Let $\mathrm{R}=\{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of elements, needed to be added in R so that R becomes an equivalence relation, is:

Let $\mathrm{X}=\mathbf{R}\times \mathbf{R}$. Define a relation R on X as : $$\left(a_1,b_1\right)R\left(a_2,b_2\right)\iff b_1=b_2$$ Statement I: $\mathrm{R}$ is an equivalence relation. Statement II : For some $(\mathrm{a},\mathrm{b})\in \mathrm{X}$, the $\mathrm{set}\mathrm{S}=\{(x,y)\in \mathrm{X}:(x,y)\mathrm{R}(\mathrm{a},\mathrm{b})\}$ represents a line parallel to $y=x$. In the light of the above statements, choose the correct answer from the options given below :

Let $\mathrm{A}=\{(x,y)\in \mathbf{R}\times \mathbf{R}:|x+y|⩾3\}$ and $\mathrm{B}=\{(x,y)\in \mathbf{R}\times \mathbf{R}:|x|+|y|\le 3\}$. If $\mathrm{C}=\{(x,y)\in \mathrm{A}\cap \mathrm{B}:x=0$ or $y=0\}$, then ${\sum }_{(x,y)\in \mathrm{C}}|x+y|$ is :

Let $\mathrm{A}=\left\{x\in (0,\pi )-\left\{\frac{\pi }{2}\right\}:{\log }_{(2/\pi )}|\sin x|+{\log }_{(2/\pi )}|\cos x|=2\right\}$ and $\mathrm{B}=\{x⩾0:\sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\}$. Then $\mathrm{n}(\mathrm{A}\cup \mathrm{B})$ is equal to :

The relation $R=\{(x,y):x,y\in \mathbb{Z}$ and $x+y$ is even $\}$ is:

Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. Then among the statements (S1): The number of elements in R is 18, and (S2): The relation R is symmetric but neither reflexive nor transitive

Let A be the set of all functions $f:\mathbf{Z}\to \mathbf{Z}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{f},\mathrm{g}):f(0)=\mathrm{g}(1)$ and $f(1)=\mathrm{g}(0)\}$. Then R is :

Let A = { ($\alpha ,\beta$) $\in \mathbb{R}\times \mathbb{R}$ : |$\alpha$ - 1| $\le 4$ and |$\beta$ - 5| $\le 6$ } and B = { ($\alpha ,\beta$) $\in \mathbb{R}\times \mathbb{R}$ : 16($\alpha$ - $2{)}^2$+ 9($\beta$ - $6{)}^2$ $\le 144$ }. Then

Let $A=\{1,2,3,\dots .,100\}$ and $R$ be a relation on $A$ such that $R=\{(a,b):a=2b+1\}$. Let $(a_1$, $a_2),\left(a_2,a_3\right),\left(a_3,a_4\right),\dots .,\left(a_k,a_{k+1}\right)$ be a sequence of $k$ elements of $R$ such that the second entry of an ordered pair is equal to the first entry of the next ordered pair. Then the largest integer k , for which such a sequence exists, is equal to :

Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$. Let R be a relation on A defined by $x\mathrm{R}y$ if and only if $0\le x^2+2y\le 4$. Let $l$ be the number of elements in R and $m$ be the minimum number of elements required to be added in R to make it a reflexive relation. Then $l+m$ is equal to

Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$ and R be a relation on A defined by $x\mathrm{R}y$ if and only if $2x-y\in \{0,1\}$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l+\mathrm{m}+\mathrm{n}$ is equal to:

Consider the sets $A=\left\{(x,y)\in \mathbb{R}\times \mathbb{R}:x^2+y^2=25\right\},B=\left\{(x,y)\in \mathbb{R}\times \mathbb{R}:x^2+9y^2=144\right\}$, $C=\left\{(x,y)\in \mathbb{Z}\times \mathbb{Z}:x^2+y^2\le 4\right\}$ and $D=A\cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:

Let $A=\{-2,-1,0,1,2,3\}$. Let R be a relation on $A$ defined by $x\mathrm{R}y$ if and only if $y=\max \{x,1\}$. Let $l$ be the number of elements in R . Let $m$ and $n$ be the minimum number of elements required to be added in R to make it reflexive and symmetric relations, respectively. Then $l+m+n$ is equal to

The value of $\underset{n\to \infty }{\lim }\left(\sum _{k=1}^n\frac{k^3+6k^2+11k+5}{(k+3)!}\right)$ is :

Let the function $f(x)=\left(x^2-1\right)\left|x^2-ax+2\right|+\cos |x|$ be not differentiable at the two points $x=\alpha =2$ and $x=\beta$. Then the distance of the point $(\alpha ,\beta )$ from the line $12 x+5 y+10=0$ is equal to :

If \sum_\limits{r=1}^n T_r=\frac{(2 n-1)(2 n+1)(2 n+3)(2 n+5)}{64}, then \lim _\limits{n \rightarrow \infty} \sum_\limits{r=1}^n\left(\frac{1}{T_r}\right) is equal to :

If \lim _\limits{x \rightarrow \infty}\left(\left(\frac{\mathrm{e}}{1-\mathrm{e}}\right)\left(\frac{1}{\mathrm{e}}-\frac{x}{1+x}\right)\right)^x=\alpha, then the value of $\frac{{\log }_{\mathrm{e}}\alpha }{1+{\log }_{\mathrm{e}}\alpha }$ equals :

If the function $$f(x)=\{\begin{array}{l}\frac{2}{x}\left\{\sin \left(k_1+1\right)x+\sin \left(k_2-1\right)x\right\},x0\end{array}$$ is continuous at $x=0$, then $k_1^2+k_2^2$ is equal to :

$\underset{x\to \infty }{\lim }\frac{\left(2x^2-3x+5\right)(3x-1{)}^{\frac{x}{2}}}{\left(3x^2+5x+4\right)\sqrt{(3x+2{)}^x}}$ is equal to :

\lim _\limits{x \rightarrow 0} \operatorname{cosec} x\left(\sqrt{2 \cos ^2 x+3 \cos x}-\sqrt{\cos ^2 x+\sin x+4}\right) is: