Sets Relations and Functions — JEE Main PYQs

Let the domain of the function
$f(\mathrm{x})={\log }_2{\log }_4{\log }_6\left(3+4\mathrm{x}-{\mathrm{x}}^2\right)$ be $(\mathrm{a},\mathrm{b})$. If …

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

Let the domains of the functions
$\mathrm{f}(\mathrm{x})={\log }_4{\log }_3{\log }_7\left(8-{\log }_2\left({\mathrm{x}}^2+4\mathrm{x}+5\right)\right)$ and
$\mathrm{g}(\mathrm{x})={\sin }^{-1}\left(\frac{7\mathrm{x}+10}{\mathrm{x}-2}\right)$ be $(\alpha ,\beta )$ and …

Let $f$, $\mathrm{g}:(1,\infty )\to \mathbb{R}$ be defined as $f(\mathrm{x})=\frac{2x+3}{5x+2}$ and $\mathrm{g}(\mathrm{x})=\frac{2-3x}{1-x}$. If the range of the function $f\circ g:[2,4]\to \mathbb{R}$ is $[\alpha ,\beta ]$, then …

Consider the sets $\mathrm{A}=\left\{(\mathrm{x},\mathrm{y})\in \mathbb{R}\times \mathbb{R}:{\mathrm{x}}^2+{\mathrm{y}}^2=25\right\}$, $\mathrm{B}=\left\{(\mathrm{x},\mathrm{y})\in \mathbb{R}\times \mathbb{R}:{\mathrm{x}}^2+9{\mathrm{y}}^2=144\right\}$, …

Let $\mathrm{A}=\{1,6,11,16,\dots \}$ and $\mathrm{B}=\{9,16,23,30,\dots \}$ be the sets consisting of the first 2025 terms of two arithmetic progressions. Then $\mathrm{n}(\mathrm{A}\cup \mathrm{B})$ is

The number of relations on the set $A=\{1,2,3\}$ containing at most 6 elements including $(1,2)$, which are reflexive and transitive but not symmetric, is

Let $\mathrm{A}=\{0,1,2,3,4,5\}$. Let R be a relation on A defined by $(x,y)\in R$ if and only if max $\{\mathrm{x},\mathrm{y}\}\in \{3,4\}$. Then among the statements
$({\mathrm{S}}_1)$ : The number of elements in R is 18, and
$({\mathrm{S}}_2)$ : The relation R is symmetric …

Let a relation $R$ on $\mathbb{N}\times \mathbb{N}$ be defined as: $(x_1,y_1)R(x_2,y_2)$ if and only if $x_1\le x_2$ or $y_1\le y_2$. Consider the two statements: (I) $R$ is reflexive but not symmetric. (II) $R$ is transitive. Then which one of …

Consider the function $f:\mathbb{R}\to \mathbb{R}$ defined by $f(x)=\frac{2x}{\sqrt{1+9x^2}}$. If the composition of $f$, $(f\circ f\circ f\circ \dots \circ f)(x)$ (10 times) $=\frac{2^{10}x}{\sqrt{1+9\alpha x^2}}$, then the value of $\sqrt{3\alpha +1}$ is equal to …

If $S=\{a\in R:|2a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72{\sum }_{a\in S}a$ is equal to ____.

Let $f,g:R\to R$ be defined as: $f(x)=|x-1|$ and $g(x)=\{\begin{array}{ll}{e}^x,& x\ge 0\\ x+1,& x\le 0\end{array}$. Then the function $f(g(x))$ is

Let $A=\{n\in [100,700]\cap N:n\text{ is neither a multiple of 3 nor a multiple of 4}\}$. Then the number of elements in $A$ is

Let the relations $R_1$ and $R_2$ on the set $X=\{1,2,3,\dots ,20\}$ be given by $R_1=\{(x,y):2x-3y=2\}$ and $R_2=\{(x,y):-5x+4y=0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the …

The function $f(x)=\frac{x^2+2x-15}{x^2-4x+9}$, $x\in R$ is

Let $A=\{1,2,3,4,5\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only $4x\le 5y$. Let $m$ be the number of elements in $R$ and $n$ be the minimum number of elements from $A\times A$ that are required to be added to $R$ to make it a symmetric relation. Then $m+n$ is …

Let $f(x)=\frac{1}{7-\sin 5x}$ be a function defined on $\mathbb{R}$. Then the range of the function $f(x)$ is equal to:

Let $[t]$ be the greatest integer less than or equal to t. Let A be the set of all prime factors of 2310 and $f:A\to \mathbb{Z}$ be the function $f(x)=\left[{\log }_2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$. The number of one-to-one functions from A to …

Let $f(\mathrm{x})=\{\begin{array}{cl}-\mathrm{a}& \text{ if }-\mathrm{a}\le \mathrm{x}\le 0\\ \mathrm{x}+\mathrm{a}& \text{ if }0<\mathrm{x}\le \mathrm{a}\end{array}$
where $\mathrm{a}>0$ and $\mathrm{g}(\mathrm{x})=((f|\mathrm{x}|)-|f(\mathrm{x})|)/2$. …

Let $\mathrm{A}=\{2,3,6,8,9,11\}$ and $\mathrm{B}=\{1,4,5,10,15\}$

Let R be a relation on $\mathrm{A}\times \mathrm{B}$ define by $(\mathrm{a},\mathrm{b})\mathrm{R}(\mathrm{c},\mathrm{d})$ if and only if $3\mathrm{ad}-7\mathrm{bc}$ is an even integer. Then the relation $R$ …

Let $A=\{2,3,6,7\}$ and $B=\{4,5,6,8\}$. Let R be a relation defined on $A\times B$ by $(a_1,b_1)R(a_2,b_2)$ if and only if $a_1+a_2=b_1+b_2$. Then the number of elements in R is

Let $\mathrm{A}=\{(\mathrm{x},\mathrm{y}):2\mathrm{x}+3\mathrm{y}=23,\mathrm{x},\mathrm{y}\in \mathrm{N}\}$ and $\mathrm{B}=\{\mathrm{x}:(\mathrm{x},\mathrm{y})\in \mathrm{A}\}$. Then the number of one-one functions from A to B is equal to ____.

If R is the smallest equivalence relation on the set $\{1,2,3,4\}$ such that $\{(1,2),(1,3)\}\subset \mathrm{R}$, then the number of elements in R is

Let the set $C=\left\{(x,y)\mid x^2-2^y=2023,x,y\in \mathbb{N}\right\}$.

Then ${\sum }_{(x,y)\in C}(x+y)$ is equal to ______.

Let $\mathrm{A}=\{1,2,3,\dots \dots \dots 100\}$. Let R be a relation on A defined by $(x,y)\in R$ if and only if $2x=3y$. Let ${\mathrm{R}}_1$ be a symmetric relation on A such that $\mathrm{R}\subset {\mathrm{R}}_1$ and the number of elements in …

A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 …

Let $A=\{1,2,3,\dots ,7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f:A\to P(A)$ such that $a\in f(a)$, $\forall a\in A$ is $m^n$, $m$ and $n\in \mathbb{N}$ and $m$ is least, then $m+n$ is equal to

Let $f:\mathbb{R}-\left\{\frac{-1}{2}\right\}\to \mathbb{R}$ and $g:\mathbb{R}-\left\{\frac{-5}{2}\right\}\to \mathbb{R}$ be defined as $f(x)=\frac{2x+3}{2x+1}$ and $g(x)=\frac{|x|+1}{2x+5}$. Then the domain of the function fog is :

Let A and B be two finite sets with m and n elements respectively. The total number of subsets of the set A is 56 more than the total number of subsets of B . Then the distance of the point $P(m,n)$ from the point $Q(-2,-3)$ is

Let $S=\{1,2,3,\dots ,10\}$. Suppose M is the set of all the subsets of S, then the relation
$R=\{(A,B):A\cap B\ne \varphi ;A,B\in M\}$ is :

The function $\mathrm{f}:\mathrm{N}-\{1\}\to \mathrm{N}$; defined by $\mathrm{f}(\mathrm{n})=$ the highest prime factor of n , is :

If the domain of the function $f(x)=\frac{\sqrt{x^2-25}}{(4-x^2)}+{\log }_{10}({\mathrm{x}}^2+2\mathrm{x}-15)$ is $(-\infty ,\alpha )\cup [\beta ,\infty )$, then ${\alpha }^2+{\beta }^3$ is equal to :

Consider the relations ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ defined as ${\mathrm{aR}}_1\mathrm{b}\iff {\mathrm{a}}^2+{\mathrm{b}}^2=1$ for all $\mathrm{a},\mathrm{b},\in \mathrm{R}$ and …

If $\mathrm{f}(\mathrm{x})=\frac{4\mathrm{x}+3}{6\mathrm{x}-4},\mathrm{x}\ne \frac{2}{3}$ and (fof) $(\mathrm{x})=\mathrm{g}(\mathrm{x})$, where $\mathrm{g}:\mathbb{R}-\left\{\frac{2}{3}\right\}\to \mathbb{R}-\left\{\frac{2}{3}\right\}$, then (gogog) (4) is e …

Let A={1,2,3,4} and R={(1,2),(2,3),(1,4)} be a relation on A. Let S be the equivalence relation on A such that R ⊂ S and the number of elements in S is n. Then, the minimum value of n is ____

Let $f:\mathbf{R}\to \mathbf{R}$ and $g:\mathbf{R}\to \mathbf{R}$ be defined as $f(x)=\{\begin{array}{ll}{\log }_{e}x& ,x>0\\ e^{-x}& ,x\le 0\end{array}$ and
$g(x)=\{\begin{array}{ccc}x& ,& x\ge 0\\ e^x& ,& x<0\end{array}$. …

Let $\mathrm{A}=\{1,2,3,\dots 20\}$. Let ${\mathrm{R}}_1$ and ${\mathrm{R}}_2$ two relation on A such that
${\mathrm{R}}_1=\{(\mathrm{a},\mathrm{b}):\mathrm{b}$ is divisible by a$\}$
${\mathrm{R}}_2=\{(\mathrm{a},\mathrm{b}):\mathrm{a}$ is an integral multiple o …

If $f(x)=\{\begin{array}{ll}2+2x,& -1\le x<0\\ 1-\frac{x}{3},& 0\le x\le 3\end{array}$; $g(x)=\{\begin{array}{ll}-x,& -3\le x\le 0\\ x,& 0<x\le 1\end{array}$, then range of $(f\circ g)(x)$ is

Let R be a relation on $\mathrm{Z}\times \mathrm{Z}$ defined by
$(\mathrm{a},\mathrm{b})\mathrm{R}(\mathrm{c},\mathrm{d})$ if and only if $\mathrm{ad}-\mathrm{bc}$ is divisible by 5.
Then R is

If the domain of the function $\mathrm{f}(\mathrm{x})={\log }_{\mathrm{e}}\left(\frac{2x+3}{4x^2+x-3}\right)+{\cos }^{-1}\left(\frac{2x-1}{x+2}\right)$ is $(\alpha ,\beta ]$, then the value of $5\beta -4\alpha$ is equal to

The number of symmetric relations defined on the set $\{1,2,3,4\}$ which are not reflexive is ______.

Let the sets A and B denote the domain and range respectively of the function $f(x)=\frac{1}{\sqrt{\lceil x\rceil -x}}$, where $\lceil x\rceil$ denotes the smallest integer greater than or equal to x. Then among the statements
(S1): $A\cap B=(1,\infty )-\mathbb{N}$ and
(S2): …

Let the number of elements in sets A and B be five and two respectively. Then the number of subsets of $\mathrm{A}\times \mathrm{B}$ each having at least 3 and at most 6 element is :

Let $A=\{0,3,4,6,7,8,9,10\}$ and $R$ be the relation defined on $A$ such that $R=\{(x,y)\in A\times A:x-y$ is odd positive integer or $x-y=2\}$. The minimum number of elements that must be added to the relation $R$, so that it is a symmetric relation, is equal to …

Let $\mathrm{A}=\{\mathrm{x}\in \mathbb{R}:[\mathrm{x}+3]+[\mathrm{x}+4]\le 3\}$,
$B=\left\{x\in \mathbb{R}:3^x{\left({\sum }_{r=1}^{\infty }\frac{3}{10^r}\right)}^{x-3}<3^{-3x}\right\}$, where $[t]$
denotes greatest integer function. Then,

Let $A=\{1,2,3,4,\dots 10\}$ and $B=\{0,1,2,3,4\}$.

The number of elements in the relation $R=\{(a,b)\in A\times A:2(a-b{)}^2+3(a-b)\in B\}$ is

Let $\mathrm{A}=\{1,2,3,4,5,6,7\}$. Then the relation $\mathrm{R}=\{(\mathrm{x},\mathrm{y})\in \mathrm{A}\times \mathrm{A}:\mathrm{x}+\mathrm{y}=7\}$ is

If domain of the function
${\log }_e\left(\frac{6x^2+5x+1}{2x-1}\right)+{\cos }^{-1}\left(\frac{2x^2-3x+4}{3x-5}\right)$ is $(\alpha ,\beta )$
$\cup (\gamma ,\delta ]$, then $18\left({\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2\right)$ is equal to ______.

Let $R=\{a,b,c,d,e\}$ and $S=\{1,2,3,4\}$. Total number of onto function $f:R\to S$ such that $f(a)\ne 1$, is equal to ______.

Let $\mathrm{A}=\{2,3,4\}$ and $\mathrm{B}=\{8,9,12\}$. Then the number of elements in the relation …