ then the number of surjections that can be defined from $$\{1, 2, 3, .......  n\}$$ onto $$\{1, 2\}$$ is<br/>
Question 24 :
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 26 :
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 27 :
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 28 :
$$f\left( x \right) =\begin{cases} x\left( \dfrac { { ae }^{ \dfrac { 1 }{ \left| x \right| } }+{ 3.e }^{ \dfrac { -1 }{ x } } }{ \left( a+2 \right) { e }^{ \dfrac { 1 }{ \left| x \right| } }-{ e }^{ \dfrac { -1 }{ x } } } \right) \\ 0 \end{cases},\begin{matrix} x\neq 0 \\ x=0 \end{matrix}$$ is differentiable at $$x=0$$ then $$[a]=$$__ ([] denotes greatest integers function )
Question 29 :
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 30 :
The function $$f: [0, 3]$$ $$\rightarrow$$ $$[1, 29]$$, defined by $$f(x) = 2x^3-15x^2 + 36x+ 1$$, is<br>
