Sets Relations and Functions — JEE Main PYQs

The number of relations, defined on the set $\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\}$, which are both reflexive and symmetric, is equal to:

Let $\mathrm{A}=\{2,3,5,7,9\}$. Let R be the relation on A defined by x Ry if and only if $2\mathrm{x}\le 3\mathrm{y}$. Let $\ell$ 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 symmetric relation. Then …

Let the relation R on the set $\mathrm{M}=\{1,2,3,\dots ,16\}$ be given by
$\mathrm{R}=\{(\mathrm{x},\mathrm{y}):4\mathrm{y}=5\mathrm{x}-3,\mathrm{x},\mathrm{y}\in \mathrm{M}\}$.
Then the minimum number of elements required to be added in R, in order to make the relation …

If the domain of the function $f(x)={\sin }^{-1}\left(\frac{5-x}{3+2x}\right)+\frac{1}{{\log }_e(10-x)}$ is $(-\infty ,\alpha ]\cup [\beta ,\gamma )-\{\delta \}$, then $6(\alpha +\beta +\gamma +\delta )$ is equal to

The number of elements in the relation $R=\{(x,y):4x^2+y^2<52,x,y\in \mathbb{Z}\}$ is

Let the domain of the function
$\mathrm{f}(\mathrm{x})={\log }_3{\log }_5\left(7-{\log }_2\left({\mathrm{x}}^2-10\mathrm{x}+85\right)\right)+{\sin }^{-1}\left(\left|\frac{3\mathrm{x}-7}{17-\mathrm{x}}\right|\right)$ be $(\alpha ,\beta ]$. Then $\alpha +\beta$ is equal to :

Let $f(x)=[x{]}^2-[x+3]-3,x\in \mathbb{R}$ where $[\bullet ]$ is the greatest integer function. Then

Let $f$ and $g$ be functions satisfying $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x})f(\mathrm{y}),\mathrm{f}(\mathrm{l})=7$ and $\mathrm{g}(\mathrm{x}+\mathrm{y})=\mathrm{g}(\mathrm{xy})$, $\mathrm{g}(\mathrm{l})=1$, for all …

Let $\mathrm{A}=\{-2,-1,0,1,2,3,4\}$. Let R be a relation on A defined by xRy if and only if $2\mathrm{x}+\mathrm{y}\le 2$. 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 …

Consider two sets $\mathrm{A}=\{\mathrm{x}\in \mathrm{Z}:|(|\mathrm{x}-3|-3)|\le 1\}$ and $B=\left\{x\in \mathbb{R}-\{1,2\}:\frac{(x-2)(x-4)}{x-1}{\log }_e(|x-2|)=0\right\}$.

Then the number of onto functions $f:\mathrm{A}\to \mathrm{B}$ is equal to :

Let $\mathrm{A}=\{0,1,2,\dots .9)$. Let R be a relation on A defined by $(\mathrm{x},\mathrm{y})\in \mathrm{R}$ if and only if $|\mathrm{x}-\mathrm{y}|$ is a multiple of 3.
Given below are two statements:
Statement I: $n(R)=36$
Statement II: $R$ is an equivalence re …

If the domain of the function
$f(x)={\log }_{\left(10x^2-17x+7\right)}\left(18x^2-11x+1\right)$
is $(-\infty ,a)\cup (b,c)\cup (d,\infty )-\{e\}$, then
$90(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e})$ equals:

Let R be a relation defined on the set $\{1,2,3,4\}\times \{1,2,3,4\}$ by
$\mathrm{R}=\{((\mathrm{a},\mathrm{b}),(\mathrm{c},\mathrm{d})):2\mathrm{a}+3\mathrm{b}=3\mathrm{c}+4\mathrm{d}\}$.
Then the number of elements in R is

Let f be a function such that $3\mathrm{f}(\mathrm{x})+2\mathrm{f}\left(\frac{\mathrm{m}}{19\mathrm{x}}\right)=5\mathrm{x}$, $x\ne 0$, where $m={\sum }_{i=1}^9(i{)}^2$. Then $f(5)-f(2)$ is equal to

If $g(x)=3x^2+2x-3,f(0)=-3$ and $4g(f(x))=3{\mathrm{x}}^2-32\mathrm{x}+72$, then $f(\mathrm{g}(2))$ is equal to:

Given below are two statements :

Statement I: The function $\mathrm{f}:\mathbf{R}\to \mathbf{R}$ defined by $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{1+|\mathrm{x}|}$ is one-one.

Statement II : The function $\mathrm{f}:\mathbf{R}\to \mathbf{R}$ defined …

The sum of all the elements in the range of $f(x)=\mathrm{Sgn}(\sin x)+\mathrm{Sgn}(\cos x)+\mathrm{Sgn}(\tan x)+\mathrm{Sgn}(\cot x)$, $\mathrm{x}\ne \frac{\mathrm{n}\pi }{2},\mathrm{n}\in \mathbf{Z}$,
where …

If the domain of the function $f(x)=\sqrt{{\log }_{(0.6)}\left(\left|\frac{2x-5}{x^2-4}\right|\right)}$ is $(-\infty ,a]\cup \{b\}\cup [c,d)\cup (e,\infty )$, then the value of $a+b+c+d+e$ is ______.

For the function $f:[1,\infty )\to [1,\infty )$ defined by $f(\mathrm{x})=(\mathrm{x}-1{)}^4+1$, among the two statements :
(I) The set $\mathrm{S}=\left\{\mathrm{x}\in [1,\infty ):f(\mathrm{x})=f^{-1}(\mathrm{x})\right\}$ contains exactly two elements, and
(II) Th …

Let for some $\alpha \in \mathbb{R},f:\mathbb{R}\to \mathbb{R}$ be a function satisfying $f(\mathrm{x}+\mathrm{y})=f(\mathrm{x})+2{\mathrm{y}}^2+\mathrm{y}+\alpha \mathrm{xy}$ for all $\mathrm{x},\mathrm{y}\in \mathbb{R}$. If $f(0)=-1$ and $f(1)=2$, then the value o …

The number of functions $\mathrm{f}:\{1,2,3,4\}\to \{\mathrm{a},\mathrm{b},\mathrm{c}\}$, which are not onto, is :

Let $\mathrm{A}=\{1,2,3,4,5,6\}$. The number of one-one functions $\mathrm{f}:\mathrm{A}\to \mathrm{A}$ such that $\mathrm{f}(1)\ge 3,\mathrm{f}(3)\le 4$ and $f(2)+f(3)=5$, is ______.

Let $\mathrm{A}=\{1,4,7\}$ and $\mathrm{B}=\{2,3,8\}$. Then the number of elements, in the relation $\mathrm{R}=\{(\left({\mathrm{a}}_1,{\mathrm{b}}_1\right)$, …

Let $f:\mathrm{R}\to \mathrm{R}$ be defined as $f(\mathrm{x})=\frac{2{\mathrm{x}}^2-3\mathrm{x}+2}{3{\mathrm{x}}^2+\mathrm{x}+3}$. Then $f$ is :

Let $\mathrm{R}=\left\{(\mathrm{x},\mathrm{y})\in \mathrm{N}\times \mathrm{N}:{\log }_{\mathrm{e}}(\mathrm{x}+\mathrm{y})\le 2\right\}$. Then the minimum number of elements, required to be added in R to make it a transitive relation, is

Let [ ] denote the greatest integer function. If the domain of the function $f(x)={\sin }^{-1}\left(\frac{x+[x]}{3}\right)$ is $[\alpha ,\beta )$, then ${\alpha }^2+{\beta }^2$ is equal to:

Consider the relation R on the set $\{-2,-1,0,1,2\}$ defined by $(\mathrm{a},\mathrm{b})\in \mathrm{R}$ if and only if $1+\mathrm{ab}>0$. Then among the statements:
I. The number of elements in R is 17
II. R is an equivalence relation

Let $\mathrm{A}=\{2,3,4,5,6\}$. Let R be a relation on the set $\mathrm{A}\times \mathrm{A}$ given by $(\mathrm{x},\mathrm{y})\mathrm{R}(\mathrm{z},\mathrm{w})$ if and only if x divides z and $\mathrm{y}\le \mathrm{w}$. Then the number of elements in R is ______.

The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :

Let $\mathrm{A}=\{1,2,3,\dots ,10\}$ and $\mathrm{B}=\left\{\frac{\mathrm{m}}{\mathrm{n}}:\mathrm{m},\mathrm{n}\in \mathrm{A},\mathrm{m}<\mathrm{n}\text{ and }\gcd (\mathrm{m},\mathrm{n})=1\right\}$. Then $\mathrm{n}(\mathrm{B})$ is equal to :

Let $\mathrm{A}=\{1,2,3,4\}$ and $\mathrm{B}=\{1,4,9,16\}$. Then the number of many-one functions $\mathrm{f}:\mathrm{A}\to \mathrm{B}$ such that $1\in \mathrm{f}(\mathrm{A})$ is equal to :

Let $\mathrm{A}=\{1,2,3\}$. The number of relations on A , containing ( 1,2 ) and ( 2,3 ), which are reflexive and transitive but not symmetric, is

Let $\mathrm{f}(\mathrm{x})={\log }_{\mathrm{e}}\mathrm{x}$ and $\mathrm{g}(\mathrm{x})=\frac{{\mathrm{x}}^4-2{\mathrm{x}}^3+3{\mathrm{x}}^2-2\mathrm{x}+2}{2{\mathrm{x}}^2-2\mathrm{x}+1}$.

Then the domain of fog is

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

Let $X=R\times R$. Define a relation $R$ on $X$ as:
$(a_1,b_1)R(a_2,b_2)\iff b_1=b_2$.
Statement-I: $R$ is an equivalence relation.
Statement-II: For some $(a,b)\in X$, the set $S=\{(x,y)\in X:(x,y)R(a,b)\}$ represents a line parallel to $y=x$. …

If the domain of the function ${\log }_5\left(18\mathrm{x}-{\mathrm{x}}^2-77\right)$ is $(\alpha ,\beta )$ and the domain of the function ${\log }_{(x-1)}\left(\frac{2x^2+3x-2}{x^2-3x-4}\right)$ is $(\gamma ,\delta )$, then ${\alpha }^2+{\beta }^2+{\gamma }^2$ is equal to :

Let $\mathrm{S}=\mathbf{N}\cup \{0\}$. Define a relation $\mathbf{R}$ from S to $\mathbf{R}$ by : …

Define a relation R on the interval $\left[0,\frac{\pi }{2}\right)$ by x R y if and only if ${\sec }^{2}x-{\tan }^{2}y=1$. Then $R$ is :

Let $f:R\to R$ be a function defined by
$f(x)=(2+3a)x^2+\left(\frac{a+2}{a-1}\right)x+b,a\ne 1$. If
$f(x+y)=f(x)+f(y)+1-\frac{2}{7}xy$, then the value of $28{\sum }_{i=1}^5|f(i)|$ is:

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

The function $f:(-\infty ,\infty )\to (-\infty ,1)$, defined by $f(x)=\frac{2^x-2^{-x}}{2^x+2^{-x}}$ is :

Let $\mathrm{f}:\mathbb{R}-\{0\}\to \mathbb{R}$ be a function such that $f(x)-6f\left(\frac{1}{x}\right)=\frac{35}{3x}-\frac{5}{2}$. If the ${\lim }_{x\to 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ;\alpha ,\beta \in \mathbb{R}$, then $\alpha +2\beta$ is equ …

Let $\mathrm{A}=\{1,2,3,\dots ,100\}$ and R be a relation on A such that $\mathrm{R}=\{(\mathrm{a},\mathrm{b}):\mathrm{a}=2\mathrm{b}+1\}$. Let $({\mathrm{a}}_1,{\mathrm{a}}_2)$, $({\mathrm{a}}_2,{\mathrm{a}}_3)$, $({\mathrm{a}}_3,{\mathrm{a}}_4)$, $\dots$, …

If the domain of the function
$f(\mathrm{x})=\frac{1}{\sqrt{10+3\mathrm{x}-{\mathrm{x}}^2}}+\frac{1}{\sqrt{\mathrm{x}+|\mathrm{x}|}}$ is $(\mathrm{a},\mathrm{b})$, then $(1+a{)}^2+b^2$ is equal to:

Let $\mathrm{A}=\{-2,-1,0,1,2,3\}$. let R be a relation on A defined by xRy if and only if $\mathrm{y}=\max \{\mathrm{x},1\}$. 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 symmetri …

Let f be a function such that $\mathrm{f}(\mathrm{x})+3\mathrm{f}\left(\frac{24}{\mathrm{x}}\right)=4\mathrm{x},\mathrm{x}\ne 0$. Then $\mathrm{f}(3)+\mathrm{f}(8)$ is equal to

If the domain of the function
$\mathrm{f}(\mathrm{x})={\log }_7\left(1-{\log }_4\left({\mathrm{x}}^2-9\mathrm{x}+18\right)\right)$ is $(\alpha ,\beta )\cup (\gamma ,\delta )$, then $\alpha +\beta +\gamma +\delta$ is equal to

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

Let $\mathrm{A}=\{-3,-2,-1,0,1,2,3\}$. Let $\mathrm{R}$ be a relation on A defined by xRy if and only if $0\le x^2+2y\le 4$. Let $\ell$ 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 relati …

If the domain of the function $f(x)={\log }_e\left(\frac{2x-3}{5+4x}\right)+{\sin }^{-1}\left(\frac{4+3x}{2-x}\right)$ is $[\alpha ,\beta )$, then ${\alpha }^2+4\beta$ is equal to